Dynamics of export share of products in

the international trades

Matthieu Barbier and Deok-Sun LeeDept. Physics, Inha University

Dr. Matthieu BarbierNow at Dept. Ecology & Evol. Biol., Princeton Univ. USA

International trades• if there are comparative advantage of importing

rather than producing a product given factors of production, politics, culture, history, etc.D. Ricardo, On the Principles of Political Economy and Taxation (London: John Murray, 1817; retrieved 2012-12-07 via Google Books)

• What products are the whole world producing and exporting?

• Any fundamental “laws” there?

Products that Korea is producing and exporting

1962 2000http://atlas.media.mit.edu/

Products the whole world is produc-ing and exporting

19623330 Petrol.oils & crude oils obt.from bitumin.minerals0711 Coffee,whether or not roasted or freed of caffeine2631 Cotton (other than linters),not carded or combed2320 Natural rubber latex; nat.rubber & sim.nat.gums2681 Seeps or lambswool,greasy or fleece-washed

20003330 Petrol.oils & crude oils obt.from bitumin.minerals7810 Passenger motor cars,for transport of pass.& good9310 Special transactions & commod.,not class.to kind5417 Medicaments(including veterinary medicaments)7788 Other elect.machinery and equipment

Statistics and dynamics of “Export share” of products

- empirical observation

- what are we interested in?

NBER-UN world trade data from 1962 to 2000

• R.C. Feenstra, R. E. Lipsey, H. Deng, A. C. Ma, and H. Mo, World Trade Flows: 1962-2000, NBER Working Paper No. 11040 (2005)

• Product ID 4-digit Standard International Trade Classification (SITC), re-vision 2

• Mainly based on the importers’ reports • Curated and supplemented by the available data of trades

of individual countries

year icode importer ecode exporter sitc4 unit dot value quantity . . . . . . . . .132627 1962 218400 USA 454100 Korea Rep. 8310 1 1 NA132628 1962 218400 USA 454100 Korea Rep. 8420 1 564 NA132629 1962 218400 USA 454100 Korea Rep. 8471 1 1 NA . . . . . . . . .

Export share of a product p in year t• Export volume (value in dollars) of a product p in

year t : • Export share (relative export volume) of a product

p in year t : • 508 products maintain non-zero volume from 1962

to 2000• Normalization , Mean •

This quantity is what we study here

Uneven distribution of export share

Power law behavior with exponent between 2 and 3 is observed for for all years.The functional form of the distribution does not change with timeThe second moment increases slightly with time with anomalous peaks af-ter oil shock (1973)

𝐶𝑡 ( 𝐴 )=𝑃𝑟𝑜𝑏 . (a   product   has   𝐴𝑝 (𝑡 )≥ 𝐴 ) 𝜎 𝐴2=𝐴𝑝

2 −𝐴𝑝2

𝑃 ( 𝐴)∼𝐴−𝛼SITC 3330 Crude oil:

Growth or decay for 39 years

• Increase or decrease? How much?

• Growth rate

• Skewed distribution

• What products increase share and what do the opposite in the international trades? What is the law governing it?

Invalid carriages (7853)

Coal and water gas(3415)

Variation of share for one year- relatively microscopic view

Gain : Loss :

Linear scaling both for gain and loss of share !

𝐴𝑝(𝑡)

Δ 𝐴𝑝(𝑡)

𝐴𝑝(𝑡 )

Δ 𝐴𝑝(𝑡)

Our viewpoint, strategy, and goal

• Construct a particle-hopping model consistent with the linear scaling between and • Can the model explain the empirical observations such as the broad distribution of share?• Can the model predict the evolution of share of individual products?

Hidalgo et al., Science (2007)

Urn model with quadratic preferential selection

- Factorized steady state

- Pseudo-condensation

Urn model (or misanthrope process) as a model for the dynamics of export share

• A complete graph of N sites, each site representing a product

• A total of M particles, each particle representing the unit of share

• Each site has particles and specifies the system’s state () • At each (microscopic) time step wo sites i and j are se-

lected with probability and a particle at site i is transferred to site j, where with

