Elements of Advanced International Trade 1

Transcription

1 Elements of Advanced Internatonal Trade 1 Treb Allen 2 and Costas Arkolaks 3 February 2016 [New verson: prelmnary] 1 Ths set of notes and the problem sets accomodatng them s a collecton of materal desgned for an nternatonal trade course at the graduate level. We are grateful to Crstna Arellano, Jonathan Eaton, Tmothy Kehoe and Samuel Kortum. We are also grateful to Steve Reddng and Peter Schott for sharng ther teachng materal and to Federco Esposto, Monca Morlacco, Ananth Ramananarayan, Olga Tmoshenko, Phlpp Stel, Alex Torgovtsky, and Ype Zhang for valuable nput and for varous comments and suggestons. All remanng errors are purely ours. 2 Northwestern Unversty and NBER 3 Yale Unversty and NBER

2 Abstract These notes are prepared for a Ph.D. level course n nternatonal trade.

3 Contents 1 An ntroducton nto deductve reasonng Inductve and deductve reasonng Employng and testng a model Internatonal Trade: The Macro Facts An ntroducton to modelng The Heckscher-Ohln model Autarky equlbrum Free Trade Equlbrum No specalzaton Specalzaton The 4 bg theorems Models wth Constant Elastcty Demand Constant Elastcty Demand The gravty setup Armngton model The model Gravty

4 3.3.3 Welfare Monopolstc Competton wth Homogeneous Frms and CES demand Setup Consumers Frms Gravty Welfare Rcardan model The two goods case Autarky Free trade Problem Sets Models wth CES demand and producton heterogenety Introducton to heterogenety: The Rcardan model wth a contnuum of goods Where DFS stop and EK start A theory of technology startng from frst prncples Order Statstcs and Varous Moments Applcaton I: Perfect competton Eaton-Kortum) Model Setup Equlbrum Gravty Welfare Dscusson of EK Applcaton II: Bertrand competton Bernand, Eaton, Jensen, Kortum)... 60

5 4.4.1 Supply sde Gravty Welfare Dscusson of BEJK Applcaton III: Monopolstc competton wth CES Chaney-Meltz) Model Set-up Equlbrum The Pareto Dstrbuton Trade wth frm heterogenety Next steps Summary Problem Sets Closng the model General Equlbrum Solvng for the Equlbrum when the Proft share s constant Endogenous Entry Labor Moblty Problem Sets Model Characterzaton The Concept of a Model Isomorphsm An Exact Isomorphsm A Partal Isomorphsm General Equlbrum: Exstence and Unqueness Analytcal Characterzaton of the Gravty Model v

6 6.4 Computng the Equlbrum Alvarez-Lucas) Conductng Counterfactuals: The Dekle-Eaton-Kortum Procedure Problem Sets Gans from Trade Trade Lberalzaton and Frm Heterogenety Trade Lberalzaton and Welfare gans Arkolaks-Costnot-Rodrguez- Clare) Ex-ante Gans Atkeson Bursten 2010) Dscusson: gans from trade and the trade heterogenety Global Gans Problem Sets Extensons: Modelng the Demand Sde Extenson I: The Nested CES demand structure Extenson II: Market penetraton costs Extenson III: Multproduct frms Gravty and Welfare Extenson IV: General Symmetrc Separable Utlty Functon Monopolstc Competton wth Homogeneous Frms Krugman 79) Consumer s problem Frm s Problem Monopolstc Competton wth Heterogeneous Frms Arkolaks- Costnot-Donaldson-Rodrguez Clare) Frm Problem Gravty v

7 Welfare Problem Sets Modelng Vertcal Producton Lnkages Each good s both fnal and ntermedate Each good has a sngle specalzed ntermedate nput Each good uses a contnuum of nputs The gravty estmator and trade cost estmates The Head and Res procedure Gravty estmators The tradtonal gravty estmator The fxed effects gravty estmator The rato gravty estmator The Anderson and van Wncoop procedure The no arbtrage condton The Eaton and Kortum 2002) approach The Donaldson 2014) approach The Allen 2012) approach Parameter Choces for Gravty Models Recoverng the trade elastcty from gravty estmates The Eaton and Kortum procedures) Gravty trade models where β = Calbraton of a frm-level model of trade Estmaton of a frm-level model The model v

8 Estmaton, smulated method of moments Some facts on dsaggregated trade flows Frm heterogenety Trade lberalzaton Trade dynamcs Appendx Dstrbutons The Fréchet Dstrbuton Fgures and Tables 198 v

9 Lst of Fgures 14.1 Sales n France from frms grouped n terms of the mnmum number of destnatons they sell to French entrants and market sze. Source: Eaton, Kortum, and Kramarz 2011) Dstrbuton of sales for Portugal and means of other destnatons group n tercles dependng on total sales of French frms there v

10 Chapter 1 An ntroducton nto deductve reasonng 1.1 Inductve and deductve reasonng Inductve or emprcal reasonng s the type of reasonng that moves from specfc observatons to broader generalzatons and theores. Deductve reasonng s the type of reasonng that moves from axoms to theorems and then apples the predctons of the theory to the specfc observatons. Inductve reasonng has faled n several occasons n economcs. Accordng to Prescott see Prescott 1998)) The reason that these nductve attempts have faled... s that the exstence of polcy nvarant laws governng the evoluton of an economc system s nconsstent wth dynamc economc theory. Ths pont s made forcefully n Lucas famous crtque of econometrc polcy evaluaton. Theores developed usng deductve reasonng must gve assertons that can be falsfed by an observaton or a physcal experment. The consensus s that f one cannot 1

11 potentally fnd an observaton that can falsfy a theory then that theory s not scentfc Popper). A general methodology of approachng a queston usng deductve reasonng s the followng: 1) Observe a set of emprcal stylzed) facts that your theory has to address and/or are relevant to the questons that you want to tackle, 2) Buld a theory, 3) Test the theory wth the data and then use your theory to answer the relevant questons, 4) Refne the theory, gong through step Employng and testng a model A vague defnton of two methodologes: calbraton and estmaton Calbraton s the process of pckng the parameters of the model to obtan a match between the observed dstrbutons of ndependent varables of the model and some key dmensons of the data. More formally, calbraton s the process of establshng the relatonshp between a measurng devce and the unts of measure. In other words, f you thnk about the model as a measurng devce calbratng t means to parameterze t to delver sensble quanttatve predctons. Estmaton s the process of pckng the parameters of the model to mnmze a functon of the errors of the predctons of the model compared to some pre-specfed targets. It s the approxmate determnaton of the parameters of the model accordng to some pre-specfed metrc of dfferences between the model and the data to be explaned. It s generally consdered a good practce to stck to the followng prncples see Prescott 1998) and the dscusson n Kydland and Prescott 1994)) when constructng 2

