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1 B Onlne Appendx not for publcaton) Ths Onlne Appendx provdes some addtonal results referenced n the paper. B.1 Recoverng the equlbrum varables from the Unversal Gravty condtons In ths subsecton, we show how the unversal gravty condtons C.1-C.5 can be combned to derve equatons 6) and 7), whch can be used to solve for equlbrum prces and prce ndces up to scale. We then show how nformaton of these prces and prce ndces up-to-scale can be used to solve for the level of real output prces {p /P } S and, combned wth the numerare n C.6, to determne the equlbrum level of ncome {Y } S, expendture {E } S, and trade flows {X }, S. Fnally, we show how all other endogenous varables can be recovered up-to-scale f the equlbrum prces and prce ndces are known up to scale. B.1.1 From Unversal Gravty C.1-C.5 to Equatons 6) and 7) We frst show Unversal Gravty C.1-C.5 mply equatons 6) and 7). Combng C.1 and C.2 n partcular the gravty equaton 10)): where recall from C.2 that the prce ndex can be wrtten as: X = P E, 31) P S 32) Combnng equaton 31) wth C.4) and C.5) yelds: p Q = S P p Q 33) Fnally, we substtute C.3 nto equaton 33) to yeld: ) p p C = ) P P p p C P S 34) Note that equatons 32) and 34) are equvalent to equatons 6) and 7). Hence, C.1-C. 5 mply equatons 6) and 7), as clamed. There are two thngs to note about equlbrum equatons 32) and 34): frst, they depend only on output prces {p }, the prce ndces {P }, and exogenous model fundamentals n partcular, they do not depend on the endogenous scalar κ); second, they are homogeneous of degree zero wth respect to {p, P }, so the scale of prces and prce ndces) are undetermned. B.1.2 From Equatons 6) and 7) to endogenous varables We now show that gven a soluton to equatons 6) and 7), we can construct all endogenous varables n the models. We dvde the dervatons nto endogenous varables determned up to scale and endogenous varables for whch the scale s known gven the choce of numerare n C.6. 41

2 Suppose that we have a set of prces {p } S and prce ndces {P } S that solve equatons 6) and 7). Note that because equatons 6) and 7) are homogeneous of degree zero wth respect to {p, P } S, for any scalar α, the normalzed prces p 1 α p and prce ndces P 1 α P contnue to satsfy equatons 6) and 7). We frst solve for the real output prce. Note that for any choce of α, the real output prce {p /P } S remans unchanged, so ts level s unaffected by the unknown scalar. We now solve for quanttes. From equaton 11), the quantty n locaton does not depend on α, but t does depend on the unknown scalar κ as follows: p Q = κc. P Hence, equlbrum quanttes are only determne up-to-scale. We now solve for ncome and expendture. From C.4 and C.5 we have: E = Y = p Q. Applyng the numerare n C.6 then yelds: Y = 1 κα S S p Q = 1 S p C p P = 1 κα = S ) 1 p p C, P whch, as clamed, pns down the product of the unknown quantty scalar and unknown prce scalar. Gven κα, we can now determne the level of ncome and expendture as follows: E = Y = p Q E = Y = p C p P S p C p P ), as clamed. We now determne the level of trade flows usng equaton 31): X = P E X = k S k p k ) p ψ p C P ) ) ψ k S p pk. kc k P k Other than real output prces {p /P } S, ncome {Y } S, expendture {E } S, and trade flows 42

3 {X }, S, all other endogenous varables are determned only up-to-scale, as they depend ether on the prce scalar α.e. output prces p, prce ndces P, blateral prces p = τ p, and the quantty traded Q = X /τ p ) or the quantty scalar κ.e. quanttes Q ). B.2 Proof of Theorem 1 part ) We frst provde a general mathematcal formulaton to ncorporate non-nteror solutons. Let the equlbrum be a duple p, Q ) R N + R N + such that for all S, Q = 1 1 k S k p k p Q 35) p, Q ) F p, Q) 36) where F s a supply condton, whch mght be a correspondence. The fact that F mght be correspondence allows us to extend the framework to allow for non-nteror solutons). In partcular, we defne F as follows: We say p, Q ) F p, Q) f and only f [ ) ] ψ p sgn ψ) Q κ 0 37) P p) p Q = κ f Q > 0, 38) P p) and where 0 0) s defned as 0. That s, f Q = 0, then we replace C.3 wth an nequalty. For example, n an economc geography model, nequalty constrant 37) corresponds to welfare equalzaton. If there are people lvng n locaton, then Q s gven by equalty 38). If not, then the welfare lvng n locaton should be lower than one obtaned n other places, whch s represented as the nequalty 37). As we mentoned n Secton 3, we restrct our attenton to non-trval equlbra where there s postve producton n at least one locaton. To show that all non-trval) equlbra are nteror, t then suffces to show that f some locatons produces nothng, then all other locatons must also produce nothng. Suppose that Q l = 0 for some l S. Then from equaton 35) for l: 0 = l 1 l k S k p k } {{ } 0 p Q, 39) whch n turn mples that for all S, l 1 l p Q = 0, 40) g where g = k S k p k. Note that there are two reasons why equaton 40) s zero for all ; ether 1) ; or 2) for all S, l = 0. We wll prove a contradcton n both cases. p Q g 43

