Notes on the Theoretical Gravity Model of. International Trade

Transcription

1 Notes on the Theoretcal Gravty Model of Internatonal Trade Ben Shepherd Nehaus Center, Prnceton Unversty & GEM, Scences Po Ths Verson Dated: November 25, 2008 Abstract I derve n detal the verson of the gravty model of trade due to Anderson and Van Wncoop (2003, 2004), whch has become the de facto standard n emprcal wor. In partcular, I emphasze the model s mcro-foundatons n terms of Dxt-Stgltz preferences and some basc macroeconomc denttes. I brefly dscuss the mplcatons of the mcro-founded gravty model s functonal form for estmaton and counter-factual smulatons n real-world settngs, and address some partcular ssues that arse when usng sectorally dsaggregated trade data. These notes are ntended as bacground for partcpants at ARTNeT s 2008 Capacty Buldng Worshop for Trade Research: Behnd the Border Gravty Modelng (Bango, December 15-19, 2008). They are amed at appled/emprcal researchers, and thus present the theory n way desgned to hghlght ts mplcatons for estmaton, nference, and smulaton. The materal draws heavly on the many helpful comments I have receved on varous appled gravty papers over the last few years. I am partcularly grateful to colleagues n the Development Research Group at the World Ban, and n the OECD s Trade and Agrculture Drectorate. Please send comments, suggestons, and correctons to: bshepher@prnceton.edu, and bashepherd@gmal.com. 1

2 1 Introducton Although t started lfe based on lttle more than common sense and a nce analogy, the gravty model has come a long way n recent years. There are now a number of gravty-le models that can be derved from sold mcro-foundatons. One of the most commonly used n emprcal wor s the model due to Anderson and Van Wncoop (2003, 2004) ( AvW ). Ths note sets out n some detal the AvW gravty model, emphaszng ts theoretcal underpnnngs and the role they play n drvng estmaton, nference, and smulaton n appled wor. One of the more appealng features of the AvW model s that t s relatvely easy to derve. There are bascally four steps. Frst, we need to develop n detal the model s consumpton sde (Secton 2). It reles on Dxt-Stgltz love of varety preferences, and we can use that lterature to state and solve the consumer s problem n order to produce a set of drect demand functons. Next, we address the model s producton sde (Secton 3). A very large number of symmetrc frms n each country engage n monopolstc competton. Snce there are so many frms, strategc nteractons dsappear and frms adopt a smple marup prcng rule. The thrd step (Secton 4) s to specfy a prce pass-through equaton, whch relates domestc and foregn prces. Ths can be done smply by ntroducng standard ceberg varable trade costs, an to an ad valorem tarff. Fnally (Secton 5), we mpose equalty of sectoral ncomes and expendtures, aggregate demand by sector, and attac the model wth some addtonal algebra, n order to produce the AvW gravty equaton. The paper then concludes wth a bref dscusson of some of the most mportant mplcatons of the AvW model for appled trade polcy wor. It s mportant to emphasze that although the AvW model s very commonly used and s treated by many researchers as a standard or baselne gravty model, t s by no means the only theoretcally grounded gravty model. What ths means, though, s that f you want to nclude a varable that AvW suggests s not relevant (le per capta GDP), or f you want to set up fxed effects n a way that does not st easly wth the AvW formulaton, then you need to justfy your approach explctly n terms of some alternatve theoretcal assumptons. 2

