UCL CENTRE FOR ADVANCED SPATIAL ANALYSIS WORKING PAPERS SERIES Paper 178 - Jan 12 A dynamc global trade model wth four sectors: food, natural resources, manufactured goods and labour ISSN 1467-1298 Centre for Advanced Spatal Analyss Unversty College London 1-19 Torrngton Place Gower St London WC1E 7HB Tel: +44 (0)20 7679 1782 casa@ucl.ac.uk www.casa.ucl.ac.uk
A dynamc global trade model wth four sectors: food, natural resources, manufactured goods and labour Hannah Fry and Alan Wlson Department of Mathematcs and the Centre for Advanced Spatal Analyss, Unversty College London, WC1E 6BT Abstract An mportant and long-standng research task s the buldng of a model of nternatonal trade flows, anchored n a model of natonal economes. A demonstraton model s presented whch ams to exhbt the prncpal phenomena of the real system. Exstng trade models, such as the Heckscher Ohln model, make certan knds of smplfyng assumptons, such as equal producton functons and wages across all countres. We seek to make the model more realstc by makng smplfyng assumptons n a dfferent way. Entropy-maxmsng spatal nteracton submodels and Lotka-Volterra-type dynamcs are deployed n a model that contans four representatve sectors: food, natural resources, manufactured goods and labour. The model uses prce as a mechansm to determne producton levels. In ths way, we ncorporate country-specfc dynamcs of labour, consumpton and ndvdual ncome, whle dstngushng between resource-rch/poor and GDP-rch/poor countres. The model s bult wth further dsaggregaton n mnd, and possble extensons are dscussed along wth some expermental results 1 1 Introducton In economcs and polcy, t s mportant to understand the mechansms of nternatonal trade, both n analysng exstng lnks and n testng the mpacts of, for example, changng technologes and network reslence. Ideally, ths would be done on the bass of nput-output models for each of the 200 or so countres, each of whch would nclude mport and export flows by sector and country of orgn or destnaton respectvely. Ths can be done n theory, for example usng the methods of [6] as appled n an nter-regonal context by [3]. However, the data sources are to say the least mperfect and t s an enormous task to assemble what s known n these terms. An objectve of research n these areas therefore, s to create a model of natonal economes and trade flows that s feasble n scale, and capable of replcatng the prncple phenomena of the full system. In recent years, several such models have been presented wth varyng success. Gravty models for example whch concentrate on the nfluence of spatal structure have been valdated emprcally, but may not take comparatve advantage nto account. The alternatve Heckscher Ohln model whch s based on comparatve advantage, s capable of handlng country specfc captal and labour markets, but cannot nclude unemployment or wage dscrepances across countres and demonstrates poor predctve power on an nternatonal level. 1 The authors acknowledge the fnancal support of the Engneerng and Physcal Scences Research Councl (EPSRC) under the grant ENFOLD-ng - Explanng, Modellng, and Forecastng Global Dynamcs, reference EP/H02185X/1. 1
Farmng Mnng Manufacturng Product Food Resources Goods Consumed by Populaton Manufacturng Populaton Unt of producton constrant Volume Volume Money Pad for by Income Goods prces Income after food purchases Cost prces depend on Labour costs Labour costs Labour & natural resources costs Fgure 1: A summary of the nput and outputs of the model. In ths paper we present a novel approach ntended to tackle some of these ssues, followng the prncples establshed n [7]. A key feature of the model s the dstncton between countres that are rch and poor n terms of Gross Domestc Product (GDP) per capta, and those that are rch and poor n terms of natural resources such as ol or precous metals. Ths produces a 2x2 classfcaton of countres and allows us to ncorporate country specfc dynamcs of labour, unemployment, ncome, and producton capacty. We assume that the populaton needs food, but that food consumpton s dfferent for each country, reflectng ncome. Indvduals also consume manufactured goods, and for the purposes of smplfcaton, we assume that the balance of expendture by the populaton on food s spent on manufactured goods. The manufacturng ndustry on the other hand s the sole consumer of natural resources, whle the three producton sectors: food, natural resources and manufacturng, all requre labour nputs. Thus, the four sectors are defned and the mplct nput-output model s very smple. A summary of the system may be seen n Fgure 1. At the core of the trade model s a set of spatal nteracton models for the three producton sectors. These are bult on entropy maxmsaton - see for example [2] as appled to