12 quanttatve models: 1. When modfyng a standard model to address a queston, the modfcaton contnues to dsplay the key facts that the standard model was capturng. 2. The ntroducton of addtonal features n the model s supported by other evdence for these partcular addtonal features. 3. The model s essentally a measurement nstrument. Thus, smply estmatng the magntude of that nstrument rather than calbratng the model can nfluence the ablty of the model to be used as a measurng nstrument. In addton the model s selecton or n partcular, parametrc specfcaton) has to depend on the specfc queston to be addressed, rather than the answer we would lke to derve. For example, f the queston s of the type, how much of fact X can be accounted for by Y, then choosng the parameter values n such a way as to make the amount accounted for as large as possble accordng to some metrc makes no sense. 4. Researchers can challenge exstng results by ntroducng new quanttatvely relevant features n the model, that alter the predctons of the model n key dmensons. 1.3 Internatonal Trade: The Macro Facts Chapter 2 of Eaton and Kortum 2011) manuscrpt 3

13 Chapter 2 An ntroducton to modelng 2.1 The Heckscher-Ohln model The Heckscher-Ohln H-O) model of nternatonal trade s a general equlbrum model that predcts that patterns of trade and producton are based on the relatve factor endowments of tradng partners. It s a perfect competton model. In ts benchmark verson t assumes two countres wth dentcal homothetc preferences and constant return to scale technologes dentcal across countres) for two goods but dfferent endowments for the two factors of producton. The model s man predcton s that countres wll export the good that uses ntensvely ther relatvely abundant factor and mport the good that does not. We wll present a very smple verson of ths model. Country s representatve consumer s problem s max a 1 log c 1 + a 2 log c 2 s.t. p 1 c 1 + p 2c 2 r k + w l 4

14 The producton technologes of good ω n the two countres are dentcal and gven by y ω = z ω k ω ) bω l ω) 1 bω,, ω = 1, 2 and where 0 < b 2 < b 1 < 1. Ths mples that good 1 s more captal ntensve than good 2. Assume for smplcty that k 1 / l 1 > k 2 / l 2. Ths mples that country 1 s captal abundant relatve to country 2. Fnally, goods, labor, and captal markets clear. One of the common assumptons for the H-O model s that there s no factor ntensty reversal whch n our example s always the case gven the Cobb-Douglas producton functon one good s always more captal ntensve than the other, wth the captal ntensty gven by b ω ) Autarky equlbrum We frst solve for the autarky equlbrum for country. Ths s easy especally f we consder the socal planner problem, but we wll compute the compettve equlbrum nstead. The Inada condtons for the consumer s utlty functon mply that both goods wll be produced n equlbrum. Thus, we ust have to take FOC for the consumer and look at cost mnmzaton for the frm. For the consumer we have max a 1 log c 1 + a 2 log c 2 s.t. p 1 c 1 + p 2 c 2 r k + w l 5

15 whch mples a 1 = λ p 1 c 1 2.1) a 2 = λ p 2 c 2 2.2) p 1 c 1 + p 2 c 2 = r k + w l 2.3) Ths gves p 2 c 2 = a 2 a 1 p 1 c ) The frm s cost mnmzaton problem mn r k ω + w lω ) s.t. y ω z ω k bω ω lω ) 1 bω mples the followng equaton, under the assumpton that both countres produce both goods, b ω 1 b ω ) l ωw = r k ω. 2.5) We can also use the goods market clearng to obtan ) c ω = z ω k bω ω lω) 1 bω = c ω = z ω lω bω w ) bω 1 b ω r. 2.6) Zero profts n equlbrum, p ωz ω k ω ) bω l ω ) 1 bω = r k ω + w l ω, combned wth 2.5), 6

16 gve us p ω = = b ω z ω r k ω r k ω ) 1 b ω ) bω w bω w b ω 1 b ω ) r w ) 1 b ω r ) b ω z ω 1 b ω ) 1 b ω b ω ) b ω We can also derve the labor used n each sector. From the consumer s FOCs, together wth the expressons for p ω and c ω derved above, we obtan: a ω = λ p ωc ω = 1 b ω ) a ω λ = w l ω ths mples that l1 1 b 2 ) a 2 = l2 1 b 1 ) a 1 We can use the labor market clearng condton and get l 2 + l 1 = l = l1 1 b 2 ) a 2 + l1 1 b 1 ) a = l = 1 l 1 = 1 b 1 ) a 1 1 b 2 ) a b 1 ) a 1 l. 2.7) The results are smlar for captal and thus, k 1 = b 1 a 1 b 1 a 1 + b 2 a 2 k, 7

17 Ths mples l k = r w ω 1 b ω ) a ω ω b ω a ω 2.8) Thus, n a labor abundant country captal s relatvely more expensve as we would expect. We can fnally use the goods market clearng condtons combned wth the optmal choces for l ω, k ω to get the values for c ω s as a functon solely of parameters and endowments, ) c bω a bω ) ω 1 ω = z ω k bω ) a 1 bω ω l. 2.9) ω b ω a ω ω 1 b ω ) a ω Free Trade Equlbrum In the two country example, free trade mples that the prce of each good s the same n both countres. Therefore, we wll denote free trade prces wthout a country superscrpt. In the two country case t s mportant to dstngush among three conceptually dfferent cases: n the frst case both countres produce both goods, n the second case one country produces both goods and the other produces only one good, and n the last case each country produces only one good. We frst defne the free trade equlbrum. A free trade equlbrum s a vector of allocatons for consumers ĉ ω,, ω = 1, 2 ), allocatons for the frm ˆk ω, ˆl ω,, ω = 1, 2), and prces ŵ ω, ˆr ω, ˆp ω,, ω = 1, 2 ) such that 1. Gven prces consumer s allocaton maxmzes her utlty for = 1, 2 2. Gven prces the allocatons of the frms solve the cost mnmzaton problem n 8