3 2 Consumpton Sde Fgurng out the consumpton sde of the AvW model means statng the consumer s problem maxmzng utlty subject to a budget constrant then solvng t to produce a set of drect demand functons. In ths model, the consumer s preferences tae the Dxt-Stgltz love of varety form. Each sector conssts of a large number of dfferentated product varetes, whch are substtutable n consumpton wth a gven (constant) elastcty. The form taen by the utlty functon s qute standard n the post-krugman trade lterature. Consder a world of C countres ndexed by. I assume from the start that countres can trade wth each other, and thus that consumers n one country can potentally purchase varetes from any other country. For the moment, trade s costless. (We ll get to costly trade later on.) Consumers are dentcal n each country, and maxmze CES utlty across a contnuum of varetes (ndex v) n K sectors (ndexed by ) wth the followng form: U = K {ˆ } 1 ] 1 x 1 1 σ σ 1 =1 v V (v) dv (1) The set V defnes the range of varetes that s consumed n country. As usual, I use x (v) to ndcate the quantty of varety v from sector consumed n country, and p (v) to ndcate ts unt prce. I use functon notaton because of the contnuum of varetes. In the verson of the model wth a dscrete number of varetes, v becomes a subscrpt, and the ntegrals are replaced wth sums. I am usng the contnuum verson of Dxt-Stgltz for two reasons. Frst, strategc nteracton on the producton sde only truly dsappears wth an uncountably large number of frms. Second, more recent versons of gravty such as HMR use contnuum notaton, and I would le to eep as close as possble to ther formulaton so that comparsons are made easer. The utlty functon s smply the sum of the sectoral sub-utltes, each of whch s weghted equally. That restrcton can easly be relaxed by aggregatng the sectoral sub-utltes va a Cobb-Douglas 3

4 utlty functon, and allowng for dfferent weghts. So long as the shares are exogenous to the model, however, the basc results stay the same. See Chaney (2008) for an example of what the alternatve expressons loo le. AvW and HMR consder, n effect, a sngle sector so as to avod clutterng up the algebra wth addtonal ndces. But nothng turns on ths, and n the present case t s useful to retan some sectoral dsaggregaton so that we can examne a couple of mportant data mplcatons that flow from the model n a mult-sector context. The budget constrant n country s E = K {ˆ } p =1 v V (v)x K (v)dv E (2) =1 where E s total expendture n that country, and E s country s total expendture n sector. The consumer s problem s to choose x (v) for all v so as to maxmze (1) subject to (2). The Lagrangan s: L = K {ˆ } 1 ] 1 x 1 1 σ σ 1 K {ˆ } =1 v V (v) dv λ p =1 v V (v)x (v)dv (3) Tang the frst order condton wth respect to quantty and settng t equal to zero gves: {ˆ L x (v) = 1 ] 1 x 1 σ 1 σ 1 v V (v) dv } 1 1 σ 1 1 ( 1 1 σ ) x (v) ] 1 σ λ p (v) = 0 (4) { Defne X = v V x (v) ] 1 1 σ dv } σ, regroup terms, and rearrange to get: v V x (v) ] 1 σ x (v) ] 1 1 σ dv X = λ p (v) (5) Now rearrange agan, multply through by prces, aggregate over all varetes n a gven sector, and 4

5 then solve for the Lagrangan multpler: ˆ v V ] {ˆ } x (v) p (v) = λ σ p 1 σ (v) (X ) σ ] σ 1 x 1 σ v V (v) dv (6) {ˆ x (v) p (v)dv E = λ σ (X ) σ λ = v V v V p (v) ] 1 σ dv E } ] σ 1 x 1 ˆ σ (v) dv p 1 σ v V (v)] dv (7) 1 σ v V X x (v) ] 1 1 σ dv To get the drect demand functon, substtute ths expresson for the multpler bac nto the frst order condton (5): (8) v V x (v) ] 1 σ x (v) ] 1 1 σ dv X = v V p (v) ] 1 σ dv E 1 σ v V X x (v) ] 1 1 σ dv p (v) (9) x (v) = v V p (v) ] σ p (v) ] 1 σ dv E (10) In the lterature these days, t s common to defne the deal CES prce ndex for sector n { country as P = p (v) ] 1 1 σ 1 σ dv}. So the demand functon then taes the slghtly less ntmdatng form of: v V { p x (v) = (v) P } σ E P (11) Compare (11) wth equatons (1) and (2) n Helpman et al. (2008), or equatons (5.24) and (5.25) n Chapter 5 of Feenstra s textboo. 5

