1 Introduction

The pages that follow will present a general overview of the gravity model in trade, with a particular focus on its origins and physical setup. Therefore, in addressing this subject, I shall assume that the reader is familiar with the historical and contemporary evolution of the model. The following bibliographic resources can be used by interested readers for an additional explanation of the concepts and formula in this article: Hartle (2003), Schutz (2003) and Misner et al. (2017). The analytical and methodological approach adopted will be strictly econophysical. Aside from reintroducing physics into economic debates regarding the gravity model, the goal is to retrieve the interdisciplinary nature of the initial model as well as the original meaning of the gravitational metaphor. Above all, some concepts regarding the econophysics model, a discipline that applies the laws of physics to the study of economics, will be explored. In contrast to econometrics, which solely entails the application of mathematical and statistical tools in the construction of models to assess the validity of hypotheses in economics, the econophysical technique largely incorporates the translation of logical reasoning and structures that characterize the world of physics. The reader might be wondering, at this point, whether academics and researchers should completely free themselves from the physical roots of economics and seek instead econometric models. However, I believe that, when conducting a study, excessively relying on or overemphasizing empirical aspects might lead to outcomes lacking a solid theoretical foundation. The primary risk, in this regard, is the potential for a drastic alteration of the original models, thereby fundamentally changing the framework they were built upon. Indeed, it is impossible to deviate from the theoretical gravitational physical rules that underlie our economic models, to take gravity as an example. In this sense, the validity of the structures and logics of theoretical physics and its subsequent application to social sciences must be presupposed. The issue could thus divide economists into two groups. The first, which we will refer to as reductionist and determinist, accepts, albeit through simplifying premises, the transposition of the laws of mechanics to economics. The second group, on the other hand, shows a more skeptical approach, characterized by a holistic and evolutionist viewpoint, critical of the deterministic perspective derived from mechanistic physics.

Although there is no conclusive solution to this problem, if we accept the use of a gravity model in social sciences, including economics, we must also accept the logics of mechanics, which serve as the fundamental premises, or as the validity conditions, for the application of gravity to social sciences and, more generally, to econophysics itself. If the opposite was true, the model itself would suffer and there would be a lack of logical coherence. Interested readers are invited to read Majorana and Mantegna (2006). The position of my paper is thus clearly revealed. If gravity is to be explored in economics, one can only do so from a deterministic perspective because it has a profoundly reductionist approach to the understanding of complex systems in physics. As a result, some degree of human action determinism must be accepted in gravity models.

Although the question might appear purely philosophical, it is crucial for economics in general and, in particular, neoclassical economics, which bases its paradigm on the concept of production function and on the work of Léon Walras and Louis Jean Baptist Bachelier (1874, 1900). They developed the idea of general economic equilibrium and they both have their roots in classical mechanics. Like Schumpeter (1908) after him, Walras believed that since economics had a close relationship to natural sciences, it should continue to be studied independently from social sciences. Furthermore, neoclassical economics bases its foundation on a mathematization of economics, on the concept of function and its subsequent maximization. A function whose definition in mathematics, as in physics, is completely deterministic.

The objective of this essay is to demonstrate that the econophysical evolution of the gravitational model of trade should take into consideration the transversal advancement of physics on the theory of gravitational interaction. To do this, it is appropriate to start by introducing the basic concepts of Newton’s universal gravitation theory, applied for the first time in economics by Walter Isard. An in-depth analysis of the scholar’s theory of international economics will also be provided, with a particular focus on Isard’s desire to combine the analysis of localization decisions and the observation of trade flows into a single analytical approach (see Isard’s superior theory). I shall also briefly discuss the subsequent multilateral developments made by other authors following Isard’s theory (1954), which are distant from his econophysics method, and assess the limitations of these studies. In particular, the objections to remoteness and to the Anderson and Van Wincoop model (2003) provide a significant in-depth analysis. Drawing inspiration from Isard (1954) and Stewart (1948) is one way to replace these multilateral models.

Therefore, this work also aims at providing a new framework in which the three principles of dynamics are translated into economics and applied to the gravitational model of trade in a consistent and coherent way. Not only does this new framework take into account the most notable variable of distance, as all simple spatial frameworks do, but it is also connected to the ideas of spatial concentration and forces of attraction. Using the physics theory of gravity and gravitational fields is, thus, necessary for this process, in accordance with an econophysics approach, to amend the gravitational law in international economics. The theory and formulas of gravitational fields in physics will be then examined, thus describing a multilateral market model that takes trade flows among several countries into account. This resulting gravitational field model represents the physical transposition of a gravitational field into a trade model. However, besides being a new model, it concerns the re-emergence of a multilateralism that has been present since Isard’s original model, which is more akin to a gravitational field in his formula than to Newton’s bilateral formula of gravity. This can clearly be observed in Stewart’s work (1950), from which Isard took inspiration (1954).