4 Frst assume that 1) 1 l = 0, whch f > 1 mples that p l =. Whle p l, Q l ) =, 0) satsfes equaton 39), t does not satsfy equaton 36). To see ths, note: 0 = Q < κ p g 1 ψ =, whch contradcts wth equaton 37) snce ψ 0. Therefore p l needs to be fnte, p l <. Now assume that 2) for all S, p Q l g = 0. Snce the prce for country l s fnte, equaton 40) s reduced to p Q l = 0 g for all S. An equvalent expresson s that for all countres connected wth l, S l = {k S; τ lk < }, p Q = 0 or g =. 41) Fx any country S l. Suppose that p, Q > 0 Then equaton 41), g =. Then for all p, Q ) R + R f ψ 0 we have = κ p g 1 ψ Q = 0, whch s a contradcton. Therefore n order to satsfy equaton 41), p or Q needs to be zero. Suppose that p = 0. Then we have 0 = κ p g 1 ψ Q. If Q > 0, then C. 3). Therefore, Q = 0. Therefore Q needs to be zero for all S l. So far, we have shown that f Q l = 0 then the connected countres S l produce nothng, Q = 0. Because of strong connectedness, any country n s connected wth l through thrd countres. Therefore, by repeatng the argument along wth the chan, we have Q n = 0 for all n S. As a result, f 1, and ψ 0 then all equlbra are nteror, as clamed. B.3 Quas-symmetrc trade frctons In ths subsecton, we show that when trade frctons are quas-symmetrc, then balanced trade mples that the orgn and destnaton fxed effects of the gravty trade flow expresson are equal up to scale. We frst formally defne quas-symmetry. We say that the set of trade frctons {τ }, S are quas-symmetrc f there exsts a set of orgn scalars { τ A } S RN ++, destnaton scalars { } τ B S RN ++, and a symmetrc matrx { τ }, S where τ = τ for all, S such that we can wrte: τ = τ A τ B τ, S. Loosely speakng, quas-symmetrc trade frctons are those that are reducble to a symmetrc component and exporter- and mporter-specfc components. Whle restrctve, t s mportant to 44

5 note that the vast maorty of papers whch estmate gravty equatons assume that trade frctons are quas-symmetrc; for example Eaton and Kortum 2002) and Waugh 2010) assume that trade frctons are composed by a symmetrc component that depends on blateral dstance and on a destnaton or orgn fxed effect. Combnng the unversal gravty condtons C. 1 and C. 2 allows us to wrte the value of blateral trade flows from to as: X = P E, whch we now re-wrte as: where we call γ effect. X = κ γ δ, 42) the orgn fxed effect and δ P E = C P ψ p 1+ψ the destnaton fxed Proposton 1. If trade frctons are quas-symmetrc, then n any model wthn the unversal gravty framework, the product of the equlbrum orgn fxed effect and the orgn scalar wll be equal to the product of the equlbrum destnaton fxed effects and the destnaton fxed effect up to scale,.e.: for some scalar λ 0, τ A ) γ = λ τ B ) δ S. Proof. We frst note that market clearng condton C.4 and balanced trade condton C.5 together mply that: S X = S X S. Combnng ths wth the gravty expresson 42) and quas-symmetry mples: κ γ δ κ γ δ }{{}}{{} =X = ) τ A γ S ) τ B = δ S X τ τ τ A τ B ) γ ) δ = S k S τ B τ k ) δ τ B k ) δk τ A τ B ) γ ) δ. It s easy to show that τ A ) γ = 1 s a soluton to ths problem for any kernel. From the Perron- τ B ) δ Frobenus theorem, the soluton s unque up to scale. Therefore we have: ) τ A ) = λ τ B δ S, 43) as requred. Proposton 1 has a number of mportant mplcatons. Frst, Proposton 1 allows one to smplfy the equlbrum system of equatons 6 and 7 nto a sngle non-lnear equaton when ψ: p 1+ψ+ ψ ) = λ) ψ C ) ψ S ) τ A 2 ψ τ B ) ψ ψ τ A ) p, S, 44) whch smplfes the characterzaton of the theoretcal and emprcal propertes of the equlbrum. 45

6 Notce that λ s an endogenous scalar. Snce 44) holds for any locaton S, λ s expressed as λ ψ = Substtutng above expresson, we obtan: S τ A τ B ) 1+ψ+ ψ ) 2 ψ ψ C. ) 1+ψ+ ψ ) 1+ψ+ ψ = S τ A τ B ) 2 ψ S τ A τ B ψ C ) 2 ψ ψ C. Notce that the system s now homogeneous degree 0. Therefore, f / { 1 2, ψ, 0}, then we can normalze λ = 1 wthout loss of generalty. Second, by showng that the orgn and destnaton fxed effects are equal up to scale, Proposton 1 provdes offers an analytcal characterzaton of the equlbrum. For example, gven the defnton of the orgn and destnaton fxed effects, Proposton 1 can equvalently be expressed as: p P τ B τ A E 1, 45).e. there s a log-lnear relatonshp between output prces, the prce ndex and total expendture n a locaton. Thrd, t s straghtforward to show that quas-symmetry mples that equlbrum trade flows wll be blaterally symmetrc,.e. X = X for all, S, allowng one to test whether trade frctons are quas-symmetrc drectly from observed trade flow data. Fnally, we should note that the results of Proposton 1 have already been used n the lterature for partcular models, albet mplctly. The most promnent example s Anderson and Van Wncoop 2003), who use the result to show the blateral resstance s equal to the prce ndex. 36 To our knowledge, Head and Mayer 2013) are the frst to recognze the mportance of balanced trade and market clearng n generatng the result for the Armngton model; however, Proposton 1 shows that the result apples more generally to any model wth quas-symmetrcal trade frctons n the unversal gravty framework. B.4 Proofs of the lemmas used n Theorem 1) There are 4 lemmas whch are not proven n the paper. In ths secton, we dscuss them carefully. Before provng these lemmas, we dscuss how we use them n the proof. In the proof, we show a fxed pont for the scaled system, not the actual system. Therefore t needs to be shown that there exsts a fxed pont for the actual system, whch s shown n Lemma 1. Then we argue that the soluton we obtan s strctly postve, whch s guaranteed by Assumpton 1. We emphasze the connectvty assumpton s crucal here. These two lemmas are used n Part ) Theorem 1. Part ) shows that there exsts an unque soluton. Durng the proof, we argue that 26 should hold wth strct nequalty. Agan the connectvty allows us to show ths result Lemma 3). After 36 The result s also used n economc geography byallen and Arkolaks 2014) to smplfy a set on non-lnear ntegral equatons nto a sngle ntegral equaton. 46