6 3 Producton Sde The producer s problem n ths model s to maxmze proft. Assumng a contnuum of frms,.e. an uncountably large number of them, maes ths problem much easer to solve. It turns out that strategc nteractons dsappear, and frms charge a constant marup. In terms of the overall model, ths secton gves us an equlbrum prcng equaton whch, wth the equlbrum demand equaton derved n the prevous secton, s just about all we need to generate gravty. Each country has a measure N of actve frms n sector. Each frm maes a unque product, so the total (worldwde) measure of products n each sector s C =1 N. In order to produce one unt of ts product, a frm must pay a fxed cost f and a varable cost a. A typcal frm s proft functon s therefore: π (v) = p (v)x (v) wa x (v) w f (12) Wth a contnuum of varetes, t does not matter at ths pont whether we assume Bertrand (prce) or Cournot (quantty) competton. If the frms play Bertrand, the frst order condton s: π (v) p (v) = x (v) + p (v) x (v) p (v) x wa (v) p = 0 (13) (v) and t can usefully be solved for prces by movng the frst and thrd terms to the rght hand sde, then dvdng through by the partal dervatve p (v) = wa x (v) x (v) p (v) (14) In order to do somethng wth that expresson, we need to now more about the x (v) term. The p (v) partal can be calculated by dfferentatng the demand functon (11) wth respect to prces. In dong so, t s mportant to note that because of the contnuum assumpton (an uncountably large number of frms), P = 0. In ths large group case, a small change n one frm s prce does not affect p (v) 6

7 the overall prce level n ts sector. In lght of ths, we can wrte: x (v) ] { } p (v) = σ p σ 1 1 σ E (v) P P σ x (v) p (v) (15) The frst order condton can therefore be rewrtten as: p(v) = wa + x (v) p (v) σ x (v) then rearranged, and solved for prces: p (v) 1 ) p (v) p (v) (1 1σ = wa (16) σ p (v) = ( ) σ wa (17) σ 1 The second term on the rght hand sde n equaton (17) s smply the frm s margnal cost of producton. The term n bracets s a constant (wthn sector) marup: snce the numerator must be greater than the denomnator, there s a postve wedge between the frm s factory gate ( mll ) prce and ts margnal cost. Snce the wedge only depends on the sectoral elastcty of substtuton, t s constant across all frms n the sector. 4 Trade Costs Thus far, we have not consdered the condtons under whch nternatonal trade taes place. At the moment, the model smply conssts of a set of demand functons and prcng condtons for all countres and sectors. As t s, the model descrbes trade n a frctonless world, n whch goods produced n country can be shpped to country j at no charge. Arbtrage ensures, therefore, dentcal prces n both countres. 7

8 To ntroduce trade frctons, we can use the common ceberg formulaton. When a frm shps goods from country to country j, t must send τ j 1 unts n order for a sngle unt to arrve. The dfference can be thought of as meltng (le an ceberg) en route to the destnaton. Equvalently, the margnal cost of producng n country a unt of a good subsequently consumed n the same country s wa, but f the same product s consumed n country j then the margnal cost s nstead τ j wa. Usng ths defnton, costless trade corresponds to τ j = 1, and τ j corresponds to one plus the ad valorem tarff rate. Snce the sze of the trade frcton assocated wth a gven ceberg coeffcent does not depend on the quantty of goods shpped, we can treat ceberg costs as beng varable (not fxed) n nature. Tang any two countres and j, the presence of ceberg trade costs means that the prce n country j of goods produced n country s (from equaton (17) above): p j (v) = ( ) σ τ σ 1 jwa = τ j p (v) (18) Ths result allows us to rewrte the country prce ndex n a more useful (and general) form: P j = {ˆ v V j } 1 ] 1 σ τ j p 1 σ (v) dv (19) Note that ths ndex ncludes varetes that are produced and consumed n the same country: all the τ terms are smply set to unty, so as to reflect the absence of nternal trade barrers. 5 Gravty Wth Gravtas These are all the ngredents requred to put together a gravty model wth gravtas. The trc s n combnng them n the rght way. 8

