A hydrodynamic model for cooperating solidary countries
Roberto De Luca, Marco Di Mauro, Angelo Falzarano, Adele Naddeo
aa r X i v : . [ q -f i n . GN ] J u l A hydrodynamic model for cooperating solidary countries
Roberto De Luca , Marco Di Mauro , Angelo Falzarano , and Adele Naddeo Dipartimento di Fisica E.R.Caianiello, Universit´a di Salerno, Fisciano (SA) - 84084, Italy Dipartimento di Scienze Economiche e Statistiche,Universit´a di Napoli Federico II, Napoli - 80126, Italy and INFN Sezione di Napoli, Napoli - 80126, Italy
The goal of international trade theories is to explain the exchange of goods and services betweendifferent countries, aiming to benefit from it. Albeit the idea is very simple and known since ancienthistory, smart policy and business strategies need to be implemented by each subject, resulting ina complex as well as not obvious interplay. In order to understand such a complexity, differenttheories have been developed since the sixteenth century and today new ideas still continue to enterthe game. Among them, the so called classical theories are country-based and range from Absoluteand Comparative Advantage theories by A. Smith and D. Ricardo to Factor Proportions theory by E.Heckscher and B. Ohlin. In this work we build a simple hydrodynamic model, able to reproduce themain conclusions of Comparative Advantage theory in its simplest setup, i.e. a two-country worldwith country A and country B exchanging two goods within a genuine exchange-based economy anda trade flow ruled only by market forces. The model is further generalized by introducing moneyin order to discuss its role in shaping trade patterns. Advantages and drawbacks of the model arealso discussed together with perspectives for its improvement. PACS numbers: 89.65Gh
I. INTRODUCTION
International trade theories help to elucidate the ba-sis for trade and the gains from trade, but also, how thegains from trade are generated, and how are they largeand how are they divided among the trading nations.Theories also explain the pattern of trade. That is, whichcommodities are traded and which are exported and im-ported by each country. Presumably a nation will volun-tarily engage in trade only if it gains from import-export.Although the idea is very simple and known since ancienttimes, the policies and intelligent strategies adopted bydifferent countries move on dynamics not always inter-pretable preventively. Even more so today, indeed, wherethe global trade is governed by multiple and changing fac-tors, often combined in a subtle way and often demandingan explanation.To understand this complex scenario, the approachesare different with different results since the sixteenthcentury. The Leitmotiv of these contributions is thelaw of comparative advantage [1], which is still one ofthe most important and unchallenged laws of economics,with many practical applications to nations, as well asto individuals. The Comparative Advantage (CA) is alsostudied in terms of the opportunity cost theory, as re-flected in production possibility frontiers or transforma-tion curves.Smith’s theory [2, 3] for the first time considered theability of a country to a better efficiency in the produc-tion of a good with respect to a second country. In asimple but ideal two-country world, that would result inan advantage for the first country, which could special-ize in producing that good. At the same time, if thesecond country would be more efficient in producing an-other good, it could specialize in this effort. In this way a better efficiency makes production to increase and tradeto flow according only to market forces with a clear bene-fit for people in both countries. In conclusion, accordingto Smith’s theory both countries gain from exchange andtheir wealth can be measured in terms of the living stan-dards of the local population. The main drawback in theabove reasoning was essentially due to the possible occur-rence of one country in the network without any absoluteadvantage in producing a good and the other efficient inproducing both goods. This issue found a satisfying solu-tion in the CA theory by Ricardo [4], whose main featurewas the international immobility of production’s factors.Indeed factors were considered as completely immobileamong countries, while goods showed a complete mobil-ity and a constant unit cost. In this context a compara-tive advantage takes place if a country can be more effi-cient in producing a particular good with respect to othergoods. As a consequence, it will specialize in the produc-tion of that good and this makes international trade todepend on a difference in the comparative cost of pro-ducing goods. The net result is that each country shouldproduce and export the goods in which it has a compar-ative advantage while importing those goods in which ithas a comparative