retal models of consumer spendng flows, [5], [1], [4] as models for nter-regonal mgraton flows, and [3] [6] appled to trade flows. A descrpton may also be found n [7], where the model presented here s both an extenson and smplfcaton of that work. Ths standard spatal nteracton model, however, requres an adaptaton to allow for varyng producton levels, and we ntroduce a varable sellng prce as a mechansm to do ths. At each tme step, each sector of each country enters the global market place wth a producton capacty: a volume whch they may not sell above, and a cost prce whch they may not sell below. The global competton then takes the form of an teratve algorthm: ncreasng the sellng prce whenever demand exceeds capacty and reducng the producton whenever a sector cannot sell at capacty. 2
Fgure 2: A schematc dagram of how varous elements of the model nteract At the end of the tradng process, the natonal government renvests any profts nto proportonally ncreasng the capacty of ther best performng sectors, thus addng a dynamc element to the model. A schematc dagram of how the varous elements of the model nteract s gven n Fgure 2. To put ths work nto context, the motvaton for ths model s as a frst step n a global demonstraton model, ncorporatng the coupled dynamcs of mgraton, nternatonal ad and securty. Thus the structure s bult wth future dsaggregaton n mnd, and desgned to cope wth varyng populatons and natonal and nternatonal nvestment. In Secton 2 we defne all the varables used to descrbe the system of nterest. The prcng and trade flows algorthm whch forms the core of the trade model s presented n Secton 3. Ths algorthm requres nputs of producton capacty, cost prces, a measure of the qualty of goods and transport costs. Theoretcally, these measures should be avalable from data and treated as exogenous. For the purposes of our demonstraton model however, we have created estmates of these quanttes, whch are outlned n Secton 4. Once these values are known, the prcng and trade flows algorthm, wth mnor modfcatons, may be appled to each of the producng sectors. The process follows n detal n Sectons 5, 6, 7. Fnally, the dynamcs of the model are demonstrated n Secton 8 and the results of an expermental run are descrbed n Secton 9. 2 Defnton of varables for system descrpton The three producng sectors, as we have seen, are food, natural resources and manufacturng. These are labelled by a superscrpt n = 1, 2, 3 respectvely. The countres are each gven a label denoted by subscrpt when the country s actng as a seller, and subscrpt j when actng as a buyer. Thus at each tme step (between t and t + δt) food, resources and manufactured goods flow from to j. The varables of nterest are as follows: {P } - Populaton of country. 3
{u } - Number of unemployed n. {ι } - Income per capta of country. {L (n) } -Work force of sector n n (so that n L(n) + u = P ) { (n) } - Capacty of producton of n n. {X (n) } - Actual producton of n n over δt. {χ (n) } - Monetary value of X (n). {Y (n) j } - Trade flows of n between and j. {ψ (n) j } - Money flows from sales of n between j and /Monetary value of Y (n) {Z (n) } - Consumpton of n n. {ζ (n) } - Monetary value of Z (n). {q (n) } - Relatve qualty of product n n {v } - Volume of food requred per person n {m } - Raw materal cost per unt of goods producton n. { φ (n) } - Cost prce per unt of n n durng δt. {φ (n) } - Sale prce per unt of n n durng δt. {D j } - Dstance between and j. {θ (n) } - Transport costs of n per unt volume per unt dstance. {C (n) } - Total cost of mantanng capacty of n n. {Π (n) } - Proft (or loss) made by sector n n over δt. - Natonal debt (f populaton cannot afford to feed tself) S - Total surplus n δt. Usng ths notaton, the dagonal of the trade flow matrx Y (n) wll gve the volume of n, consumed and produced n. In the followng we refer to ths as local consumpton. 3 The prcng and trade flows algorthm Wthn each country and sector, the dynamcs of ncome, global prcng and producton levels are all governed by the trade flows whch form the heart of the model. To smplfy the explanaton of the algorthm, the mechansm of the adapted spatal nteracton model whch derve these trade flows are descrbed n a separate secton. The prncples outlned here are dentcal for all sectors so, for ease of notaton, the superscrpts (n) are temporarly dropped. Assume, for the purpose of ths secton at least, that consumpton, Z j, producton capacty,, product qualty, q, and transport costs, θd j, of each country are known. j 4