9 The gravty model s usually concerned wth the value of blateral trade (x j ),.e. exports from country to country j of a partcular product varety. Combnng the prce equaton (18) wth the demand functon (11) gves: x j (v) = p j (v)x j (v) τ j p (v) { τ j p (v) } σ E j Pj Pj (20) { τ j p (v) P j } 1 σ E j (21) The above expresson gves us blateral exports of a sngle product varety. Compare equaton (6) n AvW (2003), and equaton (6) n HMR. To derve somethng that loos more obvously le a gravty equaton, we need to aggregate ths expresson to gve total sectoral exports from to j,.e. X j. From the producton sde of the model, t s clear that all frms n a gven country-sector are symmetrcal n terms of margnal cost, sales, prce, etc. Usng the measure N of frms actve n country, we can therefore wrte total sectoral exports very smply: { τ X j p j = N (v) } 1 σ Pj E j (22) Now comes the mportant part: ntroducng a general equlbrum accountng dentty. It must be the case that sectoral ncome n country, Y, s the ncome earned from total worldwde sales of all locally made varetes n that sector. Thus: Y = C ] X j = N p 1 σ C (v) j=1 j=1 { τ j P j } 1 σ E j (23) Solvng for N p (v) ] 1 σ gves: ] N p 1 σ Y (v) = { } C τ 1 σ j j=1 E j P j (24) 9

10 Next, substtute the above expresson bac nto the sectoral exports equaton (22): X j = C j=1 Y E j { τ j P j } 1 σ E j { } τ 1 σ j (25) P j For convenence, defne ( Π ) 1 σ = C j=1 { τ j P j } 1 σ E j Y where Y s total world output n sector. Dvdng the above expresson through by Y and substtutng for ( Π ) 1 σ gves the AvW gravty equaton (compare AvW (2003) equatons 9-11, AvW (2004) equatons 5-7, and Feenstra Chapter 5 equaton 5.35): X j = Y E j Y { τ j Π P j } 1 σ (26) or n the more common log-lnearzed form: ( log X j ) ] = logy + loge j logy + (1 σ ) logτ j logπ logpj (27) The basc logc of ths equaton accords wth that of the smple gravty model, n the sense that economc mass n the orgn and destnaton marets s postvely assocated wth blateral trade, but trade costs are negatvely assocated (snce σ > 1 by assumpton). However, there are many other nterestng and mportant thngs gong on here, and t useful to address some of them explctly. (What follows draws n partcular on Baldwn and Taglon, 2007.) 5.1 Relatve Prces Matter One of the man shortcomngs of tradtonal (non-theoretcal) gravty models s that they do not tae account of reallocatons of trade flows across tradng partners n general equlbrum. For nstance, f country lowers trade barrers affectng mports from country j only, the tradtonal gravty model predcts an effect on blateral trade between and j, but not on any other tradng 10

11 relatonshps. Ths s strongly counter-ntutve: thn trade dverson. When you do gravty wth gravtas, thngs start to loo more le they should. From (26) and the defntons of Π and Pj, t s mmedately apparent that blateral trade between and j depends on trade barrers on all worldwde routes, not just those actng drectly on j trade. AvW refer to Π as outward multlateral resstance, snce t aggregates all trade barrers facng exports from (.e., across all destnaton marets). They call P j nward multlateral resstance because t aggregates all trade barrers facng mports nto j (.e., across all orgn marets). As AvW (2003, 2004) show, leavng these terms out of an emprcal gravty model has two nasty results. Frst, the econometrc estmates are based, due to omtted varables. Second, counter-factual smulatons produce based results because they do not tae nto account the general equlbrum effects captured by the two multlateral resstance terms. (Ths bas s n addton to the bas resultng from poorly estmated parameters.) From an estmaton pont of vew, t does not overly matter whch approach s taen to account for multlateral resstance (fxed effects, nonlnear estmaton as n AvW, or a Taylor seres approxmaton as n Baer and Bergstrand, 2007). When t comes to counter-factual smulatons, however, the choce s mportant. Fxed effects wll gve unbased parameter estmates, but counter-factuals wll not tae account of general equlbrum effects transmtted through the multlateral resstance terms. Ether the nonlnear approach of AvW, or a Taylor seres approxmaton, should be used. 5.2 Fxng Fxed Effects For appled researchers, fxed effects are a very appealng alternatve to the relatvely complex estmaton procedure of AvW (2003). Care needs to be exercsed, however, n choosng approprate dmensons. Followng the pattern of subscrpts n (26), t s possble to formulate some smple recommendatons: 11

