Gravity Model and Zipf's Law: An In-Depth Study into the Nature of International Trade

Transcription

1 E-ISSN 8-46 ISSN Vol., No. 9 Gravty Model and Zpf's Law: An In-Depth Study nto the Nature of Internatonal Trade Do:0.590/ajs.03.vn9p583 Abstract Mslav Josc, Ph.D. Teachng and Research Assstant, Faculty of Economcs and Busness Zagreb, Unversty of Zagreb, Internatonal Economcs Department, Zagreb, Croata Emal: mjosc@efzg.hr, Phone: Marja Nkc Student Teachng Assstant, Faculty of Economcs and Busness Zagreb, Unversty of Zagreb, Internatonal Economcs Department, Zagreb, Croata Emal: marja.nkc.mn@gmal.com, Phone: In ths paper trade pattern based on dstances between countres was tested through gravty model approach. Ths research hghlghts some of the regulartes n nternatonal trade that arse and are n concordance wth the rank-sze rule. An emprcal study was conducted on statc data used n Croata n 0 n order to test the valdty of Zpf's law. The presence of Zpf's law was emprcally tested usng populaton data for cty proper and cty settlements as well as the export values of Croata's most mportant tradng partners. In both cases results corroborate the exstence of Zpf's law n populaton and export values. Due to the heavy talness of Pareto dstrbuton mnor adjustments to the model have been made. Gravty model approach shows sgnfcant mpact of dstance and GDP on nternatonal trade. Dummy varable common border ndcates that most of the trade n Croata s done due to preferable geographcal poston of the country. Keywords: Internatonal trade, Zpf's law, rank-sze rule, gravty model, geographcal economcs. Introducton Most of the studes conducted n the feld of nternatonal economcs are amed towards the analyss of causes and effects n nternatonal trade. Classcal theores of nternatonal trade explaned trade as a result of prce dfference based on dfferent absolute or relatve labour productvtes among countres. Later on, El Heckscher and Bertl Ohln ntroduced captal as second factor of producton havng both producton possbltes and demand preferences. New trade theores assumed several assumpton that were rather rgd n the aforementoned theores. Ths paper unpretendedly ams to recreate new trade pattern of nternatonal trade based on Zpfan or rank-sze dstrbuton. Why do ctes emerge, how do they locate n countres and how do producton nputs relocate wthn a country s stll the key queston n geographcal economcs. Power laws orgnate from mathematcs and construct relatonshp between two varables. Power law relaton s merely a functon of two varables expressed as a power of one varable. Due to the nature of power functons logarthmc decomposton can be made as well as ts exponental counterpart from reverse acton. Let's defne power law functon as follows: y x = ax () k ( ) where x and y are ndependent and dependent varables, a s estmated parameter and k s the power coeffcent. The same () expresson can be transformed n logarthmc form () as follows: log y = log a+ klog x () Prevously mentoned functons are determnstc whch s a characterstc of hard scences such as physcs and mathematcs, but not economcs. In most cases natural laws ncorporate stochastc component ε as a result of samplng/measurement error or any other uncertanty that affects the mean. In ths respect power law n socal scences s often remodeled nto nondetermnstc form (3): log y= log a+ klog x+ ε (3) 583