7 establshng ths strct nequalty, we follow the argument by Allen et al. 2014), whch requres that the largest absolute egenvalues for A are less than 1. Snce A s a 2-by-2 matrx, we can compute the egenvalues by hand and show that one of them s exactly 1, and the other s less than 1 f the condtons n Part ) are satsfed. Lemma 1. Suppose that z solves 22). Then there exsts ẑ solvng 21). Proof. Frst t s easy to show 37, S K C 1 C x a 11 y a 12 = K x a 21 y a ), S Guess a soluton ) x ) ẑ = t = 1 ) x ) ŷ ) t 1, 49) y ) ) where t =, S K C 1 C x a 21 y a a 11 a 12 ) =, S K x a 21 y a a 21 a Then t s easy to verfy that 49) solves 21); n partcular, note that x = t 1 S K C 1 C x a 11 y a 12, S K C 1 C x a 11 y a = t 1 a 11 a 12 S K C 1 C x ) a 11 ŷ a 12 12, S K C 1 C x a 11 y a 12 = S K C 1 C x a 11 ŷ a 12. We can also show that the second equatons n 21) are also solved n the same ven: S ŷ = t K x a 21 y a 22, S K x a 21 y a = t 1 a 21 a 22 S K x a 21 ŷ a 22 22, S K x a 21 y a 22 = S K x a 21 ŷ a 22. The above two equatons confrm that x and ŷ s a soluton to 21). 37 To see ths, multply C x a21 y a22 = C, to the frst equatons of 22) and sum over ; C p 1+ψ P ψ = Also multply C x a11 y a12 = C P ψ p 1+ψ S K C x a21, K C 1 y a22 C x a11 x a11 y a12 y a12. 46) to the second equatons 22) and sum over ; S S K C x a21 y a22 x a11 y a12. 47) C p 1+ψ P ψ = S, S K x a21 Notce that the LHS s the same as one n 46). Also the numerator of the RHS n 46) s the same as one n 47). Therefore the followng double sum terms should be the same:, 38 Notce that a 11 + a 12 = a 21 + a 22. K C 1 C x a11 y a12 = S, S y a22 K x a21 y a22. 47

8 Lemma 2. If {τ }, satsfes Assumpton 1, then the fxed pont for 22) s strctly postve. Proof. We need to consder four dfferent cases for the combnatons of a 11, a 12 satsfyng dfferent nequaltes. We wll consder the case a 11, a 12 > 0 snce the logc n the other cases s the same. We proceed by contradcton. Suppose that there s a soluton x to equaton 22) such that for some S x = 0. Consder an arbtrary locaton n and consder a connected path, K c n K π1... K πmn > 0 for some m ). Then, from the frst of equatons n 21) notce that x = K x a 11 y a 12 K π1 S }{{} 0 x a 11 π 1 y a 12 π 1. Note that K π1 s strctly postve due to ). Then ether x n or y n or both are zero f a 11 and a 11 > 0. Ifx n = 0 ths argument holds for anyn so ths s a contradcton wth the non-zero equlbrum proved above. Else f y n = 0 we can repeat the argument the second of the equatons n 21) to establsh another contradcton. Notce that f ether of α 11, α 12 = 0 a contradcton s also easy to establsh. Lemma 3. Equaton 26 holds wth strct nequalty. To that end, defne the set of drectly connected countres to each locaton S as S c { S : K > 0}. Then notce that equaton 24) combned wth our equalty assumpton on equaton 26) yelds x = 1 x ) α11 ) α12 ) α11 ) α12 K C 1 x y C x ) α 11 ŷ ) α 12 x y = max max. x x ŷ S x S ŷ S c Notce that gven that ˆx s a soluton, ths mples that the followng has to be true for all S c ) α11 ) α11 ) α12 ) α12 x x y y = max = max. x S x ŷ S ŷ Now notce that f α 11 0 then for all n S c,x / x = x n / x n. However, because of C. 1, we assume that there exsts an ndrectly connected path from any locaton to any other locaton, so that repeatng ths argument for all and usng the ndrect connectvty we can prove that x / x = x n / x n for all, n S.e. the solutons are the same up-to-scale, a contradcton. Lemma 4. If, ψ 0 or, ψ 1, the egenvalue for A s λ = ψ ψ, 1, and ψ ψ < 1. Proof. Notce that A = 1+ψ 1+ψ+ 1+ψ ψ+ ψ 1+ψ+ = 1+ψ 1+ψ+ 1+ψ ψ+ ψ 1+ψ+ Then we can solve the followng characterstc functons ) 1 + ψ λ ψ + + ψ λ ψ ψ 1 + ψ ψ ψ ψ + 48 ). 1 + ψ + = 0.

9 Then We need to show that ψ 1++ψ λ = ψ ψ, 1. < 1. To show t, t suffces to show g = ψ ψ > 0 Suppose that, ψ 0. Then g s strctly postve as follows: g = ψ ψ Suppose that, ψ 1. Then g s gven by If ψ, then If ψ, then whch completes the proof ψ + ψ ) = 1 > 0. g = 1 ψ ψ. g = 1 ψ + ψ = 1 2ψ 1. g = 1 ψ + ψ = 1 2ψ 1, B.5 Lemmas and Proposton used n Theorem 2 ) 39 In ths secton, we prove the lemma and proposton used n Theorem 2 ). Lemma 5. If, ψ 0 or, ψ 1, then A has strctly postve dagonal elements and s dagonal domnant n ts rows; namely, for all S A > 0, 50) A > A. 51) S Proof. Recall that A matrx s and from Lemma 4, A = Y + ψ 1 + ψ + X, ψ ψ < A smlar argument s found n Johnson and Smth 2011). 49