If the prce, φ, were also known, the flow of goods, Y j, between seller and buyer j, could be calculated drectly and wthout teraton from a sngly constraned spatal nteracton model: Y j = Z jq α exp [ β(θd j + φ )] k qα k exp [ β(θd kj + φ k )]. (3.1) Usng (3.1), the sum over all buyers, j, would then yeld the total demand n : X = j Y j. (3.2) In some cases, ths demand, X, wll exceed the producton capacty,, of. Where ths occurs, an alteraton to the trade flows s needed. Usng the usual theory of supply and demand, a varable prce φ s ntroduced as a mechansm to deal wth ths adjustment. The am s to ncrease the unt prce φ of a product n any country where demand exceeds supply untl the quantty demanded by consumers s balanced by the producton capacty: X =. All other countres may not ncrease ther prces and wll sell at, or below, capacty. Assume also, for the tme beng, that a cost prce, or prce whch a country cannot sell below, φ, s known. For any country then, ether the prce s known (as φ ) and the producton must be found (wherever X < ), or the producton s known, X =, and the prce, φ > φ, must be found. The algorthm to determne the trade flows wll proceed as follows: 1. The spatal nteracton model s run once usng cost prces, φ, to determne the demand on each country X. X = Z j q [ β(θd αexp j + φ ] ) [ j k qα k exp β(θd kj + φ ]. (3.3) k ) 2. In any country where demand exceeds supply, the optmal prces - such that demand equals capacty - are calculated separately by rearrangng (3.1) and takng X =. More formally: m s.t.x m > m, (3.4) φ m = 1 [ ] 1 β ln + 1 [ m β ln Z j q αexp [ βθd ] mj] k qα k exp [ β(θd (3.5) kj + φ k )] Whch s solved teratvely snce φ m appears n the denomnator of the last term. 3. The new prces contrbute to the latest prcng vector φ, on whch the spatal nteracton model s re-run X = Z j q αexp [ β(θd j + φ )] j k qα k exp [ β(θd kj + φ k )]. (3.6) 4. The new prces have the ablty to push the demand n other countres over the capacty of producton. If ths s the case, the process s repeated untl a global equlbrum s establshed and all producton levels and prces are known. Ths algorthm reles on knowng consumpton, Z j, producton capacty,, product qualty q, transport costs, θd j, and cost prce, φ. Once complete, t returns producton, X, sale prce, φ, and the trade flows, Y j. To obtan a complete descrpton of trade evoluton, these nputs and a mechansm for dynamcs based on the outputs must be determned for each sector. We present ths n Sectons 5, 6 and 7 for the farmng, natural resources and manufacturng sectors respectvely. However, frst we brefly dscuss some estmates of the exogenous varables requred wthn the smulaton. 5
4 Intal setup To apply the trade flows algorthm we must determne the capacty, cost prces, ncome, qualty and transport costs for each country. Theoretcally, these could all be found from data sources, partcularly f the the model was dsaggregated to apply to ndvdual products, such as ol, wheat, etc. In the present settng however, our man nterest les n demonstratng the workngs of the system, rather than n obtanng realstc results, and so we proceed by outlnng some smple functons to approxmate the above varables. Workng through the varables n the order lsted above, we take one unt of food, raw materals or goods as the amount one worker can produce or extract n a gven tme perod. Usng ths formulaton, the producton capacty of a gven sector, (n), wll be equvalent to the labour force of that sector, L (n). Knowng the populaton, P, and the number of unemployed, u, from data, an ntal guess of the work force of each sector n each country may be made, provdng an ntal (n). Cost prces are sector specfc, but may be found from the producton capacty usng a smple expresson. The exact formulatons are outlned for each sector wthn the relevant sectons. The demonstraton model does ncorporate two readly accessble data sources: Gross Domestc Product (GDP) and dstances between the countres of consderaton calculated between centrods usng the haversne formula. From these, we may approxmate the remanng varables. Assumng that only the workers draw a salary, and that ncome s unform across the populaton, an ntal mean per-capta ncome may be found: ι = GDP P u. (4.1) Ths value (4.1) wll change durng the smulaton, as countres wth a strong economy wll nvest n mprovng ncome. The method appled to facltate ths s wthn the dynamcs secton, equaton (8.9). It s worth notng at ths stage, that the smple expresson (4.1) could easly be replaced wth an ncome dstrbuton whch would not effect the workngs of the algorthm. It s for smplcty n ths case that ths partcular form has been chosen. A measure of product qualty wthn a country could also be derved from the relatve ncome. We agan propose a smple expresson whch normalses (4.1) by lowest ncome of the countres ncluded: ι q = max(ι k ). (4.2) Wth ths formulaton, k s used as a dummy varable for and 0 < q 1 for all. Thus the rchest country ncluded wthn the model has q = 1. We leave the transport costs, θ (n), as a parameter to be calbrated wthn the modellng process. Generally speakng however, to ensure the argument of the exponental n (3.1) s dmensonless and O, we would expect θ (n) to roughly take the form: θ (n) 1 βmax(d j ) As mentoned above, the dstances, D j, between each country may be found from data, whle β s another parameter n (3.1) to be determned. Physcally, β quantfes the relatve mportance of prce, rather than qualty, to the buyer. These varable defntons provde enough nformaton to proceed to the ndvdual sector algorthms and flows, although one fnal adjustments s needed for the numercs. Wth the current (4.3) 6