Therefore, this paper also aims at overcoming the limitations of previous multilateral models by proposing an innovative and more efficient tool for the analysis of international trade. In order to overcome the bilateral paradigm, it is recommended to use a vector analysis as a tool for mapping the gravitational fields where interactions between and within countries generate economic forces. As a result, it is demonstrated that the evolving vectorial theory included in the new gravitational field model is in line with Isard’s methodology and offers a means of reconciling trade theories with location theories. In this respect, the vectorial advantage, which is calculated by the superposition principle, is able to compare more than two trade attractions with each other. Greater economic magnetism in one point corresponds, other things equal, to a greater potential trade advantage for firms located there. Taking into account the effect of the third country or the rest of the world, when comparing the potential trade advantages of two economic entities, means taking into account the comparison of three forces. We move from a dual to a relative plan. This introduces a third element of evaluation into the analysis, which was not previously considered. It is thus depicted as an evolution of comparative advantages. In practice, the concept of comparative advantage is expressed here by vectors that may be compared to one another using vector calculus, which presents the quality of modularity. In this way, it is possible to easily assess international trade flows and attractive forces among many states at a time. It follows that this attempt aims to continue the work of the distinguished American economist. Besides this, the model will play a key role in explaining the phenomenon known as the “Blue Banana” or “the Mediterranean commercial slump” and other stylized facts, just to name some possible practical applications.

It’s worth noting that Nijkamp and Ratajczak (2021) have published a related paper that precedes the current study initiated in 2018. Their model offers a good basis for interaction and comes probably the closest to the full understanding of gravity discussed in this paper. Both papers utilize the concept of vector field. However, while the main reference in the other work is Tinbergen and “potential” models derived from it, the present research primarily draws inspiration from Isard (1954) and Stewart (1948), who originally presented a model of gravitational fields, which has been neglected in previous literature. The gravitational field idea proposed in this study aligns more closely to the concept in physics, since the vector forces are attractive in both directions (thus considering both imports and exports). While the previous paper adopted a multilateral approach but only considered a single country of origin and a single field between pairs of countries with the first state, this study tries to take a multilateral approach by applying the principles of vector calculus and composition, i.e., considering the resulting total field of numerous countries of origin, and focuses on markets, particularly the European market. However, the potential of applying this approach to assess the global gravitational field with high magnetism zones should not be disregarded. The limitations and barriers to using this approach are more closely related to the capabilities of the software being used, rather than the theoretical concepts discussed.

2 Isard’s Comprehensive Idea of Gravity in Trade

Contrary to what has been erroneously reported in most research papers, the first use of the gravity model in international trade is not to be attributed to Tinbergen (1962). The use of gravity in social sciences was already widespread at the time and it was used by Stewart, Zipf, Vinining, and Ulman, among others. The gravitational model was first applied to the study of international trade by Isard (1954), professor of regional sciences at MIT. However, it was only with the work of Tinbergen (1962) that the model became known in its standard bilateral form:

$$\left({F}_{ij}= A \frac{{Y}_{i}{Y}_{J}}{{D}_{ij}}\right)$$
(1)

The formula is analogous in mathematical structure to Newton’s bilateral equation of gravitation and was later revisited by several economics' scholars. Tinbergen earned his PhD in Physics from Leiden University in the Netherlands. Despite being a physicist by training and being interested in the subject, his link with the gravity model was coincidental, as he was working with a group of Finnish scholars and also studying the model at the time. Their research appeared in a book appendix with no mention of Isard’s work. This could explain why the latter was not acknowledged. The gravitational trade model predicts trade between partners (countries, regions, or companies) in an intuitive way. Transposed to international trade, Newton’s law shows how partners trade in relation to their respective economic size and mutual distance perspective (frictions of the distance), just like particles when they attract each other. The first component to be taken into account is the economic dimension; the gravity model assumes a positive relationship between international trade flows and market size. In the context of states, this role is played by GDP. Larger GDPs are consistent with larger economies. Indeed, a country with a large GDP generates a greater income from the sale of goods and services, which translates into a population with greater import power. The second element to point out is distance, understood as a geographical, spatial and economic dimension, which hinders trade. In this sense, distance is relevant because it influences transportation costs, the possibility of establishing contacts and facilitating communications, all of which have an impact on trade.

