Online Appendix for Trade and Insecure Resources

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1 B Onlne ppendx for Trade and Insecure Resources Proof of Lemma.: Followng Jones 965, we denote the shares of factor h = K, L n the cost of producng good j =, 2 by θ hj : θ Kj = r a Kj /c j and θ Lj = w a Lj /c j. Smlarly, θ KG r ψ r/ψ and θ LG w ψ w/ψ ndcate the correspondng cost shares n guns. Now denote the amount of land and labor employed n ndustry j =, 2 respectvely by Kj and L j. Then, these quanttes as a fracton of resources remanng once land and labor for producng guns have been set asde are respectvely ndcated by λ Kj K j /K and λ Lj L j /L. Fnally, let a percentage change be ndcated by a hat ˆ over the assocated varable e.g., x = dx x. Part a: Notng that c = and c 2 = p, dfferentaton of 2 and 3 totally gves c w dw + c r dr = 0 = a w dw L c w c 2 w dw + c 2 r dr = dp = a w dw L2 c 2 w + a r dr K c r = 0 + a r dr K2 c 2 r = dp p. Wth the defntons gven above, we can wrte ths system of equatons as θ L θ L2 θ K θ K2 ŵ r = 0 p. B. Now, snce h=k,l θ hj = for j =, 2 by defnton, the determnant of the coeffcent matrx above, denoted by θ, can be wrtten as θ θ K2 θ K = θ L θ L2 = ω k2 k ω + k ω + k2 0 f k2 k. Then, solvng B. for the p -nduced changes n factor prces yelds p w p w = θ K θ and p r p r = θ L θ. B.2 From B.2, then, we have p ωp/ω = p wp/w p rp/r = / θ 0 when k2 k, whch completes the proof of part a. Parts b and c: Note frst that we can combne 4 and 5 to obtan λ L k +λ L2 k 2 = k. Then, followng the strategy above n part a, we dfferentate 4 and 5 totally and solve

2 the resultng system of equatons to obtan = λ λ L2 K + λ L K2 2 = λ +λ K L λ L K + λ λ θ L2 δk + λ K2δL p λ θ λ L δk + λ KδL p, where δ K λ K θ L σ + λ K2 θ L2 σ 2 > 0 and δ L λ L θ K σ + λ L2 θ K2 σ 2 > 0, wth σ j = c 2 c j j / c j c j w r w r beng the absolute value of the elastcty of substtuton between land and labor n ndustry j; λ denotes the determnant of the coeffcent matrx obtaned from dfferentatng 4 and 5; recallng j=,2 λ hj = for h = K, L, we have λ k λ K2 λ L2 = λ L λ K = 2 k k k k k 2 k 0 f k2 k. Now, observe that k = K L, whch mples RS = 2 = λ k + δ K + δ L λ θ p. B.3 Inspecton of B.3 confrms parts b and c. Proof of Lemma.2: Denote country s land and labor shares n total net ncome R by s K r K /R and s L w L /R, and let σ G = ψ ψ wr/ψ wψ r be the absolute value of the elastcty of substtuton between land and labor n the guns sector. Total dfferentaton of 6, usng the lnear homogenety of ψ, yelds k = [ ψ θlg ] θ KG σ G G p + ψ θ R s K s L + r K 0 φ G j R s K dg j + φ dk 0 K R s K s L + dk K [ r s L ψ Parts a c & e: The proofs follow from B.4. K 0 φ G ψ r ] + θlg dg dl L. B.4 Part d: Suppose G = G j so that φ G = φ G j. Now use dg = dg j n B.4 to obtan k / G k = ψ θlg s L R s. K s L Usng the defntons of θ LG, s L, and R, along wth those for K and L n 4 and 5, 2

3 tedous algebra verfes the followng transformaton of ths relatonshp: k / G k = ψ θlg s L R s = ψ w k kg K s L K = ψ wl k kg K. B.5 L Inspecton of ths expresson confrms part d. Proof of Lemma.3: Part a: Dfferentate 9 wth respect to G and use equaton.c n ppendx to obtan V G G = µ r K 0 φ G G < 0. B.6 Part b: Dfferentaton of 9 wth respect to G j, usng equaton.d, gves V G G j = µ r K 0 φ G G j 0 f G G j. B.7 Part c: s noted n the text, ψ /r = ψω,, mplyng that ψ /r / p = ψ wω p. Then, dfferentatng 9 wth respect to prce and evaluatng the resultng expresson at the optmum gves by Lemma.a V G p = µ r ψ /r ψ p = µ w ψ w p ωp p ψ ω ψ = µ θ LG p θ 0 f k 2 k. B.8 Part d: Ths s a standard property of ndrect trade utlty functons, hghlghtng the mportant dea that, for gven guns, a country s welfare s hgher, the greater s the devaton of product prces from ther autarkc levels Dxt and Norman, 980. Proof of Lemma.4: Let σd > 0 be the elastcty of substtuton n consumpton. Focusng on percentage changes, note that RD = σ D p and that the expresson for RS s gven n B.3. Totally dfferentatng 0 and rearrangng terms gves RD = RS = σd + δ K + δ L λ θ p + λ k = 0. The above relaton and the defntons of θ and λ reveal that p s decreasng ncreasng n k f k 2 > k k 2 < k. Combnng the expresson for ˆk n B.4 wth the above 3