defnton, the prcng and trade flows algorthm (3.3) - (3.6) has no mechansm to stop global producton capacty dppng below global demand. On occasons where ths occurs, demand must be scaled back. To do so, we propose the addtonal assumpton that n cases of global shortage, t s the poorest countres whch see the largest reducton n consumpton. Reducton n global demand s found by scalng accordng to ncome. To put ths more formally, whenever: Z (n) j > (n), (4.4) j the reducton n consumpton, Z (n) j, n country j wll be relatve to: r j = 1 ι j max(ι k ). (4.5) Ths sets r = 0 for the rchest country under consderaton, and allows the new consumpton vector to be found from the orgnal consumpton mnus the relatve reducton, tmes the total reducton: ( Z (n) j = Z (n) j r j k r Z (n) l ), (4.6) k It now remans to detal the full structure of the system for each ndvdual sector. In the nterests of clarty, we do so separately for each of the three sectors, n the followng three sectons. 5 The algorthm to determne farmng trade flows The method to determne the farmng trade flows s largely based around the prcng and trade flows algorthm of Secton 3. Frst we present some estmates for the sector-specfc varables of consumpton Z j and cost prce φ. These combned wth and θ dscussed n the prevous secton, may be passed to the trade flows algorthm (3.3) - (3.6) to provde values for the actual producton, X (bounded above by ), sale prce, φ (bounded below by φ ), and the trade flows matrx Y j. To proceed wth food consumpton n country j, we defne a parameter v j to descrbe the volume consumed per capta, per unt ncome. Ths sets the total consumpton n j as: Z j = v j ι j P j. (5.1) Although v j s country specfc, takng t as a constant across all models, as we do n our expermental run, mples that rcher countres wll consume more food than ther poorer counterparts. The cost prce per unt volume of food produced n s calculated from the total labour costs of farmng nfrastructure ι L, dvded by the expected producton output or sales: l φ ι L = X (t δt). (5.2) We base the expected producton output on the actual output of the prevous tme perod. Ths removes the need for an extra teraton, and seems reasonable when the tme perods under consderaton are small. Indeed, X (t) = X (t δt) n the lmt as δt 0. As dscussed, the current formulaton defnes one unt of food producton as the amount whch one worker can produce n one tme perod. Thus, t follows, that the producton capacty of the farmng sector s equal to the labour force of the sector. = L. (5.3) 7
It s ths sector specfc quantty, producton capacty, whch determnes whether the country s resource rch or poor and allows for our 2x2 classfcaton of countres. Equaton (5.3) has the drect result that (5.2) may be rewrtten φ ι = X (t δt). (5.4) Thus, cost prce s ncome scaled by the rato of the sales of the prevous tme perod, to the capacty at the present tme. Gven ths form for (5.4), t s necessary to defne a mnmum expected producton value. Otherwse a country wth lttle or no sales n the prevous tme perod wll have a very large or nfnte cost prce, makng sector growth extremely dffcult. The now known values of Z j,, φ, θ D j are fed nto the prcng algorthm of Secton 3, provdng results for X, φ and Y j : Z j, Wth a set of farmng flows Y j costs for mports, s gven by:, φ, θ D j X, φ Y j va equatons (3.3)-(3.6). (5.5) found, the prce whch the buyer j pays, ncludng transport D j θ + φ, (5.6) and the total spent by the populaton on food may be determned from the per unt prce of each mport, tmes the volume mported and summed over all countres (ncludng ones own): ζ j = ( ) θ D j + φ Y j. (5.7) As dscussed n Secton 1, any money left over wll be spent on goods. All remanng expendture s used, regardless of the volumes of goods the money buys. Thus we may defne each country s goods consumpton for the manufacturng sector n unts of money: ζ (3) j = GDP j ζ j. (5.8) There wll be some Z (3) j consumpton term assocated wth (5.8) n volume unts, although, as may be seen n Secton 7, ths does not feature explctly n the problem. If the populaton does not have enough money to feed tself, we want to avod the computatonally expensve addtonal teratve procedure of reducng consumpton and repeatng the prcng and trade flow calculatons. To do so, we ntroduce natonal debt, whch acts to keep track of any defct: = ζ(3) (sgn(ζ (3) ) 1). (5.9) 2 Ths structure s chosen as t s zero whle ζ (3) s postve, and equal to ζ (3) f ζ (3) s negatve. Natonal debt s pad for by the profts of the three producng sectors, detals of how ths mechansm s appled are n Secton 8. 5.1 The accounts for the farmng ndustry Wth all flows and sales prces determned, t s now possble to determne all costs, takngs and profts for the farmng ndustry as follows. 8