When applied to international trade, the gravity equation basically represents the main forces that raise the demand and supply of goods and services.

To better understand the model, it is essential to have knowledge of its origin and its background. Therefore, it is important to note that Isard is also considered one of the fathers of Regional Science (Isard and Peck 1954:97–114) and focus more on his ideas. Isard reunited various scholars and academics interested in the study of regional development through a multidisciplinary approach that included economics, geography, city-planning, and the development of suburban areas, among others. This study field became known as Regional Science, a designation still used nowadays. Even though it was Isard who first brought gravity into the study of international trade, Tinbergen’s formula is generally regarded as the standard gravitational formula. Directly inspired by Newton’s gravitational equation, it is a straightforward yet effective bilateral model for the empirical study of trade between pairs of countries. It can be argued that while Tinbergen built his model for empirical purposes, Isard focused on a comprehensive theoretical approach for a thorough analysis, looking for a superior set of tools for the theoretical and empirical regional analysis in a framework shared with both localization and trade theories. Isard (1954) claimed that distance, which is far too neglected, was an important factor in the analysis of international trade. It is indeed controversial for the traditional international trade theory to assume the abstraction of transportation costs by considering them fixed or zero in bilateral trade between countries. Only by connecting and combining international trade with economic geography via space and localization theories, would it be possible to develop a “superior” and more complete theory of international economics. However, the effects caused by geographical distance and transportation costs had not been completely ignored by the classical international trade theory, although a more satisfactory analysis of these factors was offered by localization theory. On one hand, international trade theory is influenced by a powerful element of abstraction, and single regions or countries are represented as points disconnected from a space. Actually, most international trade models do not take the effect of distance into account in their calculations. Distance is understood as an instrumental variable in transportation costs (see models of international outsourcing). Therefore, the gravity model serves as an innovative example of how to use the distance variable in a trade model. On the other hand, the localization theory puts emphasis on space and, for this reason, it is “closer to reality”. The less abstract nature of this theory, however, has prevented it from developing into a comprehensive and general system of equilibrium. Isard’s studies come the closest to achieving such a combined equilibrium theory. Naturally, in neoclassical economics, the distance element is not easy to contemplate. For this reason, it is not suitable for the description of situations of general equilibrium. Theoretically, these economic balances are generally derived from initial simplifying assumptions which, as already seen, are characterized by an abstraction of distance (for more details, see Haberler 1955). The economist bases his arguments on the findings of Weber and Ohlin (1982, 1933), according to which any theory of commerce would have been constrained to an idealized universe if it had ignored the variable distance. In particular, Weber claimed that the classical theory of commerce was incomplete because it did not take distance into account, or, as he stated “the significant amount of industry which is transport-oriented”. Ohlin agreed with Weber’s position and insisted that international trade was a section of localization theory.

Isard argued that a superior theory of international economics could be formulated if two conditions were met: taking into account the factors of space and transportation costs in trade theory and encapsulating location theory into a system of interdependency (Isard and Peck 1954:114). Isard believed that international trade theory should be complemented by location theory, both for long-run and short-run analysis. Regarding long-run analysis, the author demonstrated that by merging distance inputs and transport orientation to opportunity cost, it was possible to create a superior set of tools for economic analysis. That set of tools applied to both location theories and international trade theories: to the former through the use of opportunity cost and to the latter by employing transportation cost differentials, hinting at a potential merging of the two (Isard 1954:305–20). He believed that, thanks to this fusion, it would have been possible, for instance, to simultaneously determine the location of economic activites and commodity flows.

According to the American economist, the fusion was necessary for short-run analysis too. Anticipating future studies, Isard had already recognized the limitations of a bilateral trade model and suggested moving towards a multilateral one that took into account the effects of distance, income, and occupation rate (including migratory flows caused by job-seeking) on different countries. This search would lead the scholar to the first application of the gravity model to economics: the relative income potential (Isard 1954:319). It becomes clear that, despite being a tool used for the estimation of commercial flows, the gravitational model’s original characteristics are linked to geography and spatiality. The gravitational instrument conceived by Isard is a representation of his effort and desire to incorporate the distance variable into trade and location theories.