disadvantage within an environmentcharacterized by open and free markets. This is clearly adrawback, which was addressed by Heckscher and Ohlin[5] theory. According to such a theory, countries differwith respect to production factors, such as land, laborand capital, whose cost depends on supply and demand.In this framework, a country specializes in the produc-tion and export of goods characterized by a great supplyand cheaper production factors, while establishes to im-port goods being in short supply. Another drawback ofCA theory is that it provides a static framework; in thisrespect one may wonder how trade patters change as afunction of time. To this and other issues, such as thepossibility to introduce further factors in the understand-ing of international trade flows, are devoted modern tradetheories [6]. As pointed out by Helpman [7][8] and Krug-man [9][10][11], modern theories have been developed inorder to take into account mainly the increasing of thetrade to income ratio, the concentration of trade flowsas well as the expansion of intra-industry trade amongindustrialized countries. According to Markusen [12][13]the expansion of intra-industry trade could be ascribedto the increasing of the demand for differentiated prod-ucts with respect to that for homogeneous ones, while amodel developed by Spence [14] and Dixit and Stiglitz[15] focused on the link between trade and consumption.In this way policy changes may be recognized as the driv-ing force behind the increase in trade volume. Finally themeaning and the role of CA idea in monetary economieshas been discussed as well, together with its effect onthe long-run equilibrium pattern of trade [6][16][17]. Inthis context a direct impact of exchange rates changes ontrade balance has been found and its effect on exchangerate policy widely investigated [18][19][20][21].In summary, Ricardo’s ideas represented a fundamen-tal starting point for the development of modern theoriesof international trade, even if it is clear that the matteris really complex and characterized by networks of morethan two countries and two goods to exchange and bythe interplay of a lot of different variables. Indeed Com-parative Advantage may still provide the underlying ideafor the optimal allocation of any country’s resources andthe maximization of world welfare, and the consequencewould be that the benefits of free trade outweigh the costs[22], even if today it is believed that international tradecouldn’t find a complete explanation within a single the-ory but it may require resorting to different ones at thesame time.Among the physical models that can be employed todescribe this kind of situations, there are flow models.The first such model was introduced by the French physi-cian Quesnay [23], who was inspired by the analogy withcirculatory flow of the human blood. In this model, thereis an economic equilibrium which is stationary, closed andwithout distinction between productive factors and pro-duced goods. The only source of wealth is agriculture(as appropriate for those times), and its products arethen freely distributed among the other social classes.This approach has been generalized in the 19th centuryby Fisher, who recognized that the production cycle ofwork and goods should be complemented by a second,monetary circuit. This has been the basis for more re-cent work (see e.g. [24] and references therein), whereFisher’s ideas are cast into thermodynamical language.In particular the production and the money in circle arethe analogs of work and heat exchange respectively.In this work, in the same spirit of modeling economicbehavior using general physics, we build a simple hy-drodynamic model, based on a leaking bucket analogyintroduced in previous works [25, 26], able to reproduce the main conclusions of CA theory in its simplest setup,i.e. a two-country world with country A and country B exchanging two goods within a genuine exchange-basedeconomy and a trade flow ruled only by market forces.Production and exchange costs are introduced in order toevaluate the convenience each country has in producingor sharing a good. Explicit analytic solutions of the dy-namical equations are given for different exchange paths(i.e. for different exchange functions). A fixed pointanalysis is carried out and a region in the parameters’space is found, where the time evolution of the amountsof money for both countries shows a lower bound. Thisallows us to understand how money affects trade and, ingeneral, to investigate the role of CA law within mon-etary economies. Finally, advantages and drawbacks ofthe model are discussed together with perspectives for itsextension.The paper is organized as follows.In Section 2 we introduce the model system understudy within the CA theory framework, i.e. a two-country world with country A and country B exchangingin principle different goods. We adopt a communicatingvessels scheme, with each vessel behaving as a leakingbucket [25]. The exchange of goods between the coun-tries is modeled with a simple superposition of Heavysidetheta functions in order to get an analytic solution. InSection 3 some simple examples are discussed in order toillustrate the usefulness of the model. In Section 4 themodel is augmented with the introduction of productionand exchange cost, in order to be able to evaluate theconvenience for each country in producing or sharing agood. Finally, in Section 5 conclusions and perspectivesof this work are presented. II. THE MODEL