10 Then the dagonal elements for A are postve; for all S, Also, for all S, whch s equaton 51). A l S A l A = Y + ψ 1 + ψ + X = Y ψ 1 + ψ + X > Y X 0. = Y + ψ 1 + ψ + X ψ }{{} 1 + ψ + X l l S >0 =Y + ψ 1 + ψ + X ψ 1 + ψ + Y X ) = 1 ψ 1 + ψ + Y ψ ψ + + ψ 1 + ψ + X > 0, }{{}}{{} >0 0 The next proposton plays a crucal role n the proof for Theorem 2 ). Proposton 2. If A has strctly postve dagonal elements and s domnant of ts rows, then for all, A 1 > A 1 > 0. Proof. The co-factor expanson of A 1 s 40 A 1 A 1 = det A [S ]) 1)+ det A [S, S ]) det A) det T ) = det A), where T s defned as follows: 40 Remember T = A + 0,, 0, A }{{}}{{}, I 0,, 0 }{{}. N 1) N 1 N N ) A 1 = 1) + det A [N, N ]). det A) 50

11 T s a prncpal component of T : T = T [S, S ]. If a matrx C has postve dagonal elements, and s dagonally domnant of ts rows, then det C) > Then f T has such propertes, then det T ) det A) > 0 snce A s assumed to have these propertes. Thus t suffces to show that T has postve dagonal elements and s domnant of ts rows. By constructon of T, t suffces to show A kk > 0 k S 52) A kk + A k > 0 k = 53) A kk > A kl + 1 l= A k k S 54) l S k A kk + A k > l S k A kl k =. 55) Frst we show equaton 52) and equaton 53). snce A has a strctly postve dagonal, for all k S, A kk > 0, whch s equaton 52). Also snce A s dagonal domnant, A + A > l A l + A A + A 0, whch s equaton 53). Second, we show equaton 54) and equaton 55). Fx k N. Snce A s dagonally domnant, A kk > = l S k A kl l S k l S k = l S k A kl + A k + A k A kl + A k + A k trangle nequalty) A kl + 1 l= A k, whch s equaton 54). Fx k =. Snce A has postve dagonal elements, and s dagonally 41 See also Theorem 3 of Evmorfopoulos 2012). 51

12 domnant, whch s equaton 55). A kk + A k A kk A k = A kk A k = l S k = l S k A kk A kl + A k A k A kl, l S k A kl A k B.6 Exstence and Unqueness usng Gross Substtutes Methodology a la Alvarez and Lucas 2007)) In ths subsecton, we prove the exstence and unqueness of an equlbrum n our unversal gravty framework usng the gross substtutes methodology employed by Alvarez and Lucas 2007). As we show below, the suffcent condtons here are stronger than we provde n Theorem 1 above. Proposton 3. Consder any model wthn the unversal gravty framework. If > ψ > 0 and τ 0, ) for all, S, then the excess demand system of the model satsfes gross substtutes and, as a result, the equlbrum exsts and s unque. Proof. Recall the equlbrum condtons of the unversal gravty framework from equatons 6) and ) p p C = P S P = S P p p C S 56) P S 57) Substtutng equaton 57) nto 56) yelds a sngle equlbrum system of equatons that depends only on the output prces n every locaton: p 1++ψ S ψ C = S C p 1+ψ We defne the correspondng excess demand functon as: Z p) = 1 p S k S C k C p 1+ψ 1 l S lk βp l) k S k p k k S βp k p 1+ψ k p k S S ψ C, 58) 52

13 wherep s defned by equaton 57).Ths system wrtten as such needs to satsfy 6 propertes to be an excess demand system and the gross substtute property to establsh exstence and unqueness. The sx condtons are: 1.Z p) s contnuous forp R+ N ) 2.Z p) s homogeneous of degree zero. 3.Z p) p = 0 Walras Law). 4. There exsts ak > 0 such thatz p) > k for all. 5. If there exsts a sequence p m p 0, where p 0 0 and p 0 = 0 for some, then t must be that: max {Z p m )} 59) and the gross-substtute property: 6. Gross substtutes property: Zp ) p k > 0 for all k. We verfy each of these propertes n turn. Property 1 s trval gven equaton 58) for excess demand. To see property 2, consder multplyng output prces by a scalar β > 0, whch mmedately yelds Z βp) = Z p).as requred. Property 3 can be seen as follows: Z p) p = = S Z p) p = = S k S C k S 1 l S lk p l C p 1+ψ p ψ k k S k p k p 1+ψ S ψ C =0, as requred. Property 4 can be seen as follows: snce 1 p Z p) = 1 p S k S C k Z p) > Q > Q 1 k S C k C l S lk βp l) βp k p 1+ψ ) k S ψ k p k Q l S lk βp l) = βp k S C p 1+ψ ) k S ψ k p k > 0 for all p 0 and Q Q from C. 3. Property 5 can be seen as follows: consder any p R N + ) such that there exsts anl S wherep l = 0 and anl S wherep l > 0. Consder any sequence of output prces such that p n p asn. Then we need to show that: max S Z p). 53