Example: N=14 M=35

𝑚1=4𝑚2=5

𝑚3=1𝑚4=2

𝑚5=2

𝑚6=1

𝑚7=1

𝑚8=4𝑚9=2𝑚10=1𝑚11=3

𝑚12=4

𝑚13=2

𝑚14=3

We consider the case where N is very large

Every site has at least one particle

Why is it quadratic, not linear?• To be consistent with the linear relation between

and • The annual variation of a product’s share is the sum

of plus and minus like random walk• Suppose that particle-hopping occurs times for one

year. Then

• A parameter is introduced: • One year corresponds to times of transfer of parti-

cles

or (empirical)

Urn model with quadratic preferential selection

• Different from the zero-range process in that the particle-hop probability depends on the number of particles of the destination as well as of the source

• Particle-hop probability is proportional to the square of the number of particles in the source and destination site

• Each unit of share (particle) is likely to move with probability proportional to the share of the present product and that of the destination product

• Our model is therefore capturing (only) the trend of a country’s economic policy to-wards enhancing the likelihood of profit beyond the different abundance/deficiency of factors of production from country to country.

• Related worksGodrèche, Bouchaud, Mezard, JPA 28, L603 (1995) – model A, B, CGodrèche and Luck, EPJB 23, 473 (2001) – zeta urn modelMajumdar, Evans, Zia PRL (2005); Evans, Majumdar, and Zia, JSP 123, 357 (2006) -- condensationBarabasi and Albert, Science (1999) – (linear) preferential attachment of links

Relation to empirical results• Distribution of share corresponds to the distribution of the

number of particles, which can be obtained analytically for the stationary state

• Is the broad distribution of share caused by the linear scal-ing between and ?

• Can the model predict the trajectory of share of individual product?

?Urn model based on

Factorized steady state• : Probability of a specific particle configuration at time t•

• Factorized state assumed for the stationary state with to be determined

• Detailed balance condition

• Function f: where is used.

𝑢 (𝑚 ,𝑛)∝𝑚2𝑛2

Evans and Hanney, JPA 38, R195 (2005)

Single-site particle-number distribu-tion

• • Partition function • Grand partition function with • Partition function (inverse Mellin) • Probability that a single site has m particles

Evans and Hanney, JPA 38, R195 (2005)

Partition function in our model• Logarithmic singularity of the generating function at

(J.E. Robinson, Phys. Rev. 83, 678 (1951))

• Partition function at the saddle point, which exists within the radius of convergence(otherwise, condensate is formed) in case with particle density

•

Steepest descent path

Single-site particle-number distribu-tion in our model

• with •

•

•

• A bump is formed for for the last two cases

Condensate-free𝜌=

𝑀𝑁 =2 𝜃=

𝜁 (2 )𝑒𝜌𝜁 (2 )

𝑁 =0.88 (𝑁=50 ) 0.22(𝑁=200)

Approximate

Condensate …𝜌=

𝑀𝑁 =10 𝜃=

𝜁 (2 )𝑒𝜌𝜁 (2 )

𝑁 =458180 (𝑁=50 ) 114545 (𝑁=200)

Approximate

Condensation?• Nature of condensation has been studied for the (continuum) mass transfer model

in 1D (Majumdar, Evans, Zia PRL (2005); Evans, Majumdar, and Zia, JSP (2006))

• If the particle-hop probability is given by , the single-site particle-number distribu-tion is

• If , it may happen that even for finite .

• If , can be infinite and can be equated to any finite by introducing a suitable cut-off leading to exponential decay. However, if is infinite, we should compare and , both of which are large numbers, and depending on the relation between and , a pseudo-condensate can appear

• International trade dynamics is at the edge of condensation-free phase.

Application

- Does this model explain the international trade at the aggregate level and individual level?

Yes! it does at the aggre-gate level

Simulate the model with and the initial values from the 1962 data

Δ 𝐴𝑝 (𝑡 )≃𝑐𝑔 𝐴𝑝 (𝑡 )𝛼𝑔 |Δ 𝐴𝑝 (𝑡 )|≃𝑐 𝑙 𝐴𝑝 (𝑡 )𝛼𝑙

Share distribution Second moment

Growth-rate distribution Gain versus share Loss versus share

Evolution of export share of individ-ual products

• An ensemble of simulation results• The middle 80% of the simulation values for is

shaded

Bad…

Not bad….