The aim of this Section is to reproduce the main resultsof classical CA theory in its simplest setup, i.e. a two-country world with country A and country B exchangingin principle different goods.Suppose two countries A and B are both producers ofcertain goods, and that they are willing to share their ex-cess production of the latter. More precisely, they agreeto share these goods only when their internal demand canbe satisfied and a certain reserve is secured. Notice thatthis is a simplified model, in that it doesn’t account forother macroeconomic variables, such as the consumers’different preferences, the role of brand names and prod-uct reputations in buyers’ decisions, the temporary ad-vantage gained as a consequence of the development ofa novel technology, etc. Furthermore in this Section welimit ourselves to a genuine exchange model, without in-troducing prices in the analysis. This more complicatedissue will be addressed in Section 4.This situation can be modeled, for any of the sharedgoods, using a communicating vessels scheme [25]. Sinceat this level the dynamics related to different goods are FIG. 1: Communicating vessel scheme for production, con-sumption, and exchange of a given good k for countries A and B . decoupled, we consider the case of just one good. In thisscheme, the volume of the liquid in each vessel representsthe amount of the good that each country holds. Let usdenote by P A , P B the rates of production of the goodin countries A and B respectively, and let C A , C B bethe rates of consumption. Moreover, let σ be the rateof exchange. It is assumed that each country starts ex-changing after the reserve of the good reaches a giventhreshold. In the communicating vessel scheme this isdenoted by S A h or S B h , where S A,B are the sectionsof the vessels and h is a given critical height. Since thereis no immediate significance for the sections of the vesselsin the economic context, in the following we assume forsimplicity that S A = S B = 1.The dynamics of the system is then given by the fol-lowing equations P A − C A − σF ( h A , h B ) = · h A (1) P B − C B + σF ( h A , h B ) = · h B . The function F models the exchange between the coun-tries. We choose F ( h A , h B ) = (cid:18) h A h − (cid:19) θ (cid:18) h A h − (cid:19) − (cid:18) h B h − (cid:19) θ (cid:18) h B h − (cid:19) (2)where θ ( x ) is the unitary step function, whose rˆole is toensure that the exchange is turned on only when one ofthe nations reaches the critical height. Note that this isnot an empirical type of law. Rather, with this choice weare assuming that the agreement between the countries is such that the effective exchange rate σF is linear inthe excess product if the exchange is unidirectional, andin the difference of excesses if the exchange is bilateral.We assume consumption rates to be constant. Sucha choice is not in line with a strict hydrodynamicmodel since in that case Torricelli’s law would apply, i.e. C A,B ∼ p h A,B .Before going on it is convenient to normalize all thevariables by defining˜ P A,B = P A,B h , (3)˜ C A,B = C A,B h , (4)˜ σ = σh , (5) η A,B = h A,B h . (6)Since in the following there is no risk of confusion, weshall omit the tildes. In what follows, we shall assumethat also P A,B and σ A,B are constants. In the new vari-ables, the dynamical equations (1) read: P A − C A − σf ( η A , η B ) = · η A (7) P B − C B + σf ( η A , η B ) = · η B . where f ( η A , η b ) = ( η a − θ ( η A − − ( η B − θ ( η B − f ( η A , η B ) = η A , η B < η A − η A > , η B < − ( η B −
1) if η A < , η B > η A − η B if η A > , η B > a. η A , η B < , f ( η A , η B ) = 0 . This is the simplestcase since there is no exchange and the two subsystemsare decoupled. The equations read: P A − C A = · η A (9) P B − C B = · η B (10)and the solution is: η A ( t ) = ( P A − C A ) t + A (11) η B ( t ) = ( P B − C B ) t + B , (12)where the integration constants A and B are to be de-termined at the time where this condition sets in. Thissolution is valid until η A or η B reaches the value 1, whereexchange enters the game. b. η A > , η B < , f ( η A , η B ) = η A − . In thiscase country A , being above the threshold, has startedexchanging goods with country B , whose dynamics istherefore driven by that of country A . The dynamicalequations read: P A − C A − σ ( η A −