12 1. In sngle year datasets usng total (aggregate) trade data, multlateral resstance can be captured usng exporter and mporter fxed effects. These terms capture n each case the sum of ncome or expendture and nward or outward multlateral resstance. 2. In sngle year datasets usng sectoral trade data, multlateral resstance can be captured usng exporter-sector, mporter-sector, and sector fxed effects. These terms capture n each case the sum of sectoral ncome or expendture and nward or outward multlateral resstance. 3. In mult-year datasets, nteract the fxed effects specfed n 1. or 2. above wth year effects, and nclude a full set of year effects. These terms capture the fact that ncome/expendture and multlateral resstance are lely to change over tme. 5.3 GDP, Prces, and All That Gravty models usually use GDP n the orgn and destnaton countres as proxes for ther economc mass. Alternatvely, some authors nclude populaton and GDP per capta separately. Some people prefer nomnal data, others use real (deflated) data. It can be mportant to brng some gravtas to bear on these choces as well. Frst, the AvW model does not suggest any partcular role for per capta GDP n drvng blateral trade. That does not mean that t should never be ncluded n any self-respectng gravty model. But t does mean that researchers nterested n the role of ncome levels should go bac to the theory, and fnd an explct ratonale (and functonal form) for ts ncluson. Second, the AvW model suggests that GDP (and trade flow data) should be entered n nomnal, not real, terms. To see why, consder an extenson of the above model to nclude a number of years. On the assumpton that trade costs, expendtures, or ncomes change over tme for whatever reason the multlateral resstance terms wll also change over tme. Snce these terms are already an to trade prce ndces, there s no reason for any addtonal deflators. 12

13 Thrd, t s mportant to pont out n lght of the prevous paragraph that although the multlateral resstance terms loo and functon le prce ndces, they are not equvalent to commonly avalable prce data le CPIs, PPIs, etc. The multlateral resstance terms depend on world prces, trade barrers, and the ntra-sectoral elastcty of substtuton. They are aggregated n a very partcular way. As a result, they cannot adequately be proxed by over the counter prce data. Fourth, and for exactly the same reasons as n the prevous paragraph, multlateral resstance terms cannot usually be captured by atheoretcal remoteness ndces. Although they loo le remoteness and play a smlar role n the model, they have a precse defnton that must be respected. Ffth, t s mportant to pay partcular attenton to equaton (26) and the assocated defntons when usng dsaggregated (sectoral) data. Aggregate GDP can be a decent proxy for economy-wde ncome and expendture. But n a sectoral context, what the model actually needs s sectoral ncomes and expendtures. Drect data can be hard to come by, so t wll sometmes be approprate to nteract GDP wth fxed effects n order to allow these shares to change across sectors and countres. Alternatvely, the model can be estmated separately for each sector (.e., wthout poolng). 5.4 Exports, Imports, and Blateral Trade The model derved above s expressed n terms of exports from to j. Ths would tend to suggest that gravty emprcs should use export data rather than mports or total blateral trade (exports+mports). In realty, there can n fact be good reasons for usng mport data nstead, at the prce of sacrfcng a lttle theoretcal purty. Most countres trac mports much more closely than they trac exports, because they represent a tax base. Value data, n partcular, are lely to be more accurate n the mport data than n the export data. There are also usually fewer zeros. A common technque s therefore to use ether all mport data, or export data for countres that are assumed to be good reporters combned wth mport ( mrror ) data for countres that have more serous data ssues. 13