Quantifying the typicality of empirical ob-servations among simulated trajectories• Values of return (one-year growth rate) are compared between real and

simulation for each p and t. That is, we have one and K=300 simulation values

• Normalized rank

• If a product p is well predicted by the model, a total of T=39 such normal-ized ranks at different years should be uniformly distributed over [-1/2, 1/2]

• Deviation from Uniformity : i) sort T=39 values of ’s in increasing order from the smallest to the largest such that . If they are uniform, then one would find for iii) Non-uniformity or Unpredictability of a product p is defined as

• is observed only with probability 0.05 for 39 uniformly-distributed numbers(Marhuenda, Morales, Pardo, Statistics 39, 315 (2005))

Classifying productsMean rank positive: higher returns (annual growth) than expectednegative: lower returns than expected

Rank fluctuation Larger than 0.29 : higher variability of rank than expectedSmaller than 0.29 :lower variability

Unpredictability Larger than 0.1 : deviate significantly from our model predictionSmaller than 0.1 : predictable by the model

Predictability and mean-rank

The most unpredictable products

SITC Description Ap_1962 mean rank rank fluctuation unpredictability6589 Other_made-up_articles_of_textile_materials,n.e.s 9.81E-06 0.19632 0.4202 0.258257853 Invalid_cariages,motorized_or_not,parts 5.69E-06 0.075288 0.48974 0.25156880 Uranium_depleted_in_u235_&_thorium,&_their_alloys 1.50E-05 0.049059 0.47185 0.225912239 Flours_or_meals/oil_seeds/oleag.fruit_non_defatted 1.57E-05 -0.016992 0.45334 0.203492640 Jute_&_other_textile_bast_fibres,nes,raw/processed 0.0021461 -0.20094 0.27152 0.200946122 Saddlery_and_harness,or_any_material_for_animals 2.48E-05 0.18517 0.34404 0.196792652 True_hemp,raw_or_processed,not_spun;tow_and_waste 0.00015328 -0.16539 0.36012 0.196376545 Fabrics,woven,of_jute_or_of_other_textile_bast_fib 0.0026562 -0.19545 0.26332 0.195452231 Copra 0.0023281 -0.1843 0.32578 0.191522654 Sisal_&_other_fibres_of_agave_family,raw_or_proce. 0.0013954 -0.18264 0.33049 0.185853231 Briouet.ovoids_&_sim.solid_fuels,of_coal_peat_lig. 0.00032608 -0.17393 0.34473 0.185859610 Coin(other_than_gold)_not_being_legal_tender 0.0001946 -0.049669 0.44353 0.185222742 Iron_pyrites,unroasted 0.00046455 -0.17933 0.3278 0.18282114 Goat_&_kid_skins,raw_(fresh,salted,dried,pickled) 0.00061788 -0.10918 0.41458 0.179012814 Roasted_iron_pyrites,whether_or_not_agglomerated 0.00023743 -0.13716 0.39188 0.178962235 Castor_oil_seeds 0.00020929 -0.15999 0.35708 0.178578996 Orthopaedic_appliances,surgical_belts_and_the_like 0.00023142 0.1768 0.19731 0.178252634 Cotton,carded_or_combed 3.19E-05 0.060561 0.42361 0.178058983 Gramophone_records_and_sim.sound_recordings 0.00061524 0.17707 0.20675 0.177077931 Warships_of_all_kinds 0.0003461 -0.080777 0.43397 0.17263

Raw materials and agricultural productsMostly they have their share fall behind prediction.

Products with unpredictably increased share

SITC Description Ap_1962 mean rank rank fluctuation unpredictability6589 Other_made-up_articles_of_textile_materials,n.e.s 9.81E-06 0.19632 0.4202 0.258256122 Saddlery_and_harness,or_any_material_for_animals 2.48E-05 0.18517 0.34404 0.196798983 Gramophone_records_and_sim.sound_recordings 0.00061524 0.17707 0.20675 0.177078996 Orthopaedic_appliances,surgical_belts_and_the_like 0.00023142 0.1768 0.19731 0.178256642 Optical_glass_and_elements_of_optical_glass 7.58E-05 0.16225 0.29715 0.163438710 Optical_instruments_and_apparatus 0.00072931 0.1606 0.21861 0.167028821 Chemical_products_&_flashlight_materials 0.00012313 0.16025 0.18614 0.162588959 Other_office_and_stationery_supplies 0.00019894 0.14761 0.18156 0.164915416 Glycosides;glands_or_other_organs_&_their_extracts 0.00048222 0.14543 0.20509 0.146888720 Medical_instruments_and_appliances 0.0011123 0.14526 0.16567 0.157417741 Electro-medical_apparatus 0.00023524 0.13707 0.24761 0.137075530 Perfumery,cosmetics_and_toilet_preparations 0.0012252 0.13646 0.19143 0.146158942 Childrens_toys,indoor_games,etc. 0.0025954 0.13611 0.22104 0.1435223 Milk_&_cream,fresh,not_concentrated_or_sweetened 0.00011694 0.13454 0.24441 0.13454