1) = · η A (13) P B − C B + σ ( η A −
1) = · η B . (14)The first equation is independent of B . It can be solvedto yield: η A ( t ) = P A − C A + σσ + A e − σt (15)By substituting this solution into the second equation(14) we get · η B = P B − C B + P A − C A + σA e − σt (16)whose solution is: η B ( t ) = ( P B − C B + P A − C A ) t − A e − σt + B (17)As in the previous case, the integration constants mustbe determined in the instant of time where this regimestarts. c. η A < , η B > , f ( η A , η B ) = − ( η B − . Thedynamical equations read: P A − C A + σ ( η B −
1) = · η A (18) P B − C B − σ ( η B −
1) = · η B . (19)from which it is clear that the solution can be obtainedfrom the one found in the previous case by exchanging A with B . This is to be expected since in this case itis country B that is above the threshold and shares itsexcess goods with country A . d. η A > , η B > , f ( η A , η B ) = η A − η B . In thiscase both countries are above threshold, so that exchangegoes in both ways. The dynamical equations read: P A − C A − σ ( η A − η B ) = · η A (20) P B − C B + σ ( η A − η B ) = · η B . (21)We can rewrite this system using vector notation, setting η = (cid:18) η A η B (cid:19) , P = (cid:18) P A − C A P B − C B (cid:19) (22)and M = σ (cid:18) − − (cid:19) (23)so that · η + M η = P . (24)The matrix M is singular, thus one eigenvalue is zero,the other eigenvalue being 2 σ . The system can be easilysolved, but it is useless to write down the explicit solu-tions. III. A SIMPLE EXAMPLE
In order to illustrate the usefulness of the model, inthis section we provide a simple example.Suppose that the two countries agree to maintain theproduction rates P A and P B at high enough levels so thata steady state of equations (7) is reached under conditions d . Still, the two countries would like not to overproducethe shared goods.Suppose that the production capability of country A ishigher than the production capability of country B . Thetwo countries may then agree to reach a steady statedefined by η A = 1 + ǫ, (25) η B = 1 (26)Then Eqs. (7) under steady state conditions tell us that P A = C A + ǫσ (27) P B = C B − ǫσ (28)which means that country A can make an effort to pro-duce more than what expected to reach a stationary statewith η A = η B = 1, while country B can slow down itsproduction rate. Of course, in order to compensate forthe effort of country A , country B can agree to make aneffort on the production of a different good. This possi-bility will be explored in the next section.From the above example it is clear that each countrybenefits from trade, also in the presence of a productivitygrowth in only one country. IV. INTRODUCING MONEY
In order to evaluate the convenience the countries havein producing or sharing a good we need to introduce theproduction and exchange costs. This is the topic of thepresent Section, which is devoted to a generalization ofthe previous model. In this way the role of money inshaping trade patterns will be pointed out clearly.More precisely, we denote by x A and x B the productioncosts of the given good for both countries. We assumethat x A < x B , in order to take into account the fact thatcountry A has more capability of producing the good.Also, let y A and y B be the selling prices of the good onthe internal markets of the two countries.Despite international trade is also determined by var-ious endowments of the countries, it remains implicitlysolidary. Indeed, within this cooperating solidary coun-tries environment, each country tends to export goodswhose production uses more intensively those factors thatare relatively more abundant in the country, typical of thecomparative advantage. As anticipated in the Introduc-tion, the comparative advantages are determined by therelative abundance of production factors and productiontechnologies (the relative intensity with which produc-tion factors are used in the different sectors). Each coun-try will tend to produce intensive goods in the factors ofwhich it is relatively well-equipped. The country where afactor is relatively abundant exports goods whose outputis relatively intensive in that factor and, on the contrary,it imports goods that are relatively intense in the rela-tively low production factor in the country. This complexdynamics, according to Heckscher-Ohlin’s model [6], ininternational trade leads to a convergence of the relativeprices of traded goods. So, there is a direct relationshipbetween relative prices of goods and factor prices, tradealso leads to price factor equilibrium. In some respects,similar conclusion also comes from Samuelson’s theoryof International Trade and Equalization of Factor Prices[27, 28]. On the contrary, ” price differences across