Costs of producton (only labour costs wthn ths sector) are gven by, C Money taken/sales (sale prce per unt tmes no. unts): Proft from farmng sector (sales - cost) s, 5.2 A fnal pont on the farmng flows = ι. (5.10) χ = φ X. (5.11) Π = φ X ι (5.12) It s worth notng that the total sales χ gven n (5.11) wll take one of two less general forms due to a subtlety n the prcng and trade flows algorthm of Secton 3. The mechansm of that secton reles on a two types of seller. If the frst type, country wll sell below capacty, X <, at cost prce φ = φ. In ths case, gven (5.4), the total sales of (5.11) becomes: χ = ι If the second type, wll sell at capacty X In ths case, (5.11) becomes: χ = X X (t δt) (5.13), at a hgher prce than cost, φ > φ. = φ X. (5.14) Ths mples two correspondng forms for proft, orgnally defned n (5.12). The profts of the frst type, lke (5.13) wll take the form ( ) Π X = ι X (t δt) 1, (5.15) and s postve f sales exceed those of last year, negatve otherwse. Meanwhle, the second type uses (5.14) to make (5.12): ) Π = X (φ ι, (5.16) and profts are bult from actual sales prces above the natonal ncome (theoretcally equvalent n ths case to the actual per-unt cost prce as n (5.10). In both cases (5.15) and (5.16) are lnearly related to the sze of the current nfrastructure. It s ths feature whch gves the dynamcs a logstc, or Lotka-Volterra form, as dscussed n Secton 8. 6 The algorthm to determne the natural resources trade flows The method to determne the natural resources flows largely mrrors the algorthm appled n the prevous secton to farmng flows. For the sake of clarty, we nclude a bref descrpton of the full process here. To begn, we must derve expressons for the consumpton and cost prces. Snce the sole customer of the natural resources sector s the manufacturng sector, the raw materals consumpton n country j s lnked to manufacturng producton capacty. We ntroduce m j so 9
that the consumpton s equal to the volume requred per unt producton tmes manufacturng producton capacty. Z (2) j = m (3) j j (6.1) The per-unt cost prce of raw materals n, n unts of money s the total labour cost of mantanng the mnng nfrastructure, dvded by the expected producton output (based on last years actual producton). φ (2) = X (2) ι (2) (t δt) Agan, as n the farmng flows, a mnmum expected producton value s found to be necessary to avod a large cost prce n small sectors. The trade flows are calculated n an dentcal procedure as before: Z (2) j, (2) (2), φ, θ (2) D j, are fed nto the spatal nteracton model outlned n Secton 3, provdng results for X (2), φ (2) and Y (2) j. Z (2) j, (6.2) (2) (2), φ, θ (2) D j X (2), φ (2) Y (2) j va equatons (3.3)-(3.6). (6.3) Once the flows and prces have been determned, the total per-unt prce to the buyer, ncludng transport costs s: D j θ (2) + φ (2), (6.4) whch gves the money spent on raw materals by the manufacturng ndustry n j as the prce per unt for each mport tmes the volume mported, summed over all countres ncludng one s own: ζ (2) j = ( ) D j θ (2) + φ (2) Y (2) j. (6.5) The total spend on raw materals, gven by (6.5), wll feature n the manufactured goods prcng of Secton 7. 6.1 The accounts for the natural resources sector Agan, ths follows the workngs of the farmng sector, so that costs of producton are determned only by labour costs: C (2) = ι (2). (6.6) Money taken/sales, s sale prce per unt tmes no. unts: Proft from mnng sector (sales - cost) s χ (2) = φ (2) X (2). (6.7) Π (2) = φ (2) X (2) ι (2). (6.8) As n (5.15) and (5.16), ths proft wll take one of two forms, both lnearly related to (2). 7 The algorthm to determne manufacturng trade flows Goods consumpton n country j s calculated from the per capta ncome mnus food purchases. We assume that people spend all ther dsposable ncome on goods, regardless of what volume of goods ths buys them. The result, n unts of money s gven n equaton (5.8) and repeated here for clarty: ζ (3) j = GDP j ζ j. (7.1) 10