4 expresson gves p = [ k / G ] λ k dg + k / Gj k dg j + φ dk 0 K + dk K dl L, B.9 where σd + δ K +δ L λ θ + ψ θlg θ KG σ G G > 0. The proofs to parts a d now follow from λ θ R s K s L B.9 and Lemma.2 n ppendx. Proof of Theorem.: Exstence: We establsh exstence of equlbrum n pure strateges, by showng that each country s payoff functon V s strctly quas-concave n ts strategy, G. To do so, t s suffcent to show ether that V s strctly monotonc n G or that V s frst strctly ncreasng and then strctly decreasng over the agent s strategy space. Let F KG, L G be the producton functon for guns that s dual to the unt cost functon ψw, r and defne G F K, L as the level of guns produced wth the country s entre secure endowments of land and labor. Country s strategy space s [0, G ]. For any G j [0, G j ], f G = G, country j wll not be able to produce ether of the consumpton goods; therefore, V G, G j < V G, G j for any G [0, G whch mples that, under autarky, no country wll use all of ts resources to produce arms. Furthermore, snce lm G 0 f G = by assumpton, we must have V / G > 0 as G 0. By the contnuty of V n G, there wll exst a best response functon for each country, B Gj mn{g 0, G V / G = 0}, wth the property that V / G > 0 G < B Gj. Thus, to establsh strct quas-concavty of V n G we need only to prove that V / G < 0, G > B Gj. Suppose, to the contrary, that V / G 0. Snce V must eventually fall to V G, G j, ths supposton mples that V must attan a local mnmum at some G > B G j, whch would mply that 2 V / G 2 > 0. We now establsh that ths s not possble. Recallng that p = p G, G j under autarky and that ω = ωp, we dfferentate 9 wth respect to G and apply 9 to the resultng expresson to obtan 2 V G 2 = [ V G G ]p =p + ± [V G p ] p =p p G < 0. B.0 By Lemma.3a, the frst term n the RHS of the above expresson s negatve regardless of the rankng of factor ntenstes. Furthermore, by Lemmas.3c and.4b, the product of the expressons n the second term wll also be negatve. It follows that 2 V / G 2 < 0 In ths expresson and below, the top sgns n ± and apply when k 2 > k and the bottom sgns apply when k 2 < k. 4

5 at any G where V / G = 0 regardless of the rankng of factor ntenstes. Ths proves B Gj s unque and establshes the exstence of a pure-strategy equlbrum. Unqueness: Havng already establshed that guns producton s bounded.e., B Gj 0, G for =, 2 j, we can now establsh unqueness of equlbrum by showng that, at any equlbrum pont, the determnant of the Jacoban of the net margnal payoffs n 9 s postve.e., J = 2 V G 2 2 V 2 G V G G 2 2 V 2 G 2 G > 0 Kolstad and Mathesen, 987. Consder an equlbrum pont where G = B G2 and G2 = B G. From the expresson for J, t can be seen that, f the product of the slopes of the two countres best response functons, B / G2 B 2 / G, s less than at G, G2, then J > 0, mplyng that ths equlbrum s unque. The slope of country s best-response functon can be wrtten as B G j = 2 V / G G j 2 V / G 2 = [ V G G j ]p =p + [ V G G ]p =p + ± [V G p ] [V G p ] p =p p =p ± p G j p. G B. Snce 2 V / G 2 < 0 as shown n B.0, the sgn of B / Gj s determned by the sgn of 2 V / G G j shown n the numerator of B.. Now, by Lemmas.3c and.4b, the second term of the numerator of the RHS of ths expresson s always postve. By Lemma.3b, the frst term n the numerator s postve f B Gj > G j also see equaton B.7, n whch case G s a strategc complement for G j. However, f B Gj < G j, then the frst term s negatve. Thus, when B Gj s suffcently smaller than G j, G can become a strategc substtute for G j. Furthermore, snce φ G G = φ 2 2 G 2 G see.d, t follows from B.7 that sgn [ VG ] G = sgn [ V 2 ] 2 G 2 G. Therefore, we have p =p p 2 =p 2 two possbltes to consder. Ether B / Gj > 0 and B j / G 0 for =, 2 j ; or, B / Gj > 0 for =, 2 j. It s easy to check that, n case, B / G 2 B 2 / G < and therefore J > 0. Turnng to case, we now establsh the exstence of suffcent condtons that ensure B / G2 B 2 / G < and thus J > 0. 2 To proceed, use B.4 and B.9, applyng 9, to obtan p p ψ θ LG G = λ R s K s L and p G j = p ψ λ R s K s L φ G j φ G The above expressons together wth 9, B.6, B.7, and B.8 can be substtuted nto 2 Note that, n case, J > 0 s also the condton for local stablty of equlbrum. s L. 5