2 E-ISSN 8-46 ISSN Vol., No. 9 where ε represents devaton factor or error term. There are several varetes of power laws avalable and ts prevalence s n mathematcs, physcs, bology, economcs and many other socal scences n partcular. In ths paper we analyse specfc form of power law that s know as Zpf's law and s of great mportance for economsts. Frst proposton regardng cty-sze relaton was made by Felx Auerbach (93). Fne tunng of Auerbach's fndngs was done by Amercan researcher George Kngsley Zpf (949) n a rule that states that frequency of some occurrence s nversely related to the rank of the same occurrence. The latter concluson was formed out of analyss of word occurrence n text where one could conclude that the frequency of the word occurrence s nversely related to ts rank n frequency table. Cobb-Douglas producton functon s also a power law functon wth two parameters n the exponent (α and α ).. Applcaton of Zpf's Law n Economcs How can Zpf's law (or rule) be appled n economcs today? Due to ts nonlnearty (convexty) Zpf's law s very useful n convexed analyss of a sngle varable calculus. Well known Pareto dstrbuton has a power law tal and s very smlar to Zpf's dstrbuton. Pareto dstrbuton s strctly used for contnuous functons whle Zpf's one s for dscrete cases. In ths respect they are somewhat dfferent and much alke at the same tme. One of the frst applcaton of Zpf's law was n geographcal economcs. Populaton of the ctes and ther rank follow Zpf's law. The second bggest cty n a country s approxmately ½ of the largest cty sze, and thrd bggest cty s /3 of the largest cty sze, etc. The rank-sze rule appled n geographcal economcs emerges from Auerbach's paper from the begnnng of 0th century and can be expressed n Pareto dstrbuton (4): k y( x) = ax (4) whch s equvalent to (5): log y = log a klog x (5) where x s populaton sze, y s the number of ctes larger than x, and ak, as constants. Zpf's work on Auerbach's dea goes further and assumes that dstrbuton should have the followng values of parameters: k = and a = the sze of the bggest cty n order for data to ft. Several papers have tested Zpf's law usng dfferent approaches. Let us menton few of the most promsng: Ioanndes and Overman (000) emprcally examned valdty of Zpf's law for metro areas n the Unted States on a large set of data rangng from They used nonparametrc test n order to estmate varabltes n endogeneous varables. Results have showed that Zpf's law holds for a large range of cty szes. General nvestgaton n the area of power laws was systematcally lad out by Newman (004) showng remarkable concordance wth Zpf's law. It was found n varatons of the followng varables: word frequency, ctatons, web hts, books sold, telephone calls receved, earthquake magntude, crater dameter n km, net worth n US dollars, name frequency and most notably populaton of ctes. A cross country nvestgaton of Zpf's law was agan tested by Kwok Tong Soo (005) usng OLS and Hll's estmator. Wth the use of OLS method Zpf's law was rejected for majorty of countres whle Hll's estmator rejected Zpf's law for mnorty of countres. Overall concluson was that poltcal economy varables better explan varatons n parameters than geographcal ones. Hnloopen and van Marrewjk (008) concluded that Balassa ndex of revealed comparatve advantage also follows Zpf's law usng estmator developed by Gabax and Ibragmov (006). Despte the general acceptance of Zpf's law n data there are dfferences between countres, sectors, populaton sze, etc. Recent comprehensve study n the area of Economcs and Fnance was conducted by Xaver Gabax (009) leavng several questons open for dscusson, e.g. why do orgns of ctes, frms, nternatonal trade, CEO pay or market fluctuatons follow power laws. 3. Data and Methodology In order to test the presence of Zpf's law an approprate populaton data for cty proper and cty agglomeraton was taken from Croatan Bureau of Statstcs based on census survey n 0. Populaton was matched for top 00 ctes or Sometmes referred as zeta dstrbuton. 584