coun-tries are determined by trade barriers and by a country’sspecialization in production. ” [29].After the two countries start exchanging the good,these two relative prices converge to a common value y ,to which also the price of the good in international tradetends. We consider it to be the case that such a conver-gence has already happened. We assume moreover thatinternational agreements are such that x A < y < x B (29)i.e. country B finds itself in the situation that the pro-duction cost of the good exceeds the cost of the good onthe international market. Then country B can decide tostop producing the good and cover its necessities by im-porting it from country A . In this case we are clearly incase b , i.e. η A > η B <
1. The situation is described byfive variables, P A , P B , η A , η B and σ . Moreover there arethe fixed parameters x A , x B , y , C A and C B . Supposingwe are at a fixed point, consistently with our hypothesisthat all transients are gone, the equations for the quan-tities η A and η B are P A − C A − σ ( η A −
1) = 0 , (30) P B − C B + σ ( η A −
1) = 0 . (31)These two equations reduce the number of parameters ofthe problem from 5 to 3. In the following we consider η A , η B and σ as independent variables (with the constraint η A > η B < P A and P B are determined by theabove fixed point equations.Let us denote by m A and m B the amounts of moneyfor countries A and B . The equations which describetheir variations are: dm A dt = − x A P A + y [ C A + σ ( η A − dm B dt = − x B P B + y [ C B − σ ( η A − . (33)The conditions we seek are dm A dt ≥ dm B dt ≥ − x A P A + y [ C A + σ ( η A − ≥ − x B P B + y [ C B − σ ( η A − ≥ . (35) Solving the fixed point equations (30) and (31) with re-spect to the P ’s and substituting in our conditions weget ( y − x A )[ C A + σ ( η A − ≥ y − x B )[ C B − σ ( η A − ≥ . (37)Since C A + σ ( η A − ≥
0, and by hypothesis − x A + y ≥ C B − σ ( η A − ≤
0, since by hypothesis − x B + y ≤
0. On the other hand the left hand side ofthis inequality is a non negative quantity being nothingbut P B , therefore our condition can be satisfied only if C B = σ ( η A −
1) (38)i.e. if P B = 0. Thus the only way for country B not tolose money is to completely stop production of the good,importing from country A all the amount consumed. Ifcountry B wants to have a positive income, i.e. dm B dt ≥ x A > y > x B (39)so that it is country A that imports the second good fromcountry B , so that a symmetric situation is achieved,described by the equations dm A dt = − x A P A + y [ C A + σ ( η A − − x A P A + y [ C A − σ ( η B − , (40) dm B dt = − x B P B + y [ C B − σ ( η A − − x B P B + y [ C B + σ ( η B − . (41)for the amounts of money, while the fixed point equationsare P A = C A + σ ( η A − , (42) P A = C A − σ ( η B − , (43) P B = C B − σ ( η A − , (44) P B = C B + σ ( η B − , (45)where the indices 1 and 2 refer to the two different goods.Thus the time evolution of the amounts of money for thetwo countries at the fixed point are described by dm A dt = ( − x A + y ) P A + ( − x A + y ) P A , (46) dm B dt = ( − x B + y ) P B + ( − x B + y ) P B . (47)Putting α = − x A + y > , (48) α = − x A + y < , (49) β = − x B + y < , (50) β = − x B + y > . (51)the inequalities dm A dt ≥ dm B dt ≥ α − α ≥ P A P A , (52) β − β ≥ P B P B (53)Explicitly, the two inequalities read C A − σ ( η B − C A + σ ( η A − ≤ − y − x A y − x A , (54) C B − σ ( η A − C B + σ ( η B − ≤ − y − x B y − x B . (55)Actually, in international trade agreements participat-ing countries are required to match their respective tradebalances. In the case of a two side agreement the tradebalances are equal and opposite, thus the requirement isthat they both be equal to zero. In formulas, we require B A + B A = − B B − B B = 0 (56)where B A = y σ ( η A − , (57) B A = − y σ ( η B − , (58) B B = − y σ ( η A − , (59) B B = y σ ( η B − . (60)This requirement translates in the following relationwhich links the exchange rates of the two goods: σ = η A − η B − y y σ . (61)Substituting this relation in the inequalities (54, 55), weobtain C A − σ ( η A − y y C A + σ ( η A − ≤ − y − x A y − x A , (62) C B − σ ( η A − C B + σ ( η A − y y ≤ − y − x B y − x B . (63)which remarkably depend only on two variables, σ and η A . These inequalities must be satisfied together withthe conditions P A = C A − σ ( η A − y y ≥ P B = C B − σ ( η A − ≥
0. These two inequalities requirethat the consumption rates of the imported goods in thetwo countries be not too small and that the exchange rate σ be not too large. The figure shows the region in the σ − η A plane where all inequalities are satisfied. As afinal remark, our model shows clearly that both countriesmay benefit from free exchange. Therefore, as a generalrule, one may say that, given this occurrence, any traderestriction should be avoided. Σ Η A1 FIG. 2: Locations of the points where all four inequali-ties are satisfied. The used values of the parameters are C A = 1 , C A = 5 , C B = 7 , C B = 2 , x A = 1 , y = 2 , x B =3 , x A = 5 , y = 4 , x B = 2. We considered σ ≥ . η A ≥ . σ ≥ η A ≥ V. DISCUSSION AND CONCLUSIONS