6 B. to obtan B / Gj = φ G j /φ G Γ, where Γ = [ φ G G j φ G j + ψ θ LG s L λ θ R s K s L ] / [ φ G G φ G + ψ θ LG 2 λ θ R s K s L ]. B.2 From equatons.a and.b n ppendx, we have φ G 2 /φ G φ 2 G /φ 2 G 2 =, mplyng B / G2 B 2 / G = Γ Γ2 ; therefore, f Γ 0, for =, 2, then J > 0. In case, both the numerator and the denomnator of Γ are postve, so Γ > 0. Now defne η G [ φ /φ + φ /φ ]. From.a.d, η = G [f G G G G G j G j /f f /f ] > 0. Then, subtractng the numerator of Γ from ts denomnator whle usng the defnton of shown below B.9 gves the followng: η G σd + δ K + δ L λ θ + ψ θ LG λ θ R s K s L θ LG + θkgσ Gη s L. B.3 Clearly, a suffcent condton for Γ < s that B.3 s postve, whch s almost always true. In partcular, snce the frst term and the coeffcent n front of the second term are unambguously postve, a suffcent but hardly necessary condton for Γ < s θlg + θ KG σ G η s L 0 or equvalently θ LG s L + θ KG σ G η s L 0. Ths condton s satsfed under a wde range of crcumstances, 3 ncludng: σg η s L, whch requres arms nputs not to be close complements; and θlg s L or, by B.5, k > kg, whch requres the guns sector to be suffcently labor ntensve, regardless of the degree of substtutablty between nputs n arms. condtons establshed above, ensures unqueness of equlbrum. Ether condton, along wth the boundary Proof of Theorem.2: The proofs for exstence and unqueness of equlbrum, whch buld on parts a and b of Lemma.3, are smlar to and smpler than those n the case of autarky see the proof of Theorem. above, and are thus omtted here. The man text provdes a dscusson of the logc underlyng the symmetry results. Proof of Proposton.: Under free trade wth p = π, each dentcal country s excess demand for good 2 s M = D 2 2 = D 2 π, Rπ, K, L R π π, K, L. B.4 The effect of conflct on a country s mports and hence exports can be dentfed by dfferentatng the expresson above wth respect to guns. We thus move from Nrvana to a stuaton of some conflct. Frst, total dfferentaton of B.4 keepng the world prce 3 If, for example, the producton functon for guns s Cobb-Douglas and the conflct technology takes the Tullock form.e., fg = G γ, γ 0, ], then σ G = and η =, thus mplyng that the suffcent condton smplfes to s L 0, whch s always satsfed. 6