18 = 1, 2, ) b ω p ω z ω k bω 1 ω lω) 1 bω r, wth equalty f y ω > 0 ) 1 b ω ) p ω z ω k bω ω lω) bω w, wth equalty f y ω > 0 3. Markets clear ω ω ĉ ω = ŷ ω, ω = 1, 2 ˆk ω = k for each = 1, 2 ˆl ω = l for each = 1, No specalzaton We analyze the three cases separately. Frst, let s thnk of the case n whch both countres produce both goods. max a 1 log c 1 + a 2 log c 2 s.t. p 1 c 1 + p 2c 2 r k + w l a 1 = λ p 1 c ) a 2 = λ p 2 c ) p 1 c 1 + p 2c 2 = r k + w l 2.12) 9

19 Ths mples agan that p 2 c 2 = a 2 a 1 p 1 c ) When both countres produce both goods the frms cost mnmzaton problem mples the followng two equaltes, ) b ω p ω z ω k bω 1 ω lω) 1 bω = r, ) 1 b ω ) p ω z ω k bω ω lω) bω = w, whch n turn mply b ω l ω ) w 1 b ω ) r = k ω. 2.14) Addtonally, from zero profts, p ω = r ) b ω w ) 1 b ω z ω b ω ) b ω 1 b ω ) 1 b ω 2.15) and, of course, technologes by assumpton) and prces due to free trade) are the same n the two countres. Notce that the equalty 2.15) s true for = 1, 2 ths mples that r 1) b ω w 1) 1 b ω = r 2 ) b ω w 2 ) 1 b ω ω = 1, 2 ) r 1 bω ) w 2 1 bω = ω = 1, 2 r 2 w 1 Notcng that the above expresson holds for ω = 1, 2 and replacng these two equa- 10

20 tons n one another we have w 2 w 1 ) 1 b 2)b 1 b 2 1+b 1 = 1 = w 2 = w 1 and of course r 2 = r 1. Ths shows that we have factor prce equalzaton FPE) n the free trade equlbrum. From the cost mnmzaton of the frm we have b ω p ω z ω k ω ) bω l ω) 1 bω = r k ω = b ω p ω y ω = r k ω = p ω y ω = r k ω b ω. Summng up over and usng FPE we have ) p ω y ω = r b ω k ω ). 2.16) The equatons 2.10) and 2.11) mply a ω λ 1 + a ) ω λ 2 = p ω c 1 ω + c 2 ω 2.17) 11

21 Usng goods market clearng, c ω = y ω, we have b ω a ω a ) ω λ = p ω c 1 ω + c 2 ω 1 λ = r k ω = = p ω y ω = r b ω k ω = ) 1 λ ω b ω a ω = r ω k ω = 1 λ = r k1 + k 2) 2.18) ω b ω a ω and n a smlar manner 1 λ = w l1 + l 2). 2.19) ω 1 b ω ) a ω Usng 2.18) and 2.19) we can determne the w/r rato l 1 + l 2 k 1 + k = r ω 1 b ω ) a ω ) w ω b ω a ω Assumng that one country s more captal abundant than the other say k 1 / l 1 > k 2 / l 2 ), the equlbrum factor prce rato r/w under free trade les n between the autarky factor prces of the two countres determned n equaton 2.8). Usng the relatonshps for the captal labor rato 2.14) together wth the above expresson and factor market clearng condtons we can derve the equlbrum labor used from each country n each sector. Usng the captal labor ratos for both goods and for 12

22 both countres we get: w r [ l ) l2 b ) 1 1 b 1 ) + l2 ] b 2 = 1 b 2 ) k l2 = 1 b 2) 1 b 1 ) l b 2 b 1 l2 = 1 b 2) 1 b 1 ) l b 1 b 2 r k b ) 1 w l 1 b 1 ) b1 1 b 1 ) ω b ω a l1 ω + l 2 ) k ω 1 b ω ) a ω k 1 + k 2 l You may notce two thngs n ths expresson. Frst, f ntal endowments of the two countres are nsde a relatve range, there s dversfcaton snce l > 0. If the endowments of a country for a gven good are not n ths range, then a country specalzes n the other good ths range of endowments that mples dversfcaton n producton s commonly referred to as the cone of dversfcaton). Second, condtonal on dversfcaton labor abundant countres use relatvely more labor n the labor ntensve sector. What s the share of consumpton for each country? We can use the FOC from the consumer s problem to obtan p 1 c a ) 2 = r k + w l = a 1 c1 = r k + w l ) = p a 2 a 1 c 1 = 1 1 b ω ) w l p a 2 a 1 ) 2.21) where n the last equvalence we used equaton 2.14). Obtanng the rest of the allocatons and prces s straghtforward. In fact, you can show that f the producton functon exhbts CRS and the captal-labor rato for both countres s fxed n a gven sector), total 13

23 producton can be represented by 1 ) bω y ω = z ω k ω 1 bω lω). We can determne k ω, l ω by combnng expresson 2.18) wth 2.16), 2.17) and usng the market clearng condton. Ths gves b ω a ω ω b ω a ω = k ω k, 2.22) and smlarly for labor Specalzaton [HW] 1 Assume that k 1 /l 1 = k 2 /l 2. We only have to prove that gven ths assumpton A k 1 + k 2) b l 1 + l 2) 1 b = A k 1) b l 1) 1 b + A k 2) b l 2) 1 b. Usng repeatedly the condton we have that ) b ) b k 1 + k 2 k A l 1 + l 2 = l 1 k A l 1 l 1 + l ) b A l 2 l 2 l 1 + l 2 ) b ) b k 1 + k 2 k l 1 + l 2 = l 1 k 2 l 1) ) ) /l 2 + k 2 k l 1 + l 2 = l 1 k 2 l 2 = k1 l 1, whch holds by assumpton completng the proof. 14