5161 Ethers,alcohol_peroxides,ether_perox.,epoxides_etc 0.00031245 0.13332 0.20459 0.138148841 Lenses,prisms,mirrors,other_optical_elements 0.00034402 0.13254 0.21238 0.148098310 Travel_goods,handbags,brief-cases,purses,sheaths 0.0010566 0.13254 0.18475 0.142311110 Non_alcoholic_beverages,n.e.s. 0.00020163 0.13114 0.23304 0.131146647 Safety_glass_consisting_of_toughened/laminat.glass 0.00019306 0.12949 0.24531 0.13226123 Parts_of_footwear 0.00023262 0.12949 0.23223 0.1318

Medical appliances, toys, cosmeticsThey are not exclusively subject to economic demands

Products with the largest fluctuation of rank

SITC Description Ap_1962 mean rank rank fluctuation unpredictability7853 Invalid_cariages,motorized_or_not,parts 5.69E-06 0.075288 0.48974 0.25156880 Uranium_depleted_in_u235_&_thorium,&_their_alloys 1.50E-05 0.049059 0.47185 0.225912239 Flours_or_meals/oil_seeds/oleag.fruit_non_defatted 1.57E-05 -0.016992 0.45334 0.203499610 Coin(other_than_gold)_not_being_legal_tender 0.0001946 -0.049669 0.44353 0.185227931 Warships_of_all_kinds 0.0003461 -0.080777 0.43397 0.172637911 Rail_locomotives,electric 0.00026424 -0.012199 0.4247 0.150952634 Cotton,carded_or_combed 3.19E-05 0.060561 0.42361 0.178057913 Railway_&_tramway_coaches,vans,trucks_etc. 0.00023165 0.045399 0.4203 0.154116589 Other_made-up_articles_of_textile_materials,n.e.s 9.81E-06 0.19632 0.4202 0.258252714 Potassium_salts,natural,crude 0.00030787 -0.046532 0.41845 0.155656724 Puddled_bars_and_pilings;ingots,blocks,lumps_etc. 0.00075701 -0.039735 0.4175 0.149732860 Ores_and_concentrates_of_uranium_and_thorium 0.0026274 -0.0030498 0.41522 0.135352114 Goat_&_kid_skins,raw_(fresh,salted,dried,pickled) 0.00061788 -0.10918 0.41458 0.179017914 Railway_&_tramway_passenger_coaches_&_luggage_van 0.00048688 -0.059167 0.41353 0.147727915 Rail&tramway_freight_and_maintenance_cars 0.00070004 -0.061258 0.41318 0.163957187 Nuclear_reactors_and_parts 3.72E-05 0.047839 0.40814 0.152787933 Ships,boats_and_other_vessels_for_breaking_up 0.00027311 -0.023789 0.40432 0.12718451 Rye,unmilled 0.00060056 -0.076246 0.40347 0.14527541 Potatoes 0.0024874 -0.030063 0.40127 0.12806

4235 Olive_oil 0.0012997 0.016556 0.39846 0.13502

Railways, warships, uranium, Nuclear reactorsOffer and demand are highly variable in time and historically determined

Summary and Discussion• Time-evolution of the market share of products in the world trade has been studied by

data analysis and model study• Urn model with quadratic preferential selection reproduces linear scaling of annual

gain and loss of share and the power-law distribution of share with exponent 2• The model represents the pressure of directing a country’s investment towards more

popular products in the global economy• The quadratic preferential selection leads the world trade market to the edge of con-

densation • The condition for the emergence of pseudo condensate has been found.• The model explains the empirical observations very successfully at the aggregate

level• The share trajectory of more than 60% products are predicted by the model capturing

the pressure towards enhancing the likelihood of profit.

• Nature of unpredictable products provides the reason of deviation from model predic-tion

• For more realistic and predictive model, one should consider the network structure of product space– hopping in product space does not happen randomly but depending on the proximity of two products.