In the stuaton where a country cannot afford to feed themselves, there wll be no remanng ncome to spend on manufactured goods. In such a case, the volume of food requred s bought regardless, and a natonal debt term s ntroduced (see equaton (5.9)). To allow for ths scenaro, we rewrte the amount of money each country has to spend on goods as: ζ (3) j = ( GDP j ζ j where, by defnton, debt,, may only be zero or postve. ) (1 sgn( )), (7.2) Wthn the manufacturng sector, the consumpton (7.2) s expressed n terms of money, so that volumes do not feature explctly n the problem. Thus, nstead of applyng the prcng and trade flows algorthm to solve the system ((3.3)-(3.6)) as before, an adapted verson must be appled to take nto account ths specal case of consumpton n unts of money. A per-unt cost prce s stll requred however, where the form dffers from that seen n farmng (5.4) and mnng (6.2) to allow for the natural resources costs to mantan the manufacturng nfrastructure. Ths cost was found n the mnng flows n (6.5). Thus the per-unt cost prce of goods n, n money unts, s: φ (3) = ι (3) + ζ (2) X (3) (t δt). (7.3) The adapted prcng and trade flows algorthm determnes an ntal demand on sales, by applyng the standard spatal nteracton model: χ (3) = j ζ (3) j q α exp[ β(θ(3) D j + k qα exp[ β(θ(3) D k + φ (3) )] (7.4) (3) φ k )]. Any countres where demand exceeds capacty are enttled to rase ther sales prces. Ths prce adjustment takes the form: m s.t χ (3) (3) (3) m > φ m m (7.5) [ ] φ (3) m = 1 β ln 1 + 1 φ (3) (3) m β ln ζ (3) j q α exp[ βθ(3) D j ] j k qα exp[ β(θ(3) (3) D k + φ k )], (7.6) whch s solved teratvely as φ (3) m appears on the rght-hand sde of (7.6). These adjusted prces form part of the latest prcng vector {φ (3) }, on whch the spatal nteracton model s re-run: χ (3) = j ζ (3) j q α exp[ β(θ(3) D j + φ (3) )] (7.7) k qα exp[ β(θ(3) D k + φ (3) k )]. The new prces {φ (3) } have the ablty to push other countres over producton capacty, thus the process s repeated untl a global equlbrum s establshed and all prces are known. Fnally, the flow n money unts from j to s gven from: ψ (3) j j q α exp[ β(θ(3) D j + φ (3) )] (7.8) k qα exp[ β(θ(3) D k + φ (3) k )], = ζ(3) and actual producton may be found from X (3) = χ(3) φ (3). (7.9) 11
Despte beng a money flow, the prce term and qualty term allow the flows to dfferentate between large volume, cheap, low qualty exports and expensve, low volume hgh qualty exports - the buyer s always drven towards the best deal. What s meant by best of course, s dctated by the calbraton of the Lagrangan multplers α and β. 7.1 The accounts for the manufacturng ndustry Costs of producton (wthn ths sector costs nclude both labour and raw materals): Money taken/sales (here of course, χ (3) nonetheless): C (3) Proft from manufacturng sector (sales - cost): = ι (3) + ζ (2). (7.10) s known from the flows, but the followng should balance χ (3) = φ (3) X (3). (7.11) Π (3) = φ (3) X (3) ι (3) ζ (2). (7.12) The last term here s not qute lnearly related to and the profts of the manufacturng sector do not take the same form as those seen prevously. 8 The Dynamcs The proft of all sectors s now known and t remans to renvest any profts to ncrease the capacty of each naton s best performng sectors. Before we do so however, any natonal debt accrued must be accounted for, as other countres have already been pad for the food whch the populaton of bought. The total surplus of all ndustry n s the proft from each sector (whch could be negatve) plus any natonal debt: (3) S = Π + Π (2) + Π (3). (8.1) Although we use the term surplus, S could be negatve. As all surplus s renvested, t would be possble at ths stage to ntroduce natonal or nternatonal nvestment, and the man structure of the dynamcs would reman unchanged. For the purposes of our demonstraton model, we neglect ths addtonal varable and leave surplus as defned n (8.1). Assume that each country, and all sectors wthn t, s governed by one central body whch renvests n a manner best for everyone by splttng the surplus between sectors accordng to the performance n the latest tme step. Ths renvestment takes the form of both addtonal employment and wage ncrease. Such assumptons could well be adjusted at a later date, to nclude shareholders or allow for frms, for example. However ths smplfyng assumpton works well for the purposes of our demonstraton model. Wth statc populatons, a country may not employ more people than the current unemployed and and cannot fre more people than the employed. So, the extra number of people who may be employed (or fred) s gven by E = 1 [ ] 2 (1 + sgn(s ɛ1 S )) mn, u 1 [ ] ι 2 (1 sgn(s ɛ2 S )) mn, P u, (8.2) ι where, ɛ 1 and ɛ 2 are parameters to be calbrated. 12