24 2.1.5 The 4 bg theorems. In ths fnal secton for the H-O model we wll state man theorems that hold n the benchmark model wth two countres and two goods. Varants of these theorems hold under less or more restrctve assumptons. Our approach wll stll be as parsmonous as possble. 2 Theorem 1. Assume countres engage n free trade, there s no specalzaton thus there s dversfcaton) n equlbrum and there s no factor ntensty-reversal, then factor prces equalze across countres. Proof. See man text Theorem 2. Rybczynsk) Assume that the economes reman always ncompletely specalzed. An ncrease n the relatve endowment of a factor wll ncrease the rato of producton of the good that uses the factor ntensvely. 3 Proof. TBD Theorem 3. Stolper-Samuelson) Assume that the economes reman always ncompletely specalzed. An ncrease n the relatve prce of a good ncreases the real return to the factor used ntensvely n the producton of that good and reduces the real return to the other factor. Proof. TBD Theorem 4. Hekchser-Ohln) Each country wll produce the good whch uses ts abundant factor of producton more ntensvely. Proof. TBD 2 For a detaled treament you can look at the books of Feenstra 2003) and Bhagwat, Panagarya, and Srnvasan 1998). 3 If prces were fxed a stronger verson of the theorem can be proved. 15

25 Chapter 3 Models wth Constant Elastcty Demand Suppose there s a compact set S of countres. For now, we assume that S s dscrete, although havng a contnuum of countres does not change thngs much. Whenever t s possble, we refer to an orgn country as and a destnaton country as and order the subndces such that X s the blateral trade from to. We defne as X the total spendng of country. We further denote by L the populaton of country and let each consumer have a sngle labor unt that s nelastcally suppled. There are three common assumptons made about the market structure by trade theorsts. The frst s that markets n every country are perfectly compettve, so the prce of a good s smply equal to ts margnal cost. The second s that there s Bertrand competton so that the prce of a good depends on the margnal cost of the least cost producer as well, potentally, on the cost of the second cheapest producer. The thrd s that producton s monopolstcally compettve so that the frm does not perceve any mmedate compettor but t s affected by the overall level of competton. We wll consder each below. 16

26 We also assume throughout that labor s the only factor of producton we wll add ntermedate nputs later on). We also assume that there are ceberg trade costs {τ }, S. Ths means that n order from one unt of a good to arrve n destnaton, destnaton must shp τ unts. Iceberg trade costs are so called because a fracton τ 1 melts on ts way from to, much as f you were towng an ceberg. We almost always assume that τ 1 and usually assume that τ = 1 for all S,.e. trade wth oneself s costless. Furthermore, we sometmes assume that the followng trangle nequalty holds: for all,, k S: τ τ k τ k. The trangle nequalty says that t s never cheaper to shp a good va an ntermedate locaton rather than sell drectly to a destnaton. 3.1 Constant Elastcty Demand We frst ntroduce one of the most remarkably smple as well as versatle demand functons that wll be the bass of our analyss for the next two chapters, the Constant Elastcty of Substtuton CES) demand functon. Why do we do so? CES preferences have a number of attractve propertes: 1) they are homothetc; 2) they nest a number of specal demand systems e.g. Cobb-Douglas); and 3) they are extremely tractable. Trade economsts often do not beleve that CES preferences are a good representaton of actual preferences but are seduced nto makng frequent use of them due to ther analytcal convenence. In partcular, assume that the representatve consumer n country derves utlty U from a set of varetes Ω the consumpton of goods shpped from all countres S: U = ) σ σ 1 a ω) σ 1 q ω) σ 1 σ, 3.1) ω Ω where σ 0 s the elastcty of substtuton and a ω) s an exogenous preference shfter. 17

27 A couple of thngs to note: frst, q ω) s the quantty of a good shpped from that arrves n the amount shpped s τ q ω)); second, the fact that there s a representatve consumer s not partcularly mportant: we can always assume that workers wth dentcal preferences) are the ones consumng the goods and U can be nterpreted as the total welfare of country n ths case homothetcty s, of course, crucal for ths property to be true). We now solve the representatve consumer s utlty maxmzaton problem. Gven the mportance of CES n the class we wll proceed to do the full dervaton for any gven good ω Ω. Let the spendng of country be denoted X and let the prce of a good net of trade costs) from country n country be p. Then the utlty maxmzaton problem s: max {q ω)} ω Ω ) σ a ω) σ 1 q ω) σ 1 σ 1 σ Ω s.t. q ω) p ω) X, ω Ω where I gnore the constrant that q ω) > 0 why s ths okay?). The Lagrangan s: L : ) σ ) a ω) σ 1 q ω) σ 1 σ 1 σ λ q ω) p ω) X ω Ω ω Ω Frst order condtons FOCs) mply: L q ω) = 0 ) 1 a ω) σ 1 q ω) σ 1 σ 1 σ a ω) σ 1 q ω) σ 1 ω Ω = λp ω) L λ = 0 X = q ω) p ω) ω Ω 18

28 Usng the FOCs for two dfferent goods, ω and ω : a ω) a ω ) = pσ ω) q ω) p σ ω ) q ω ) Rearrangng and multplyng both sdes by p ω) yelds: q ω ) p ω ) = 1 a ω) q ω) p ω) σ ω) a ω ) p 1 σ ω ) Summng over all ω Ω yelds: q ω ) p ω ) = 1 a ω Ω ω) q ω) p ω) σ a ω ) p ω ) 1 σ ω Ω X = 1 a ω) q ω) p ω) σ P 1 σ where the last lne used the second FOC and P ω Ω a ω ) p ω ) 1 σ) 1 1 σ s known as the Dxt-Stgltz prce ndex. It s easy to show that U = X P,.e. dvdng ncome by the prce ndex gves the total welfare of country. Rearrangng the last lne yelds the CES demand functon: q ω) = a ω) p σ ω) X P σ 1, 3.2) Equaton 3.2) mples that the quantty consumed n of a good produced n wll be ncreasng wth s preference for the good a ), decreasng wth the prce of the good p ), ncreasng wth s spendng X ), and ncreasng wth s prce ndex. Note that the value of total trade s smply equal to the prce tmes quantty. In what follows, let us denote the value of trade of good ω from country to country as X ω) 19