14 In any case, there s no partcular reason for usng total blateral trade, or the average of exports and mports. 5.5 Cross-Sectoral Heterogenety and Trade Cost Elastctes Most gravty models these days are estmated usng data dsaggregated at the sectoral level. Ths s good n the sense that t can result n more precse parameter estmates. But t s mportant to note a feature of gravty models le (26): the reduced form elastcty of trade wth respect to trade costs can vary across sectors due to changes n the ntra-sectoral elastcty of substtuton σ. A coeffcent of -1.2 on dstance n fact represents the product of two structural parameters: the elastcty of trade costs wth respect to dstance, and (1 σ ). It s thus not entrely correct to nterpret a reduced form coeffcent of -1.2 as a dstance elastcty. To do that, t would be necessary to dvde through by (1 σ ). In addton, ths dscusson maes clear that sectoral gravty models should usually allow reduced form trade elastctes to vary by sector. Why? Because the (1 σ ) term vares by sector even f the structural trade cost elastcty does not. Ths s another good reason for usng extensve sectoral nteracton terms, or estmatng the model separately for each sector. 5.6 Zeros n the Blateral Trade Matrx Ths s not the place for a detaled dscusson of the zero problem. See HMR. It s mportant to note, however, that estmatng a log-lnear gravty model when the trade matrx ncludes zeros mght ntroduce some serous bas to the extent that there s useful nformaton n the part of the matrx that s dropped. The model set out n these notes does not say anythng n partcular about zero entres n the blateral trade matrx, but the heterogeneous frms extenson n HMR does. Possble approaches to dealng wth zeros nclude: 14

15 1. Ad hoc fxes, such as addng a small postve constant to all zero flows. 2. Econometrc fxes that allow the zero entres to be ncluded n the estmaton sample, such as the Tobt or Posson estmators. 3. Theoretcal and econometrc fxes that 1) provde a ratonale for the exstence of zeros, and 2) correct for ther presence, for nstance through a modfed verson of the Hecman sample selecton model. 6 Concluson The theoretcal structure set out by AvW, whle very general, s only one of many that can produce somethng loong le a gravty model. However, ther formulaton s currently one of the more mportant benchmars. As a result, emprcal wor should eep as closely as possble to the constrants mpled by ths model n terms of the estmatng equaton. Devatons from that benchmar need to be explaned, and preferably justfed by reference to an explct (and credble) theory. The AvW model has mportant mplcatons for how appled researchers estmate and nterpret gravty models, partcularly n panel data contexts (multple years and/or multple sectors). The selecton of dependent and ndependent varables, the confguraton of fxed effects, the decson whether or not to pool data, and the nterpretaton of reduced-form elastctes, all depend ntmately on the theoretcal results derved above. Moreover, the model s general equlbrum bass, as captured through the multlateral resstance terms, has mportant mplcatons for runnng counter-factual smulatons a partcularly mportant pont for polcy-orented wor. 15

16 References and Further Readng Anderson, James E., and Erc Van Wncoop, 2003, "Gravty wth Gravtas: A Soluton to the Border Puzzle", The Amercan Economc Revew, 93(1), ( AvW 2003 ) Anderson, James E. and Erc Van Wncoop, 2004, Trade Costs, Journal of Economc Lterature, 42(3), ( AvW 2004 ) Baldwn, Rchard E. and Dara Taglon, 2007, Trade Effects of the Euro: A Comparson of Dfferent Estmators, Journal of Economc Integraton, 22(4), Baldwn, Rchard E., et al., 2005, All You Wanted to Know About Dxt-Stgltz but were Afrad to As, Baer, Scott, and Jeffrey H. Bergstrand, 2007, Bonus Vetus OLS: A Smple Method for Approxmatng Internatonal Trade-Cost Effects usng the Gravty Equaton, Worng Paper, Chaney, Thomas, 2008, "Dstorted Gravty: The Intensve and Extensve Margns of Internatonal Trade", Amercan Economc Revew, 98(4), Dngel, Jonathan I, 2008, The Bascs of Dxt-Stgltz Lte, Feenstra, Robert, Advanced Internatonal Trade, Chapter 5. Helpman, Elhanan, Marc Meltz, and Yona Rubnsten, 2008, "Estmatng Trade Flows: Tradng Partners and Tradng Volumes", Quarterly Journal of Economcs, 123(2), ( HMR ) 16