14 To see ths note that: max Z p n ) = max S S max Z p n p ) > max S, S 1 p S τ p ) C p 1+ψ ψ k S τ kp k ) ) p Hence, f t s the case that max, S p k S C k C p ψ k S C k p l S lk p l k S k p k l S lk p l p ψ k p ψ k ) C p ψ k S ψ k p k k S C k l S lk p l p ψ k Q. s bounded below t, t must be that max S Z p n ) as well. Note that: Q =, then because max S Z p n ) p max, S p C p ψ k S C k ) k S ψ k p k l S lk p l p ψ k p > max, S p C k S C k > C mn l S p ψ) l, p ψ k S k p mn ) ) ψ l S lk pmn ) ) = ψ p max where p mn mn l S p l,p max max l S, and C C k S ψ k p mn ) ). Snce k S C k l S lk pmn ) p max as n, then we have max S Z p n ) > ψ > 0 and there exsts anl S such that p n l as well. Fnally, we verfy gross-substtutes. Wthout loss of generalty, we dfferentate only the bracketed term as the term outsde the bracket wll be multpled by zero snce the bracket term s equal to zero n the equlbrum). We have: Z p) p = p S = 1 + ψ) ψ) 1 C p 1+ψ C l S p ψ l k S C l k S k p k k p k p ψ l k S + kl k p 1+ψ S 1 + ψ 1 p 1+ψ ψ C S ψ 1 > 0 because > ψ > 0 and prces, trade frctons, and supply shfters C l are strctly postve. Because propertes 1-6 hold, by Propostons 17.B.2, 17.C.1 and 17.F.3 of Mas-Colell et al. 1995), the equlbrum exsts and unque. Note that n the case where ψ > > 0 whch s the orderng we fnd when we estmate the gravty constants n Secton 5 Theorem 1 stll proves exstence and unqueness of the equlbrum. The followng example shows that gross substtutes may not be satsfed n ths case. 54

15 Example 1. Gross substtuton) Consder the three locaton economy. Take p 3 as the numerare The gross substtute s volated f there exsts p 1 such that Z 1 p 1, p 2, 1) s not monotonc w.r.t. p 2. Consder the followng parameter values:, ψ) = 2, 5) τ = 1 for, {1, 2, 3} C =.9,.6,.1) T. Fgure 12 shows that wth these parameter values, Z 1 p 1, p 2, 1) s not monotonc w.r.t. p 2 when p 1 =.5. B.7 Examples of multplcty n two locaton world In ths subsecton, we derve the equlbrum condtons of a two locaton world and provde examples for dfferent combnatons of the gravty constants.e. the demand elastcty and supply elastcty ψ). We frst derve equatons for the demand and supply of the representatve good n each locaton as a functon of parameters and prces n all other locatons. Combnng C. 2 aggregate demand) and C. 3 market clearng) yelds the followng aggregate demand equaton: Q d = p 1+) S k p Q d k p, 60) k where we denote the quantty of the representatve good demanded n locaton as Q d. Smlarly, C. 3 aggregate supply) yelds the followng aggregate supply equaton: Q s = κc p S p1 p 2 ) +1 p1 p 2 ) +τ, 61) where we denote the quantty of the representatve good suppled n locaton as Q d. Now consder the two-locaton case.e. S {1, 2}) where τ 12 = τ 21 = τ 1 and C 1 = C 2 = 1. Dvdng Q d 1 by Qd 2 usng equaton 60) delvers the followng relatve demand equaton: ) p1 +1 ) p Q d ) ) 2 1+) p1 +τ p 1 p 2 Qd 1 + Q d 1 p1 2 p Q d = 2 ) ) 62) 2 p 2 p 1 p 2 Qd Q d 2 Smlarly, dvdng Q s 1 by Qs 2 delvers the followng relatve supply equaton: Q s 1 Q s 2 = p1 p 2 ) ) ψ p1 p ) p1 p 2 + ψ 63) Note that gven the trade frcton τ and gravty constants, the relatve demand and relatve supply 55

16 can be solved solely as a functon of relatve output prce p 1 p 2 usng equatons 62) and 63), allowng us to analytcally characterze the equlbra usng standard relatve) supply and demand curves. Fgure 3 depcts example equlbra possble for dfferent combnatons of gravty constants; the ponts where the two curves ntersect are possble equlbra. The top left fgure shows that when the supply and demand elastctes are both postve correspondng to a case where the relatve aggregate supply s ncreasng and the relatve aggregate demand s decreasng), there s a unque equlbrum. The top rght fgure shows that when the supply elastcty s postve but the demand elastcty s negatve, both the relatve aggregate supply and demands are ncreasng, potentally resultng n multple equlbra. Smlarly, the bottom left fgure shows that when the supply elastcty s negatve and the demand elastcty s postve, both the relatve aggregate supply and demand curves are decreasng, also potentally resultng n multple equlbra. Fnally, the bottom rght fgure shows that when both the supply and demand elastctes are negatve and sutably large n magntude, the relatve aggregate supply curve s downward slopng and the relatve aggregate demand curve s upward slopng, allowng for a unque equlbra albet one wthout much economc relevance). These examples are consstent wth the suffcent condtons for unqueness presented n Theorem 1. B.8 Tarffs n the unversal gravty framework In ths subsecton, we show how one can use the tools developed above to analyze the effect of tarffs n a smple Armngton trade model. Because tarffs ntroduce an addtonal source of revenue, they are are not strctly contaned wthn the unversal gravty framework. However, t turns out that the equlbrum structure of an Armngton trade model wth tarffs s mathematcally equvalent to the equlbrum structure of the unversal gravty framework. As a result, we can apply Theorems 1 and 2 almost mmedately to the case of tarffs n ths model. To see ths, consder a smple Armngton trade model wth N locatons. 42 Each locaton S s endowed wth ts own dfferentated varety and L workers who supply ther unt labor nelastcally and consume varetes from all locatons wth CES preferences and an elastcty of substtuton σ. Suppose that trade s subect to technologcal ceberg trade frctons τ 1 and ad-valorem tarffs t 0. Defne t 1 + t. Then we can wrte the value of trade flows from to excludng the tarffs) as: X = τ 1 σ t σ Aσ 1 w 1 σ P σ 1 E, 64) where A s the productvty n locaton S, w s the wage, P s the deal Dxt-Stgltz prce ndex, and E s expendture. Income n locaton from trade s equal to ts total sales excludng tarffs): Y = S X. 65) Total ncome and hence expendture) also ncludes the revenue earned from tarffs T : E = Y + T, 66) 42 We consder an Armngton model n order to have an explct welfare functon, the results that follow wll hold for any general equlbrum model where the aggregate supply elastctyψ = 0. 56