29 p ω) q ω). Then we have: X ω) = a ω) p 1 σ ω) X P σ ) The only thng left to construct blateral trade s to solve for the optmal prce and aggregate across varetes, whch we wll do after a bref dscusson of the gravty equaton. 3.2 The gravty setup It s helpful to provde a bref motvaton of why we are nterested n wrtng down a flexble model n the frst place. Classcal trade theores Rcardo, Heckscher-Ohln), whle extremely useful n hghlghtng the economc forces behnd trade, are very dffcult to generalze to a set-up wth many tradng partners and blateral trade costs. Because the real world clearly has both of these, the classcal theores do not provde much gudance n dong emprcal work. Because of ths dffculty, those dong emprcal work n trade began usng a statstcal.e. a-theoretc) model known as the gravty equaton due to ts smlarty the Newton s law of gravtaton. The gravty equaton states that total trade flows from country to country, X, are proportonal to the product of the orgn country s GDP Y and destnaton country s GDP Y and nversely proportonal to the dstance between the two countres, D : 1 X = α Y Y D. 3.4) For a varety of reasons whch we wll go nto later on n the course), ths gravty equaton s often estmated n a more general form, whch we refer to as the generalzed gravty 1 Ths s actually n contrast to Newton s law of gravtaton, where the force of gravty s nversely proportonal to the square of the dstance. 20

30 equaton: X = K γ δ, 3.5) where K s a measure of the resstance of trade between and, γ measures the orgn sze and δ measures the destnaton sze note that each country has two dfferent measures of sze).. The gravty equaton 3.4) and ts generalzaton 3.5) have proven to be enormously successful at explanng a large fracton of the varaton n observed blateral trade flows; ndeed, t s probably not too much of an exaggeraton to say that the gravty equaton s one of the most successful emprcal relatonshps n all of economcs. Because t was orgnally proposed as a statstcal relatonshp, however, the absence of a theory ustfyng the relatonshp made t very dffcult to ask any meanngful counterfactual questons; e.g. what would happen to trade between and f the tarff was lowered between and k? 3.3 Armngton model The Armngton model Armngton, 1969) s based on the premse that each country produces a dfferent good and consumers would lke to consume at least some of each country s goods. Ths assumpton s of course ad hoc, and t completely gnores the classcal trade forces such as ncreased specalzaton due to comparatve advantage. However, as we wll see, the model when combned wth Constant Elastcty of Substtuton CES) preferences as n Anderson, 1979)) provdes a nce characterzaton of trade flows between many countres. 2 The Armngton model as formulated by Anderson, 1979)) was mportant because 2 Actually, n the man text, Anderson 1979) consders Cobb-Douglass preferences and wrtes that there s lttle pont n the exercse of generalzng to CES preferences, dong so only n an appendx. Despte hs reluctance to do so, the paper has been cted thousands of tmes as the example of an Armngton model wth CES preferences. 21

31 t provded the frst theoretcal foundaton for the gravty relatonshp. It s also a great place to start our course, as one of the great surprses of the nternatonal trade lterature over the past ffteen years has been how robust the results frst present n the Armngton model are across dfferent quanttatve trade models. By now, as we wll dscuss n ths chapter, models that yeld the gravty relatonshp 3.5) are ubqutous and much of the rest of what follows wll focus on analyzng ther common propertes The model We now turn to the detals of the Armngton models and n partcular to the supply sde of ths model, gven CES demand. The Armngton assumpton s that each country S produces a dstnct varety of a good. Because countres map one-to-one to varetes, we ndex the varetes by ther country names ths wll not be true for Bertrand and monopolstc competton when we have to keep track both of varetes and countres). Suppose that the market for each country/good s perfectly compettve, so that the prce of a good s smply the margnal cost. Suppose each worker can produce A unts of her country s good and let w be the wage of a worker. Then the margnal cost of producton s smply w A. Ths mples that the prce at the factory door.e. wthout shppng costs) s p = w A. What about wth trade costs? Recall that wth the ceberg formulaton, τ 1 unts have to be shpped n order for one unt to arrve. Ths means that τ 1 unts have to be produced n country n order for one unt to be consumed n country. Hence the prce n country of consumng one unt from country s: p = τ w A. 3.6) 22

32 Note that ths mples that: p p = τ, 3.7).e. the rato of the prce n any destnaton relatve to the prce at the factory door s smply equal to the ceberg trade cost. Equaton 3.7) s called a no-arbtrage equaton, as t means that there s no way for an ndvdual to proft by buyng a good n country and sell n country or vce versa). Note, however, that there may stll be proftable tradng opportuntes between trplets of countres even f equaton 3.7) holds when the trangle nequalty s not satsfed Gravty Assumng that each country produces a dfferent good ω, and substtutng equaton 3.6) nto equaton 3.3) yelds a gravty equaton for blateral trade flows: X = a τ 1 σ w A ) 1 σ X P σ ) To the extent that trade costs are ncreasng n dstance, the value of blateral trade flows wll declne as long as σ > 1. We can actually use equaton to get a lttle close to the true gravty equaton. The total ncome n a country s equal to ts total sales: Y = X = a τ 1 σ w A ) 1 σ X P σ 1 w A ) 1 σ = Y / a τ 1 σ X P σ 1 23

33 Replacng ths expresson n the equaton 3.8) yelds: where Π 1 σ foreshadowng here]. X = a τ 1 σ Y ) X Π 1 σ P 1 σ ), 3.9) a τ 1 σ X P σ 1 bears a strkng resemblance to the prce ndex [nsert Equaton 3.9) whch shows that the blateral trade spendng s related to the product of the GDPs of the two countres gravty!!), the dstance/tradecost and a GE component. Equaton 3.9) s actually about as close as we wll ever get to the orgnal gravty equaton. Ths s because all of our theores say that blateral trade flows depend on more than ust the blateral trade costs and the ncomes of the exporter and mporter; what also matters s so-called blateral resstance : ntutvely, the greater the cost of exportng n general, the smaller the Π 1 σ ; conversely, the greater the cost of mportng n general, the smaller the P 1 σ. Ths means that trade between any two countres depends not only on the ncomes of those two countres but also the cost of tradng between those countres relatve to tradng wth all other countres. Ths pont was made n the enormously famous and nfluental paper Gravty wth Gravtas: A Soluton of the Border Puzzle Anderson and Van Wncoop, 2003) Welfare We wll now show that welfare n relatonshp to trade s gven by a smple equaton nvolvng the trade to GDP rato and parameters of the model but no other equlbrum varables). We wll be revstng ths relatonshp multple tmes n these notes. To begn 24