3 E-ISSN 8-46 ISSN Vol., No. 9 settlements ranked by sze. Along wth classc relatonshp between rank and sze of the ctes presented by G. K. Zpf, ths paper goes further on establshng the lnk between export values and ts rank among top 00 mportng countres of Croatan products. The followng regresson equatons for cty propers and cty agglomeratons were used n testng the valdty of Zpf's law: log Rank = β0 + βlog Cty Pr oper + ε (6) for each -th cty proper rangng from largest to smallest (Zagreb to Zlatar), = 00. and log Rank = β0 + βlog CtyAgglomeraton + ε (7) for each -th cty agglomeraton rangng from largest to smallest (Zagreb to Buje), = 00. Expermental test of power law, more specfcally Zpf's law, present n export values of Croata's top 00 trade partners was carred out very smlar: log Rank = β0 + βlog Exports + ε (8) for each -th mportng partner country of Croata rangng from largest to smallest (Italy to New Zealand), = 00. In all three log-lnear regresson equatons (6)-(8) rank values were adjusted usng Gabax-Ibragmov estmator due to the fact that most of the data were grouped on the rght sde of the Pareto dstrbuton. Emprcal results n the next secton should confrm model's goodness of ft (hgh R value) and estmated β close to -. Gravty emprcs was tested usng 0 export values (n EUR) for Croata's tradng partner countres, agan frst 00 countres. Data was collected from Croatan Bureau of Statstcs based on values for foregn trade n goods n 0. Partner GDP values were downloaded from Eurostat and World Bank sources for year 0. Aeral dstance was calculated as a shortest dstance between two captal ctes of partner countres engaged n blateral trade. Proposed statcal gravty model s as follows: log Exportsjt = β0 + β log GDPjt + β log Ds tan cej + β3common _ borderj + ε (9) j where = - reference country (Croata), j = 00 - partner country (Italy to New Zealand), t = 0 - base year, Exports - exports from Croata to j-th partner country n year t (n EUR), jt GDP - j-th partner country GDP n year t (n EUR), jt Ds tan ce - dstance from Croata's captal Zagreb to j-th country captal cty, common _ border j j - 0 f countres do not share common border, otherwse ε - nose/error term j If practce (emprcs) comples wth theory then expected sgns of regresson parameters β < 0 and β 3 > 0 along wth hgh R value and statstcal sgnfcance of all ndependent varables. 4. Emprcal Results β should be: β > 0, Wth the methodology already mentoned n prevous secton the presence of Zpf's law s tested for cty proper and cty agglomeraton values based on 0 data. Graphcal relatons are shown at Fgure and Fgure. Expermental data analyss provded for export values n Croata's goods trade yelds somewhat dfferent results, but stll wthn the theoretcal bounds, whch can be carefully observed at Fgure

4 E-ISSN 8-46 ISSN Vol., No. 9 Fgure : Zpf's law for cty proper n Croata n logctyproper y = x R² = log(rank-/) Fgure : Zpf's law for cty agglomeraton n Croata n logctyagglomeraton y = x +.85 R² = log(rank-/) As shown at Fgure and Fgure observed data scatter neatly along the lne representng log-lnear ft correspondng wth Zpf's fndngs. Both R values ( and ) are very hgh mplyng the correct use of an economc model. The slopes of the regresson lnes n both cases are negatve wth coeffcents close to -. Observed data for cty agglomeraton ft more better Zpf's model wth β =.0369 as opposed to cty proper data where β = Croata's goods export values and the rank of the mportng countres also follow Zpf's law but not perfectly as shown n two prevous cases. Fgure 3: Zpf's law for Croata's export partners n logexports y = x R² = log(rank-/) 586

5 E-ISSN 8-46 ISSN Vol., No. 9 Coeffcent of determnaton s stll very hgh n ths case also ( R = ) whle the regresson lne s more steeply (negatvely) nclned wth β = Overall concluson s that populaton of cty proper, cty agglomeraton and export values to partner countres follow Zpf's law. As a complement to Zpf's law gravty model assumes several mportant facts that should be accounted for future nvestgaton of trade patterns and trade flows. Most of the nternatonal trade among countres s affected by locaton and geographcal poston of a country as a result of dfferences n transport costs. Due to Croata's Medterranean orentaton and ts close connecton wth the European Unon most of the trade s created wth countres n near proxmty, most notably Italy, Bosna and Herzegovna, Germany, Slovena and Austra whle the rest of the non-europe (extra) trade s created wth Russan Federaton and Unted States. Trade dstrbuton over contnents along wth the sze of a countres measured by GDP can be closely examned at Fgure 4. Fgure 4: Croata's exports and dstance over contnents Regresson analyss on expresson (9) corroborates theoretcal assumptons regardng the mpact of geographcal dstance of countres and transport costs on blateral trade flows. Estmated regresson equaton based on expresson (9) yelds the followng results: Table : Regresson output n gravty model for Croata, 0 data Independent varable β coeffcent t-statstc p-value Constant *** LogGDP *** LogDstance *** LogCommon_border.63759*** R F-statstc Prob(F-statstcs) 0 *** - varable sgnfcant at % level ** - varable sgnfcant at 5% level * - varable sgnfcant at 0% level From Table t s clear that all ndependent varables are statstcally sgnfcant at %, 5% and 0% level usng t-statstc. F-statstc shows that at least one of the varables n group test s statstcally sgnfcant as well. The sgns of β 587