17 where tarff revenue s equal to the blateral tarff charged on all trade beng sent 43 : T = S t X. 67) The total expendture by consumers n locaton s also equal to ts total mports plus the tarffs ncurred: E = ) 1 + t X. 68) S Combnng equatons 66), 67), 68), we can demonstrate that trade flows are balanced: E = S 1 + t ) X Y + S t X = S 1 + t ) X Y = S X 69) Fnally, total expendture s equal to the payment to workers plus tarff revenue: E = w L + T Y = w L 70) Defne K τ 1 σ t σ as the blateral kernel, B A L as the ncome shfter, γ A σ 1 w 1 σ as the orgn fxed effect, δ P σ 1 E as the destnaton fxed effect, and α 1 1 σ. Combnng equatons 65), 69), and 70) yelds the followng system of equlbrum equatons: w L = S X B γ α = S K γ δ 71) w L = S X B γ α = S K γ δ. 72) Equatons 71) and 72) can be ontly solved to recover the equlbrum {γ } S and {δ } S ; gven {γ } S and {δ } S, n turn, we can solve for all endogenous varables, as wages can be wrtten 1 1 σ ) 1 as w = γ A, the prce ndex can be wrtten as P = S τ 1 σ t 1 σ 1 σ γ, expendture can ) be wrtten as E = δ S τ 1 σ t 1 σ γ, and real expendture can be wrtten as W E P = δ S τ 1 σ ) σ t 1 σ σ 1 γ. As we note at the begnnng of Secton 3, ths equlbrum system s 43 If we had nstead supposed that tarffs are only leved on goods that actually arrve, we would havet = t τ X, whch does not change the followng analyss n any substantve way. 57

18 dentcal n mathematcal structure to the unversal gravty equlbrum equatons 6 and 7. Hence, Theorem 1 apples drectly wth exstence as long as σ 0 and unqueness as long as σ 1). Moreover, a smlar methodology as employed n Theorem 2 can be used to determne how the equlbrum varables γ and δ respond to shocks that alter the kernel K be they due to changes n ceberg trade frctons or tarffs). In partcular: ln γ ) l = X A + ln K l, + A+ N+l, c 73) ln δ ) l = X A + ln K N+l, + A+ l, c, 74) where à 1, s the, element of the 2N 2N matrx the pseudo) nverse à 1 : 44 à 1 = σ 1 σ Y 1 1 σ Y XT X ) 1, 75) Y Because all endogenous varables n the model are smple functons {γ } S and {δ } S, one can apply equatons 73) and 74) to mmedately derve any elastcty of nterest, e.g. the effect of welfare n locaton l from changng the tarffs mpose on goods comng from. B.9 Global shocks In ths subsecton we show that the exact hat algebra poneered by Dekle et al. 2008) and extended by Costnot and Rodrguez-Clare 2013) can be appled to any model n the unversal gravty framework to calculate the effect of any possbly large) trade shock. Note that Secton 4 nstead showed how to calculate the elastcty of endogenous varables to any trade frcton shock). We show that the key takeaway from Secton 4 holds for all trade shocks: Gven observed data, all the gravty models wth the same gravty constants mply the same counterfactual predctons for all endogenous varables.e. output prces, prce ndces, nomnal ncomes, real expendtures, and trade flows). Consder an arbtrary change n the trade frcton matrx {τ } S S. In what follows, we denote wth a hat the rato of the counterfactual to ntal value of the varable,.e. ˆx xcounterfactual. x ntal The followng proposton provdes an analytcal expresson relatng the change n the output prce and the assocated prce ndex to the change n trade frctons and the ntal observed trade flows: Proposton 4. Consder any gven set of observed trade flowsx, gravty constants andψ, and change n the trade { frcton } matrx τ. Then the percentage change n the exporter and mporter shfters, {ˆp } and ˆP, f t exsts, wll solve the followng system of equatons: p 1++ψ P ψ = S X P Y p p and P = P S X E ) ˆ ˆ, S 76) Proof. We frst note that equlbrum equatons 10) and 7) must hold for both the ntal and 44 The psuedo-nverse can be calculated smply by removng the frst row and column and takng the nverse; see footnote

19 counterfactual equlbra. Takng ratos of the counterfactual to ntal values yelds: p 1++ψ P ψ = P = ) ) S τ P p p C P S S τ P p p C ) p S ) P, S S where we denote the counterfactual equlbrum varables wth a prme and the ntal equlbrum varables as unadorned. Note that from the gravty equaton 10) and C. 3 - C. 5) we have X = ) P p p ψ ) p ψ C P, where p C P = E, so that the above equatons become: p 1++ψ P ψ = P = ) S τ S τ P p ) P ) p p C P S S X ) p 1 E S X, S Fnally, note that from C. 2 and C. 4 we have E = S X and Y = S X, respectvely. Then usng our defnton ˆx xcounterfactual x ntal as requred. p 1++ψ P ψ P = S = S x counterfactual X Y X E = ˆx x ntal ) ˆ ˆP ˆp p S P ) ˆ ˆ S, we have: Note that equaton 76) nherts the same mathematcal structure as equatons 6) and 7). As a result, part ) of Theorem 1 proves that there wll exst a soluton to equaton 76) and part ) of Theorem 1 provdes condtons for ts unqueness. B.10 Identfcaton In ths subsecton, we show how one can always choose a set of blateral trade frctons to match observed trade flows for any choce of gravty constants, own trade frctons, and supply shfters. We frst state the result as a proposton before provdng a proof. Proposton 5. Take as gven the set of observed trade flows {X }, an assumed set of supply shfters {C }, an aggregate scalar κ, and own trade frctons {τ }, and the gravty constants and ψ. Then there exsts a unque set of trade frctons {τ }, output prces {p }, prce ndces {P }, and output {Q } such that the followng equlbrum condtons hold: 59