34 defne λ as the fracton of expendture n spent on goods arrvng from locaton : λ X k X k. From equaton 3.8) we have: λ = a τ 1 σ k a k τ 1 σ k ) 1 σ w A ) 1 σ wk A k λ = a τ 1 σ A σ 1 w P ) 1 σ 3.10) snce P 1 σ k a k τ 1 σ k wk A k ) 1 σ. Remember from the CES dervatons above that the utlty of the representatve agent s the real wage,.e. W = X P. Assume that τ = 1. Then by choosng =, equaton 3.10) mples that welfare can be wrtten as: U = X λ 1 w 1 σ σ 1 a 1 A, 3.11).e. welfare depends only on changes n the trade to GDP rato, λ, wth an elastcty of 1/ σ 1) whch s the nverse of the trade elastcty. 3.4 Monopolstc Competton wth Homogeneous Frms and CES demand Wth the Armngton model, we saw how we could ustfy the gravty relatonshp n trade usng the ad-hoc assumpton that every country produces a unque good as well as the assumpton that consumers have a love of varety.e. they want to consume at least a lttle bt of every one of the goods). In ths secton, we wll dspense of the frst assumpton 25

35 by ntroducng frms nto the model. However, we wll contnue to rely heavly on the second assumpton by assumng that each frm produces a unque varety and consumers would lke to consume at least a lttle bt of every varety. The model consdered today was ntroduced by Krugman 1980) and was an mportant part of the reason he won a Nobel prze. A key feature of the Krugman 1980) model s that there are ncreasng returns to scale,.e. the average cost of producton s lower the more that s produced. All else equal, ths wll lead to gans from trade, snce by takng advantage of demand from multple countres, frms can lower ther average costs. To succnctly model the ncreasng returns from scale, we suppose that a frm has to ncur a fxed entry cost f e n order to produce. The e mght seem lke unnecessary notaton; however, we keep t here because n future models there wll be both an entry cost and a fxed cost of servng a partcular destnaton). We assume the fxed cost of entry lke the margnal cost) s pad to domestc workers so that f e s the number of workers employed n the entry sector thnk of them as the workers who buld the frm). Some of the results toward the end of ths secton are based on the subsequent analyss of Arkolaks, Demdova, Klenow, and Rodríguez-Clare 2008) Setup The man departure from the perfect competton paradgm s that n monopolstc competton each dfferentated varety s produced potentally) by a dfferent frm, where there s a measure N of frms n country. Ths number of frms s determned n equlbrum by allowng frms to enter after ncurrng a fxed cost of entry n terms of domestc labor, f e. 26

36 3.4.2 Consumers As n the Armngton model, we assume that consumers have CES preferences over varetes. Hence a representatve consumer n country S gets utlty U from the consumpton of goods shpped by all other frms n all other countres, where: U = S ˆ q ω) σ 1 σ Ω dω ) σ σ 1, 3.12) where q ω) s the quantty consumed n country of varety ω. Note that for smplcty, I no longer nclude a preference shfter although one could easly be ncorporated) so that consumers treat all frms n all countres equally. The consumer s utlty maxmzaton problem s very smlar to the Armngton model whch shouldn t be partcularly surprsng, gven preferences are vrtually the same). In partcular, a consumer n country S optmal quantty demanded of good ω Ω s: q ω) = p ω) σ X P σ 1, where: P S ˆ Ω p ω) 1 σ dω ) 1 1 σ 3.13) s the Dxt-Stgltz prce ndex. The amount spent on varety ω s smply the product of the quantty and the prce: x ω) = p ω) 1 σ X P σ ) Note that we derved a very smlar expresson n the Armngton model, from whch the gravty equaton followed almost mmedately. In ths model however, ths s the amount spent on the goods from a partcular frm, so we now need to aggregate across all frms n 27

37 country to determne blateral trade flows between and,.e.: Frms ˆ X x ω) dω = X P σ 1 Ω ˆ Ω p ω) 1 σ dω. 3.15) All frms n country have a common productvty, z, and produce one unt of the good usng 1 z unts of labor. The optmzaton problem faced by a frm ω from country s: ) τ max p ω) q ω) w q ω) w f e s.t. q ω) = p ω) σ X P σ 1 {p ω)} z S S We can substtute the constrant nto the maxmand and wrte the equvalent unconstraned problem of choosng the prce to sell to each locaton as: max p 1 σ ω) X P σ 1 τ w p σ {p ω)} z S S ) ω) X P σ 1 w f e Note that the constant margnal cost assumpton mples that the country can treat each destnaton as a separate optmzaton problem ths wll come n helpful n models we wll see later on). 3 Proft maxmzaton mples that optmal prcng for a frm sellng from country to country s p z ) = σ τ w, 3.16) σ 1 z 3 Notce that here we haven t ntroduced fxed costs of exportng. Introducng these costs wll change the analyss n that we may have countres for whch all the frms chose not to export dependng on values of the fxed costs and other varables. More extreme predctons can be delvered f the producton cost f s only a cost to produce domestcally and ndependent of the exportng cost. However, n order to create a true extensve margn of frms.e. more frms exportng when trade costs decrease) requres heterogenety ether n the productvtes of frms as we wll do later on n the notes) or n the fxed costs of sellng to a market see Romer 1994)). 28

38 and snce all frm decson wll depend on parameter s and frm productvty we drop the ω notaton from here on. We wll make the notaton a bt cumbersome by carryng around the z s n order to allow for drect comparson of our results wth the heterogeneous frms example that wll be studed later on Gravty Because every frm s chargng the same prce, we can substtute the prce equaton 3.16) nto the gravty equaton 3.15) to yeld: ˆ X = X P σ 1 X = Ω ) σ 1 σ τ 1 σ σ 1 ) σ w 1 σ τ σ dω 1 z w z ) 1 σ N X P σ ) where N Ω dω s the measure of frms producng n country. Comparng ths equaton to the one derved for the Armngton model wth monopolstc, we see that the two expressons are nearly dentcal - the only dfference here s that we have to keep track of the mass of frms M and all trade flows are smaller f σ > 1) as a result of the markups Welfare It turns out welfare can be wrtten smlarly to the Armngton model. Frst, note that substtutng equaton 3.16) for the equlbrum prce charged nto the prce ndex equaton 3.13) yelds: P 1 σ σ 1 σ σ 1) k τ 1 σ k wk z k ) 1 σ N k. As above, defne λ X k X k to be the fracton of expendture of country on goods sent from country. Then usng equaton 3.17), we can wrte λ as a functon of the prce 29