The algorthm to assgn ths change n labour force to each sector s based on the sectors contrbuton to the overall profts δl (n) = δt Π (n) k Π(k) E, (8.3) although, naturally ths must be adjusted f t yelds any negatve results snce a country cannot have a negatve work force. Ths translates drectly to a change n producton capacty n the current model, snce one unt of volume s equvalent the amount one worker can produce n a tme perod. Thus: δ (n) = δl (n). (8.4) Note that the extra employment, E, n (8.2) takes the same sgn as the country s overall profts k Π(k). If both are postve - that s f the country s n proft overall - any sector wth losses Π (n) wll stll see a shrnkng workforce, and vce-versa. Even f a country does badly, the mechansm (8.3) allows growth n a thrvng sector. In addton, gven the two forms of profts dscussed n Secton 5, equatons (5.15) and (5.16), the dynamcs of producton capacty 8.3 wll take one of two forms wthn the farmng sector. These wll be, δ δ = δtι = δt ( X X (t δt) 1 ) (φ ι E k Π(k) ) E k Π(k) to correspond wth (5.15) and (5.16) respectvely. Both of these, (8.5) and (8.6), are therefore n Lotka-Volterra form. Ths s also true for the natural resources sector, as (5.15) and (5.16) apply when n = 2. For manufacturng however, the change n unt for the constrant on producton capacty, and the nteracton wth the farmng sector leads to a slghtly dfferent form. Specfcally, combnng (7.3) (7.12) and (8.3) gves the two possble forms as: δ δ (3) (3) = δt = δt E k Π(k) E k Π(k) { ι (3) { (3) ( X (3) X (3) (t δt) 1 ) } (φ (3) ι ζ (2) These expressons may be consdered as forced Lotka-Volterra. ) + ζ (2) X (3) (t δt) The natonal ncome also has a dynamc element whch n turn affects both consumpton and qualty of products: δι = δtɛ 3 S (8.9) where ɛ 3 s a fnal parameter to be calbrated. Fnally, latest GDP may be found as the sum of all ncomes GDP = ι L (n) (8.10) and relatve qualty q may then be recalculated. The entre process may be repeated for subsequent tme steps. n } (8.5) (8.6) (8.7) (8.8) 13
(a) (b) Fgure 3: A quanttatve measure of beneft to a potental buyer aganst the parameter α. See equaton (9.1). 9 Expermental results The model generates an nterestng balance of factors between countres. A poor country has the beneft of low cost of producton due to lower wages, but (wth the current assumptons on qualty (4.2)) lower qualty goods. The relatve mportance of these two factors to the buyer s determned by the weghtng parameters α and β. In lne wth the dervaton of the spatal nteracton model of (3.1), ths beneft to a potental buyer may be quantfed as follows: Beneft = exp (α ln q j βφ j βθd j ). (9.1) Explorng the relatonshp between prce and qualty further, some plots are presented n Fgure 3 of total beneft for a range of α values, at a fxed β. In the plots, we have taken θ = 0, so that dstance s deemed unmportant to the buyer. In realty, these curves would be dstorted by θ, and dfferent for each potental buyer. The examples chosen llustrate that there s a value of α below whch cheaper tems are valued, and above whch, qualty becomes more mportant. The countres selected form two pars: Brazl and Australa, whch are both bg exporters of coal, tmber and ron ore; and Ngera and UAE whch both export petroleum products. The mean ncome of UAE s 18 tmes that of Ngera, whle Australan ctzens earn an average of 3.55 tmes more than Brazlans. Thus the four demonstrate examples of GDP rch (Australa, UAE) and poor countres (Brazl, Ngera). In contrast, the producton capacty (or equvalently, work force) of Brazl s fve tmes that of Australa: resource rch and resource poor countres respectvely. By our measure of relatve qualty gven n (4.2) Australa has q = 0.7, Brazl q = 0.20, UAE q = 0.7 and Ngera q = 0.03. Of course, a large varance n the qualty of ol such as these s unrealstc, although n the context of our toy model the results serve the purpose of demonstratng the nterestng potental bfurcatons whch the algorthm generates. In all plots, beneft tends to zero wth large α, although ths happens much faster for low qualty products, such as those from Ngera and Brazl. Beneft s not the only factor n the trade model. Asde from dstance between buyer and seller and cost to transport goods, consumpton - both local and global - plays a key role n contrbutng to the demand on a natons produce. In Fgure 4, we present a plot of the demand (as a rato of capacty) on the natural resources products of the same four countres lsted above, when offerng goods at cost prce. Ths s equvalent to the sum over j of the trade flows at the 14