20 1. For all locatons S, ncome s equal to the product of the output prce and theoutput: Y = p Q 2. For all locaton pars, S, the value of trade flows from to can be wrtten n the followng gravty equaton form: X = P E 3. For all locatons S, output satsfes the followng supply condton: p Q = κc P Proof. Frst, note that the ncome Y = S X, expendture E = S X, and own expendture share λ X E, are all mmedately derved from the observed trade flow data. Second, let us defne our unknown parameters and endogenous varables as functons of data and known parameters. The trade frctons are defned follows: τ = τ Y Y for all, S such that. The output prces are defned as ) λ λ C C p = Y λ τ ) τ X τ /κc for all S. Gven the output prces and trade frctons, the prce ndex s defned as: for all S, P = S 1. X ) 1 Fnally, the output n each locaton s defned as: for all S, p Q = κc. P It s frst helpful to note that gven the above defntons of the trade frctons and output prce ndces, we have the followng convenent relatonshp between own expendture shares and prces: λ = ) τ P 60

21 To see ths, note that we can wrte: Y /C ) λ τ λ = X E = p = p = ) τ P p S ) X E X p = S p p = S = S E S ) ) Y λ τ Y λ X E X E X E S ) Y /C ) λ τ ) ) C ψ τ X Y /C ) λ τ ) Y /C ) λ τ p ) p C τ p X ) 1 ) E = S X, whch s the defnton of E. We now confrm each of the three equlbrum condtons. To see that ncome s equal to the product of the output prce and the output, we wrte: p Q = Y p Q = Y ) λ τ ψ ) /κc Q ) 1 p κc Q P p Q = Y Q Q p Q = Y, as requred. To see that the value of trade flows can be wrtten n the gravty equaton form, we wrte the gravty equaton as follows: P E = ) Y λ τ Y λ ) ) C ψ τ X C ) P E Y /C ) λ ψ = X τ ψ p Y /C ) λ ψ τ ψ p τ ) X p X ) 1 ) P E P E 61

22 Recall from above that we have the followng relatonshp between prces and own expendture shares: ) λ = τ P so that: P E = X Y ) p P C ) Y ) p P C ) p Furthermore, recall that we have defned our quanttes as follows: p Q = κc, P p ) p X P E whch mples that: P E = X Y /Q ) Y /Q ) ) p p ) p X P E We have shown above that p Q = Y, so that we have: P E = X p X P E We clam that ths mples that observed trade flows are explaned by the gravty equaton,.e.: To see ths, suppose not. Then we have X = P E P E = X p X P E but X P E. Then wthout loss of generalty we can wrte X = P E ε, where ε 1. P E = ) P E ε 1 = ε ε ε = ε ε S p p P E ) P E ε whch then mples that we have: X = P E ε 62

23 however, we know that: S X = E S P E ε = E S S = 1 ε ε = 1, whch s a contradcton. Hence, the observed trade flows are explaned by the gravty equaton. Fnally, we note that the thrd equlbrum condton trvally holds by the defnton of Q : p Q = κc. P Hence, gven our defntons, we have found a unque set of trade frctons {τ }, output prces {p } S, prce ndces {P } S, and output {Q } S such that the equlbrum condtons hold for any set of observed trade flows {X }, S, an assumed set of supply shfters {C } S and own trade frctons {τ } S, and the gravty constants, ψ). B.11 Real output prces, welfare, and the openness to trade In ths secton, we explore the relatonshp between the real output E /P and real output prce p /P n the unversal gravty framework and the welfare n a number of semnal models. We then show how the real output prce n the unversal gravty framework relates to the observed own expendture share. Combnng the two results allow ones to wrte the welfare n each of these models as a functon of observed own expendture share, as n Arkolaks et al. 2012a). B.11.1 Real output prces and welfare In ths subsecton, we provde a mappng between real output prces and the welfare of a unt of labor for the trade ntroduced and the economc geography model n Secton 2. The trade model In the trade model, the output prce p s w ζ P 1 ζ /A. As a result we have the welfare of each worker Ω can be expressed as a functon of the real output prce n the unversal gravty framework as follows: w P = p A P 1 γ ) 1 ζ } {{ } =w 1 = A P 1 γ p P ) 1 ζ. Or equvalently, we can express the welfare n terms of the supply elastcty. w = A 1+ψ P p P ) 1+ψ. 63

24 The economc geography model In the economc geography model, the welfare s w P u,and the prce p s. Therefore the welfare s w A L a Ω = A u L a+b p Welfare equalzaton and the labor market clearng condton mples S P ). Ω = L ) [ a+b [ )] 1 ] a+b) p a+b A u. P B.11.2 Real expendture, real output prces and the openness to trade In ths subsecton, we show we can express real expendture and real output prces n any model wthn the unversal gravty framework as a functon of openness to trade and the gravty constants, as n Arkolaks et al. 2012a). We begn by defnng λ X E can express the real output prce p P as locaton s own expendture share. From equaton 10), we n a locaton as a functon of ts own expendture share: X = p k S p k 1 E = = λ. 77) P Moreover, gven C. 3, C. 4 and C. 5, we can wrte total real expendture W E P ts own expendture share as well: as a functon of W = E W = W = P p P p P ) Q ) ) p κc P Combnng equatons 77) and 78) yelds: ) 1+ψ p W = κc. 78) P W = κc λ ) 1+ψ. Note that a postve aggregate supply elastcty ψ > 0) ncreases the elastcty of total real expendture to own expendture share, thereby amplfyng the gans from trade. Note too that the 64

25 dervatons above mply that: ) ln W ln p P = ψ + 1) + ln κ, ln τ ln τ ln τ.e. we can recover the elastcty of the total real expendture to-scale) to the trade frcton shock from the elastcty of the real output prce to the trade frcton shock by smply multplyng by ψ + 1. B.12 Addtonal Fgures 65

26 Fgure 3: Examples of multplcty and unqueness n two locatons a) Postve supply and demand elastctes b) Postve supply elastcty, negatve demand elastcty c) Postve demand elastcty, negatve supply elastcty 5 d) Negatve supply and demand elastctes both 1) Notes: Ths fgure shows examples of relatve supply curve and relatve demand curves for a two locaton world for dfferent combnatons of supply and demand elastctes; see Secton B.7 for a dscusson.