39 ndex n : λ = σ ) ) 1 σ 1 σ σ 1 τ 1 σ w z N X P σ 1 σ ) ) 1 σ 1 σ k σ 1 τ 1 σ wk z Nk k X P σ 1 k = τ1 σ = k τ 1 σ k Ths equaton can be rewrtten as ) 1 σ w z N ) 1 σ wk z Nk k ) 1 σ w z N σ ) 1 σ τ 1 σ σ 1 P = P 1 σ ) ) σ w τ σ 1 z N 1 1 σ σ 1 λ ). 3.19) Snce equaton 3.19) holds for any and, we can focus on the partcular case where =. Then assumng τ = 1, we can wrte equaton 3.19) as: ) ) σ w P = σ 1 z ) w σ 1 σ 1 = z N 1 P σ N 1 1 σ 1 σ λ 1 σ 1 λ 1, 3.20).e. the real wage s declnng n λ or equvalently, ncreasng n trade openness. Note, however, that unlke the Armngton model, frms are makng postve profts, so that the real wage no longer captures the welfare of a locaton. To deal wth ths ssue, we ntroduce a free entry condton. 30

40 3.5 Rcardan model The Rcardan model s a model of perfect competton where countres produce the same goods usng dfferent technologes. The Rcardan model predcts that countres may specalze n the producton of certan ranges of goods The two goods case We consder the smple verson of the model wth two countres and two goods. In order to get as much ntuton as possble we wll frst consder the case where both countres specalze n the producton of one good. The producton technologes n the two countres = 1, 2 are dfferent for the two goods ω = 1, 2 and gven by y ω) = z ω) l ω),, ω = 1, 2. Assume that country 1 has absolute advantage n the producton of both goods z 2 1) < z 1 1), z 2 2) < z 1 2). Assume that country 1 has comparatve advantage n the producton of good 1 and country 2 n good 2 z 1 2) z 2 2) < z1 1) z 2 1). 3.21) 31

41 Assume Cobb-Douglas preferences. The consumer s problem s max a 1) log c 1) + a 2) log c 2) s.t. p 1) c 1) + p 2) c 2) w l. Consumer optmzaton mples that p 2) c 2) = a 2) a 1) p 1) c 1) 3.22) p 1) c 1) + p 2) c 2) = w l 3.23) Autarky Usng frms cost mnmzaton and the Inada condtons that ensure that the consumer actually wants to consume both goods) from the consumer problem we drectly obtan that p 1) z 1) = w = p 2) z 2). Usng the goods market clearng c ω) = y ω) for ω = 1, 2, together wth labor market clearng l ω) = a ω) l, we get labor allocated to each good. Usng the producton functon and goods market 32

42 clearng we can obtan the rest of the allocatons Free trade Under free trade nternatonal prces equalze. Relatve productvty patterns wll determne specalzaton. There can be three possble specalzaton patterns, two where one country specalzes and the other dversfes and one where both countres specalze. Proposton 1. [Specalzaton]Under the assumptons stated, at least one country specalzes n the free trade equlbrum. Proof. If not then the frm s cost mnmzaton together wth the consumer FOCs would mply z 1 2) z 2 2) = z1 1) z 2 1), a contradcton. In the three dfferent equlbra that can emerge the countres export what they have comparatve advantage on specalzaton nto exportng). Under free trade ths relatve prce has to be n the range gven the Inada condtons n consumpton): z 1 2) z 2 2) p 2) p 1) z1 1) z 2 1) To consder an example of how the wages are determned notce that for the country that s under ncomplete specalzaton equatons cost mnmzaton mples p 1) z 1) p 2) z 2) = w w = p 1) p 2) = z 2) z 1),.e. ths country sets the relatve prce of the two goods. Now assume that country 1 s 33

43 ncompletely specalzed whch means that country 2 specalzes n good 2 and normalze w 1 = 1. Because of free trade and perfect competton t must be the case that the cost of producng good 2 n both countres s the same,.e. w 1 z 1 2) = w2 z 2 2) = w2 = z2 2) z 1 2) < 1 = w1. Notce that usng the wages and the zero proft condtons for country 1 we now get p 1) z 1 1) = 1 and p 2) z 1 2) = 1 z 1 1) z 1 2) = p 2) p 1). Fnally usng the budget constrants of the ndvdual we can determne the levels of consumpton and verfy that the equlbrum s consstent wth our ntal assumpton for the patterns of specalzaton.e. ndeed country 2 exports good 2 and country 1 exports good 1) 3.6 Problem Sets 1. Dxt-Stgltz Preferences. Suppose that a consumer has wealth W, consumes from a set of dfferentated varetes ω Ω, and solves the followng CES maxmzaton problem: ˆ ) σ max U = q ω) σ 1 σ 1 σ dω {qω)} Ω s.t. ˆ Ω p ω) q ω) dω X, 3.24) where σ > 0, q ω) s the quantty consumed of varety ω and p ω) the prce of varety ω. 34

44 a) Fnd a prce ndex P such that n equlbrum U = W P. b) Derve the optmal q ω) as a functon of W, P and p ω). c) Show that σ s the elastcty of substtuton,.e. for any ω, ω Ω, σ = ) qω) d ln qω ) U/ qω d ln ) U/ qω) ). d) What happens as σ? σ 1? σ 0? 35

45 Chapter 4 Models wth CES demand and producton heterogenety The purpose of ths chapter s to develop a general model for producton heterogenety n whch dfferent assumptons on technology and competton wll gve us dfferent workhorse frameworks mportant for the quanttatve analyss of trade. Our analyss of the general framework s based on the exposton of Eaton and Kortum 2011)) and earler results of Kortum 1997)) and Eaton and Kortum 2002)). We start wth a smple extenson of the Rcardan model wth ntra-sector heterogenety. 4.1 Introducton to heterogenety: The Rcardan model wth a contnuum of goods The model of Dornbusch, Fscher, and Samuelson 1977) s based on the Rcardan model where trade and specalzaton patterns are determned by dfferent productvtes. 1 There 1 The notes n ths chapter are partally based on Eaton and Kortum 2011). 36