Although most nations claim to be in favor of freetrade, it seems paradoxical that today, most of them con-tinue to impose many restrictions on international trade.As a result, they advocated restrictions on imports, in-centives for exports, and strict government regulation ofall economic activities. Indeed, the list of protected prod-ucts is long and varied. By so doing, both nations endup consuming more of both commodities than withouttrade.Despite modern protectionists claim that the compar-ative cost doctrine has little contemporary validity, andmight apply more in a static world in which capital andlabor would be fixed in quantity and immobile interna-tionally - and trade restrictions are needed because theeconomy is not sufficiently adaptable to changing com-parative advantages, the international mobility of capitaland managerial resources can quickly alter factor propor-tions, thereby raising the risks of specialization and thecosts of adjustment - the economic lesson of comparativeadvantage demonstrates, and this work too, that bothinternational trading partners are best served withouttrade restrictions. While not denying that comparativeadvantages change more rapidly today than in past, thecontemporary economy has sufficient flexibility to adjustto such changes.Comparative advantage shows tariffs and trade quo-tas protect inefficient firms, harm consumers and lowertotal productivity. The fact that the country A gainsmuch more than the country B is not important. Whatis important is that both nations can gain from special-ization in production and trade. With complete special-ization, the equilibrium-relative commodity prices will bebetween the pretrade-relative commodity prices prevail-ing in each nation.According to the law of comparative advantage, even ifone nation is less efficient than (has an absolute disadvan-tage with respect to) the other nation in the production ofboth commodities, there is still a basis for mutually ben-eficial trade. The less efficient nation should specialize inthe production and export of the commodity in which itsabsolute disadvantage is less. This is the commodity ofits comparative advantage. The principle of comparativeadvantage remains as cogent today as it was in Ricardo’stime.In this work, building on a leaking bucket analogy anda communicating vessels scheme [25], we introduced asimple hydrodynamic model, able to reproduce the mainconclusions of CA theory in its simplest setup, i.e. a two-country world with country A and country B exchang-ing two goods within a genuine exchange-based economyand a trade flow ruled only by market forces. We mod-eled the exchange flux between countries by means ofa simple but non trivial exchange function in order toget an analytic solution while retaining all the main phe-nomenology. Finally, production and exchange costs areincluded in this framework in order to evaluate the conve-nience each country has in producing or sharing a good.A fixed point analysis has been carried out and a regionin the parameters’ space has been found, where the time evolution of the amounts of money for both countriesshows a lower bound. That allowed us to assess the roleof money in shaping trade patterns.Our results reproduce the main features of CA the-ory. In a free exchange world all countries benefit fromtrade; that happens also in the presence of productivitygrowth only in one country. More in general, countriesgain because they export goods whose prices are rela-tively higher while import goods whose prices are rela-tively lower. The focus is on cooperation rather thanon competition and on the free exchange flow betweencountries. We would like to emphasize the rˆole played bythe trade balance constraint, which represents the onlyaction required from the government.We have to point out that our model doesn’t take intoaccount the role of other macroeconomic variables, suchas the consumers’ different preferences, the role of brandnames and product reputations in buyers’ decisions, thetemporary advantage gained as a consequence of the de-velopment of a novel technology, etc.. In fact the leakingbucket model is based on the concept of representativeagent and, as such, is subjected to all its limitations [30].Finally, it would be interesting to recast the problemin the language of cooperative Game Theory [31]. In thiscontext a general N -player game could be envisaged withcooperation as the mandatory choice in order to reach afree trade regime. Author contribution statement
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