27 Fgure 4: Correlaton between observed ncome and own expendture shares and the equlbrum values from the gravty model Income log) Observed JPN HKG DEU FRA GBR SGP IDN THA KOR ITA TWN NLD MYS BEL CHE CHN ESP PHL IND SWE NOR AUT DNK BGD FIN POL GRC PRT IRL RUSZAF VNMTUN AUS BGR CZE HUN NGA LKA IRN EGY NZLPAK TUR MLT SVK SVN MUS HRV KHM ROU TZA CYP UGA MAR UKR BLRLTU KAZ MOZ LVA EST MMR AZE ALB MDG ZMBBWA SENLAO ARM ETH GEO MWI ZWE KGZ LUX BRA USA VEN MEX CAN COL PER ECU CRI GTM PAN ARG BOL NICPRY URY CHL Predcted Observed IDN MYS SGPTHA PHL KHM Own expendture share log) BGD ALB BGR GTM HKG JPN PAN ARM GBR LAO CHE DNK NLD LKA GRC PAK PRT COL PER SVN TZA UGA HRV ITA ESP EGY NIC KOR POL NORFRA BOL AUS BRA CRIGEO ECU CYP DEU FINLTULVA VEN NZL SEN IND KGZ ZAF SWE USA ZMB ETH AUT CZE BLR MEX IRN MOZ MAR MUS SVK ROU PRY URY TUN TURCHL ARGTWN MDG AZE CAN HUN CHN MMR KAZ MWI RUS UKR BWA VNM BEL EST ZWE MLT IRL LUX NGA Predcted Notes: Ths fgure shows the relatonshp between the observed and predcted ncome and own expendture shares, respectvely. The predcted ncomes and own expendture shares are the equlbrum values from the general equlbrum gravty model where blateral frctons are those estmated from a fxed effects gravty regresson and the supply shfters are estmated from a regresson of log ncome on geographc and nsttutonal controls. The scatter plots are plots of the resduals after controllng for the drect effect of the geographc, hstorcal, and nsttutonal observables. 67

28 Fgure 5: The network effect of a U.S.-Chna trade war: Degree 0 Notes: Ths fgure depcts the degree 0 effect of an ncrease n the blateral trade frctons between the U.S. and Chna a trade war ) n all countres. The degree 0 effect s the drect mpact of the trade war on the U.S. and Chna, holdng constant the prce and output n all other countres. Note that output prces, output, and the prce ndex effects are dentfed only to scale, whereas the level of ncome and real output prces are known see the dscusson n Secton 2). 68

29 Fgure 6: The network effect of a U.S.-Chna trade war: Degree 1 Notes: Ths fgure depcts the degree 1 effect of an ncrease n the blateral trade frctons between the U.S. and Chna a trade war ) n all countres. The degree 1 effect s the mpact of the degree 0 shock on all countres through the trade network, holdng constant the prces and output of ther tradng partners. Note that output prces, output, and the prce ndex effects are dentfed only to scale, whereas the level of ncome and real output prces are known see the dscusson n Secton 2). 69

30 Fgure 7: The network effect of a U.S.-Chna trade war: Degree 2 Notes: Ths fgure depcts the degree 2 effect of an ncrease n the blateral trade frctons between the U.S. and Chna a trade war ) n all countres. The degree 2 effect s the mpact of the degree 1 shock on all countres through the trade network, holdng constant the prces and output of ther tradng partners. Note that output prces, output, and the prce ndex effects are dentfed only to scale, whereas the level of ncome and real output prces are known see the dscusson n Secton 2). 70

31 Fgure 8: The network effect of a U.S.-Chna trade war: Degrees >2 Notes: Ths fgure depcts the cumulatve effect of all degrees greater than two of an ncrease n the blateral trade frctons between the U.S. and Chna a trade war ) n all countres. A degree k effect s the mpact of a degree k 1 shock on all countres through the trade network, holdng constant the prces and output of ther tradng partners. Note that output prces, output, and the prce ndex effects are dentfed only to scale, whereas the level of ncome and real output prces are known see the dscusson n Secton 2). 71

32 Fgure 9: The network effect of a U.S.-Chna trade war: Total effect Notes: Ths fgure depcts the total effect of an ncrease n the blateral trade frctons between the U.S. and Chna a trade war ) n all countres. Ths s the nfnte sum of all degree k effects. Note that output prces, output, and the prce ndex effects are dentfed only to scale, whereas the level of ncome and real output prces are known see the dscusson n Secton 2). 72

33 Fgure 10: Local versus global effects of a U.S.-Chna trade war Notes: Ths fgure depcts the correlaton of the local nfntesmal) elastctes and the global 50% ncrease) mpacts of a trade war on the real output prce n each country. 73

34 Fgure 11: The effect of a U.S.-Chna trade war on real output prces n the U.S. and Chna: Robustness US Supply elastcty Supply elastcty Demand elastcty Chna Demand elastcty Notes: Ths fgure depcts the elastcty of real output prces to an ncrease blateral trade frctons between the U.S. and Chna a trade war ) for many constellatons of demand and supply elastctes and ψ, respectvely. The star ndcates the estmated supply and demand elastcty constellaton, and the red box outlnes the 95% confdence nterval of the two parameters. 74

35 Fgure 12: Excess non-monotonc demand functon for 1, Z 1 p 2 )