7 fxed yelds dm = D 2 R [R KdK + R L dl ] R πk dk R πl dl. B.5 The frst term n the RHS of the above expresson s an ncome effect. Snce our focus here s on the case of dentcal countres, an ncrease n G would be matched by an equal ncrease n G j, and so would result n a decrease n ncome and thus a decrease n the demand for good 2. The last two terms combned represent a producton effect, as an ncrease n G and an equal ncrease n G j nfluence each country s resdual factor endowments. To proceed, recall that D 2 = α D R/p where α D πµ p /µ 0, s each country s expendture share on good 2. Under the assumpton that the country trades freely so that p = π, these relatonshps mply D 2 / R = α D /π. Furthermore, by the propertes of the revenue functon wth p = π, we have R K = r, R L = w, R πk = R Kπ = r π and R πl = w π for =, 2. ssumng dg = dg j = dg whch mples φ G = φ G j and keepng n mnd that K = K + φk 0 ψ r G and L = L ψ w G, equaton B.5 can be wrtten as dm = [ ] π α D ψ + r π ψ r + w π ψ w dg. B.6 Next, recall from the proof of Lemma.a, the defntons of the cost shares of land and labor n the producton of consumpton goods j =, 2 and guns: θ Kj = ra Kj /c j and θ Lj = wa Lj /c j, for j =, 2; and θ KG rψ r /ψ and θ LG wψ w /ψ. Then, tedous algebra usng equaton B.2 wth p = π shows that equaton B.6 can be rewrtten as follows: dm = ψ π θ [ α Dθ L + α D θ L2 θ LG ] dg, B.7 where θ = θ K2 θ K = θ L θ L2 0 as k 2 k. s shown n B.7, the key n determnng the sgn of dm s the sgn of the term n brackets, whch we denote by Λ: Λ α D θ L + α D θ L2 θ LG. We nterpret Λ as reflectng the degree of land ntensty of guns producton. In partcular, f Λ < 0, then we say that guns producton s suffcently labor ntensve; and f Λ > 0, then we say that guns producton s suffcently land ntensve. 4 To fx deas suppose that good 2 s produced ntensvely wth land k 2 > k, whch mples that θ > 0. In ths case, f guns producton s suffcently labor ntensve Λ < 0, then dm < 0, mplyng that there s a postve bas n the export of good 2 relatve to 4 One can show that these condtons, when evaluated n the autarkc equlbrum, are dentcal to requrng respectvely k Gω < k and k Gω > k. 7

8 the hypothetcal case of no conflct. 5 lternatvely, f guns producton s suffcently land ntensve Λ > 0, then dm > 0, mplyng that there s a postve bas n the mport of the land-ntensve good. Proof of Proposton.2: Consder an allocaton n the ES subset of S, where G G 2 and G F = G2 F > assumng world prces n the neghborhood of p. Under the mantaned assumpton that k2 > k, p > p2 holds see Lemma b n ppendx. Now suppose, just for the sake of argument, that both countres move to free trade, but face dfferent world prces: π = p and π2 = p 2. The resultng outcome s smply the autarkc equlbrum. fxed, and suppose that π2 ncreases to π = p. Ths ncrease To proceed, keep π = p n π 2 nduces country 2 to ncrease ts arms, whch n turn nduces country to ncrease ts arms. However, the ncrease n country s armng wll be proportonately less, so that n the free trade equlbrum armng s equalzed across countres: BF 2 G F ; p = B F G2 F ; p > G > G2. ssumng that θ LG s not too large relatve to s L so that dk /dg2 G =BF G2 < 0 see equaton.5, π = p < π and hence country s comparatve advantage s dstorted n ths equlbrum: MF p > 0. Snce a shft from autarky to free trade can be vewed as an exogenous change n the effectve prce, the welfare effects of such a shft can generally be decomposed nto the terms-of-trade effect and the strategc welfare effect as shown n. Of course, π has not changed n ths experment, mplyng the terms-of-trade effect on country s welfare s zero. Thus, the effect of both countres movng from autarky to free trade on country s welfare at π = p wll be captured by the negatve strategc welfare effect alone.e., the second term n equaton nduced by the ncrease n π 2. Note that, whle drven solely by the strategc welfare effect, ths adverse consequence for country s welfare reflects the dstorton n country s comparatve advantage at π = p. fall n the world prce below p nduces both a terms-of-trade mprovement and a postve strategc effect for country, and thus causes an ncrease n country s welfare. s such, there exsts a world prce less than p, for whch country s welfare under free trade wll be equal to that under autarky. s π rses above p approachng country s trade elmnatng prce π, both the terms of trade effect and the strategc welfare effect for country are negatve. Further ncreases n π above π contnue to mply a negatve strategc welfare effect, but now also a postve terms-of-trade effect. Nevertheless, for π suffcently close to π, the former effect wll domnate. 5 Note that f the representatve country were to export good 2 n the hypothetcal case of no conflct, ths bas would mply an expanson of the volume of trade under conflct. But, f the country were to mport good 2 n the case of no conflct, ths bas means that the volume of trade shrnks n the presence of conflct. 8

9 References Dxt,.K. and Norman, V., 980 Theory of Internatonal Trade, Cambrdge, England: Cambrdge Unversty Press. Jones, R.W., 965 The Structure of Smple General Equlbrum Models, Journal of Poltcal Economy 736, Kolstad, C.. and Mathesen, L., 987 Necessary and Suffcent Condtons for Unqueness of a Cournot Equlbrum, Revew of Economc Studes 544,