6 E-ISSN 8-46 ISSN Vol., No. 9 coeffcents are consstent wth economc theory,.e. classcal gravty model approach. Coeffcent of determnaton s not too hgh thus reducng the rsk of multcollnearty n OLS model, but stll explans a great porton of varablty n dependent varable. 5. Conclusons Internatonal economcs today stll doesn't have one Theory of everythng, unfed theory that should explan causes and effects of nternatonal trade. Trade effects are pretty much known to us, but the causes of trade stll reman a mystery. Geographcal economcs as a part of nternatonal economcs analyses mcro and macro aspects of trade explanng why do ctes emerge, and most mportantly where do they locate. Local trade creates mcro trade patterns that also reflect n macro trade patterns, the one that exst between countres. In ths paper we tested Zpf's law, one of the power laws based on Pareto dstrbuton whch puts n relaton rank and sze of the ctes and settlements n a country. Postonng of ctes greatly affects local economy due to re(a)locaton of nputs and outputs thus leadng to new optmal equlbrum. Emprcal results from ths paper suggest that the sze of cty proper and cty agglomeraton n Croata strctly follows Zpf's law based on very strong log-lnear relatonshp n varables. Surprsngly, Croata's export values also follow Zpf's law wth less explaned varance n dependent varable. In addton to Zpf's law, gravty model was constructed based on classcal model by Jan Tnbergen. Estmated parameters of log-lnear regresson output are consstent wth mpled (theoretcal) values and strongly support the thess of trade based on economc sze measured by GDP and dstance between countres. Common border as a qualtatve regressor varable also postvely affects nternatonal trade gvng preference to adjacent countres. References Auerbach, F. (93) Das Gesetz der Bevolkerungskoncentraton, Petermanns Geographsche Mttelungen 59: Croatan Bureau of Statstcs: Census of Populaton, Households and Dwellngs 0. Avalable at: [ /publcaton/0/si-44.pdf] Croatan Bureau of Statstcs: Foregn Trade n Goods of the Republc of Croata, January December 0. Avalable at: [ 0.htm] Gabax, X. (009) Power Laws n Economcs and Fnance, Annual Revew of Economcs, Annual Revews, Vol. (), Hnloopen J. and van Marrewjk C. (008) Comparatve advantage, the rank-sze rule, and Zpf s law. Workng Paper, Tnbergen Insttute. Ioanndes, Y. M. and Overman H. G. (000) Zpf's law for ctes: An emprcal examnaton, Regonal Scence and Urban Economcs, Vol 33, (), Newman, M. E. J. (004) Power laws, Pareto dstrbutons and Zpf's law, Contemporary Physcs, Vol. 46, (5), Soo, Kwok Tong (005) Zpf's law for ctes: a cross country nvestgaton, Regonal Scence and Urban Economcs, Vol. 35 (3), World Bank database: World Development Indcators (WDI) 0. Avalable at [ Zpf, G. K. (949) Human Behavour and the Prncple of Least Effort, Readng, MA, Addson-Wesley. 588