46 s absolute advantage due to hgher productvty n producng certan goods, but also comparatve advantage due to lower opportunty cost of producng some goods. The man drawback of the smple Rcardan model, smlar to that of the Heckscher-Ohln model, s n the complexty of solvng for the patterns of specalzaton for a large number of ndustres. Breakthrough: Dornbusch, Fscher, and Samuelson 1977) used a contnuum of sectors. The characterzaton of the equlbrum ended up beng very easy. Perfect competton 2 countres H, F) Contnuum of goods ω [0, 1] CRS technology labor only) Cobb-Douglas Preferences wth equal share n each good Iceberg trade costs τ HF, τ FH We normalze the domestc wage to 1. We want to characterze the set of goods produced and exported from each country. Wthout loss of generalty we wll characterze producton and exportng for country F. We frst need to compare the prce of a good ω potentally offered by country H to country F to the correspondng prce of the good produced by F n order to determne the set of goods produced by country F n equlbrum. For ths purpose, we wll order the goods n a decreasng order of domestc to foregn productvty and defne ω as the good wth the lowest productvty produced n the foregn country. Thus, the foregn country produces goods [ω, 1] whle the domestc [0, ω]. When trade costs exst then the two sets wll overlap, ω > ω, but f τ HF = τ FH then ω = ω. A smple condton that determnes whch are the goods that wll be produced by 37

47 country F dctates that the prce of these goods n country F has to be lower than the prce of mported goods,.e. w F z F ω) < τ HF z H ω) = A ω) z H ω) z F ω) > τ HF. w F Therefore, we can defne A ω) = w F τ HF. 4.1) whch determnes that only products that wll be produced by country F. To fnd the set of goods that F wll be exportng we need to determne set of goods produced by country H. Usng a smlar logc ths smply entals fndng the ω that satsfes A ω) = τ FH w F 4.2) and [0, ω] s the set of goods produced by the home country. Thus, F produces goods [ω, 1] and exports [ ω, 1] snce the domestc does not produce any of the goods n that last set. In order to get sensble relatonshps from the model, DFS parametrze z Fω) by usng z H ω) a monotonc functon. In ths last case we can nvert A and get the exact range of goods produced by each country,.e. effectvely determne ω and ω as a functon of parameters and w F. Subsequently, we can solve for the equlbrum wage, usng the labor market clearng L H = ω w F ) w F L F + ω w F ) L H Where DFS stop and EK start Eaton and Kortum 2002) henceforth EK) treat productvtes z ω) as an ndependent realzaton of a random varable Z ndependently dstrbuted accordng to the same ds- 38

48 trbuton F for each good ω n country. Gven the contnuum of goods usng a LLN argument) we can determne wth certanty the fracton of goods produced by each country. Ths way EK are able to overcome the complcatons faced by the standard Rcardan framework and go much further n developng an analytcal quanttatve trade framework. Assume that the random varable Z follows the Frechet dstrbuton 2 : [ Pr Z z) = exp A z θ]. The parameter A > 0 governs country s overall level of effcency absolute advantage) wth more productve countres havng hgher A s). The parameter θ > 1 governs varaton n productvty across dfferent goods comparatve advantage) hgher θ less dspersed). Now we wll splt the [0, 1] nterval by thnkng of ω as the probablty that the relatve productvty of F to H s less than Ã, where à can ether be defned by 4.2). Therefore, n order to determne ω whch s defned as the share of goods that the domestc country produces we smply compute the probablty that the domestc country s the cheapest provder of the good across all the range of productvtes. For example usng 4.2) for the 2 See the appendx for the propertes of the Frechet dstrbuton and the next chapter for a dervaton from frst prncples. 39

49 defnton of à we can derve ω = λ HH [ ] zf = Pr à z H = Pr [ z F Ãz H ] = = = ˆ + 0 ˆ + 0 Ãz ) ] θ exp [ A F }{{} exp A H A H + A F à θ Przω) à zh ω)) df H z) }{{} densty of z H ω) Ãz ) ] [ θ [ A F θa H z) θ 1 exp A H z) θ] dz Country H s spendng 1 ω) w H L H on mports gven Cobb-Douglas) whch mples X FH = A F w F τ HF ) θ A H + A F w F τ HF ) θ w HL H Notce that ths relatonshp s smlar to the one derved wth the assumpton of the Armngton aggregator, equaton 3.10), but wth an exponent θ for the elastcty of trade wth respect to trade costs. A lower value of θ generates more heterogenety. Ths means that the comparatve advantage exerts a stronger force for trade aganst resstance mpose by the geographc barrer τ n. In other words wth low θ there are many outlers that overcome dfferences n geographc barrers and prces overall) so that changes n w s and τ s are not so mportant for determnng trade. 40

50 4.2 A theory of technology startng from frst prncples We start wth a very general technologcal framework under the followng assumptons. Tme s contnuous and there s a contnuum of goods wth measure µ Ω). Ideas for good ω ways to produce the same good wth dfferent effcency) arrve at locaton at date t at a Posson rate wth ntensty ār ω, t) where we thnk of ā as research productvty and R as research effort. The qualty of deas s a realzaton from a random varable Q drawn ndependently from a Pareto dstrbuton wth θ > 1, so that Pr [Q > q] = q/q ) θ, q q where s a lower bound of productvtes. Note that the probablty of an dea beng bgger than q condtonal on deas beng bgger than a threshold, s also Pareto see appendx q for the propertes of the Pareto dstrbuton). The above assumptons together mply that the arrval rate of an dea of effcency Q q s ) θ ār ω, t). q/q normalze ths wth q 0, ā + such that āq θ 1 n order to consder all the deas n 0, + )). We also assume that there s no forgettng of deas. Thus, we can summarze the hstory of deas for good ω by A ω, t) = ˆ t R ω, τ) dτ. The number of deas wth effcency Q > q s therefore dstrbuted Posson wth a parameter A ω, t) q ) θ usng the prevous normalzaton). 41