Fgure 4: The demand on the natural resources produce for varous values of alpha, normalsed by producton capacty. In the above plots, θ = 25, β = 5x10 6. very frst step n the prcng and trade flows algorthm - equaton (3.1). In ths example, θ = 25 and so dstance s taken nto account n the buyng process. The plots agan demonstrate that hgh values of α correspond to qualty beng favoured, wth the products of UAE and Australa ncreasng n demand. Gven that qualty and cost prce of UAE and Australa are smlar, the bgger demand on UAE s probably due to the large dstances between Australa and ts potental consumers. The relatve demand of Ngera and Brazl s at a maxmum when α = 0, and prce domnates the decson process. The demand n both cases, however tends to zero wth ncreasng α. If demand exceeds supply, the countres may rase ther sale prce untl producton equals capacty. Our fnal plot of ths type then s gven n Fgure 5 and shows the rato of sale prce to cost prce for the same range of α and values of β, θ. As demand on Brazl never exceeds capacty n ths smulaton, they contnue to sell at cost prce, regardless of the value of α. Ths s not the case for Ngera, where other countres are prepared to pay up to 140 tmes the cost prce for low values of α. As the mportance of qualty, along wth α ncreases, demand n Ngera drops to below capacty, and they must sell at cost prce. For UAE and Australa however, the relatve sale prce, lke demand ncreases wth α. To emprcally valdate ths model, ts prcng algorthm, sector accounts and dynamcs these parameters α, β and θ wll have to be calbrated aganst data. In practce, the dynamcs wll be determned by a number of adjustments; the tme path wll be very much a functon of the ntal condtons, and varables at each stage. To explore the system, along wth model refnements and extensons s our next task, but to demonstrate the goal of the demonstraton model we nclude a map of smulated flows for the farmng sector n Fgure 6. The key s gven wthn the capton of that fgure. As one would expect, countres wth hgher GDP tend to produce a smaller percentage of ther own food, and consume more per capta. In addton, trade lnks tend to form over shorter dstances whch matches wth the results of other models. For example the gravty model. In general then, a great deal of work needs to be done to explore the feasble solutons, however, ths frst stage smulaton looks to be very promsng. 15
Fgure 5: The actual sales prce of natural resources for varous values of α. 10 Future work We beleve that the model of trade flows presented here offers a good alternatve to exstng models of trade, not least for the varous extensons and adaptatons whch the method can handle. Perhaps most mportantly, the formulaton allows us to pose the dynamc programmng queston: f we add, say, World Bank nvestment by country, and an objectve f formulated to ncrease GDP per capta n a number of poor countres, what s the optmal path towards the objectve? Indeed, s there a feasble soluton? The method also allows for varable mgraton, and connectng to an equally well developed model of mgraton should be relatvely straghtforward. Gven the growng mpact of pracy on trade flows, the model here could also be adjusted to test varous scenaros, ncludng the reslence of global flows to attacks on a gven shppng route. Along these lnes, the total money spent on shppng wthn the model s gven by the relatvely smple expresson: j Y j D j θ + Y (2) j D j θ (2) + ψ(3) j φ (3) θ n D j (10.1) so that ntroducng a global shppng company nto the model would be possble. Indeed, any number of frms or stakeholders could be ntroduced - the only adjustment beng n the renvestment dynamcs of Secton 8. These extensons demonstrate the flexblty of the model, however the next step remans clear: to emprcally valdate ths model, calbrate the parameters aganst real data and to explore the dependence of ntal condtons on the dynamcs. References [1] Rees P. Champon T. Kalogrou S. Fotherngham, A. S. and A. R. Tremayne. The development of a mgraton model for england and wales: overvew and modellng out-mgraton. Envronment and Plannng A, 36:1633 72, 2004. [2] B. Harrs and A. G. Wlson. Equlbrum values and dynamcs of attractveness terms n producton-constraned spatal-nteracton models. Envronment and plannng. A, 10:371 88, 1978. 16
Fgure 6: Some frst results of smulaton for the food flows. Here the node sze s the consumpton per person, edge wdth s the sze of the nter-country flows, and node colour s the percentage of consumpton met locally (see scale) 17
[3] J. H. Rho, D. E. Boyce, and T. J. Km. Comparson of soluton methods for wlson s nterregonal commodty flow model. Geographcal Analyss, 21(3):259 267, 1989. [4] J. Stllwell. Interzonal mgraton: some hstorcal tests of spatal-nteracton models. Envronment and Plannng A, 10:1187 200, 1978. [5] F. Wllekens. Modelng approaches to the ndrect estmaton of mgraton flows: from entropy to em. Mathematcal Populaton Studes, 7:239 78, 1999. [6] A. G Wlson. Interregonal commodty flows: Entropy maxmzng approaches. Geographcal Analyss, 2(3):255 282, July 1970. [7] A. G. Wlson. Urban and regonal dynamcs from the global to the local: herarches, dna, and genetc plannng. Envronment and plannng B, Plannng and Desgn, 37(5):823 837, 2010. 18