The core tenet of Isard’s theories is that trade is encouraged by geographical proximity for a variety of reasons, including low transportation costs and institutional and linguistic similarities between states. Therefore, he concluded that in any model that considers distance as a direct factor in increasing costs, a gravitational relationship must exist (Isard 1954:308). According to Isard, “the distance variables act in much the same manner with respect to the social world as to the natural world”. However, Isard believed that such assumptions, as they were presented, posed a risk for the study of international commerce. He argued that further empirical research was necessary. The economist developed the concept of income potential in international economics (1954) in the short term, drawing on Stewart’s concept of demographic gravitation (1947), which calculated the total possibility of interaction between an individual at one point and the population of all other areas under consideration. The study of human population has been crucial for the development of economic models taking into account the dynamics that underly the pull factors of attraction in the migration process, which are also connected to Stewart’s model. Isard used this demographic foundation to apply the theory of gravity to international economics for the first time, taking into account the wealth of the countries under consideration (a transposition of masses in physics), and the attenuating influence of distance upon the physical volume of trade in the short term.

According to the following formula:

$$iV= \sum\limits_{j=1}^{n}\ {_i{{V}_{j}}}=\sum\limits_{j=1}^{n}k\frac{{Y}_{j}}{{d}_{ij}^{a}}$$
(2)

where \({Y}_{j}\) is the income of nation (region) \(j\), \({d}_{ij}\) is the average effective distance (i.e., the distance adjusted to the level of transport rates) between nations \(i\) and \(j\), a is a constant power to which \({d}_{ij}\) is elevated, \(k\) is a constant similar to the gravitational constant, \(_i{V}_{j}\) is the income potential produced by nation j upon nation i. Finally, \(iV\) is the income potential produced by all nations upon nation i.

It is important to note that several scholars attempted to advance Tinbergen’s standard bilateral gravitational model from 1962 by using a “new” multilateral approach. However, a few economists noticed that Isard’s studies from 1954 already contained this multilateral model. In Isard’s formula, the income potential produced by a given nation on nation i varies inversely with the intermediate distance. Between two nations, similar in resources and technological development, the one closest to nation i will have a greater income potential on i.

When comparing Isard’s formula (2) with Tinbergen’s more popular formula (1) from 1962, it is clear that the first one already had a multilateral character, since it considered the effects that a group of countries had on a nation i, instead of focusing exclusively on two countries. Despite the fact that the need for a multilateral trade model that took distance into account may appear as a modern concept, it was already clear to scholars such as Machlup (1943), Frisch (1947), Metzler (1950), Samuelson (1949, 1952), and Hansson (1952) as early as 1954 (Isard 1954:307-8). This is because Isard deemed the traditional bilateral trade model (either with fixed transport costs or with costs near zero) useless and stressed the importance of a multilateral model that included different spatial resistances for each pair of nations or regions that were object of analysis. Furthermore, Tinbergen’s standard formula (1962) is basically a direct mathematical transposition of Newton’s gravitational law formula into economics, while Isard’s is more closely related to the formula of gravity fields in physics. More generally, a significant similarity can be found in the concept of mass potential in international trade. This should be clear while reading the economist’s ideas, who, as mentioned, for the theorization of gravity in international trade took inspiration from Stewart’s demographic studies (Stewart 1947:461–85, 1948:31–58, 1950:239–53). Stewart identified valid, linear empirical links between distance and factors such as the value of agricultural land per acre, road and rail mileage levels, and population density.

Isard deduced that a higher potential of said factors leads to a higher gravitational magnetism by looking at Stewart’s figure (Fig. 1), which shows equal contour lines of population of the United States in the 1940s having their density peak in the New York area. By contrast, their potential tends to decrease to the west. Therefore, the potential is inversely proportional to population proximity at any point. Moreover, the figure on the left (Fig. 2), provides a graphic representation of the interaction between two gravity fields.

Fig. 1
figure 1

Stewart’s Contours of equal population potential for the United States, 1940. Source: W. Isard (1954)

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Fig. 2
figure 2

Personal elaboration of a gravity field between Earth and the Moon. Source: personal elaboration

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One represents the gravitational field of the Earth, while the other that of the Moon, making the similarities between the two representations clear. According to Stewart, the formula for the potential at any point produced by the entire population was:

$$V=\int \frac{1}{r} {D}_{P}dS$$
(3)

where \({D}_{p}\) is the density population in the area \(dS\), and \(r\) is the distance. Since populations are not noted for infinitesimal elements of an area but rather for comparisons between large sections of an area, it is only possible to approximate potential.

Isard’s original model differed from Tinbergen’s standard formula in that it did not only define a mathematical relation for the empirical assessment of commercial bilateral flows in three independent variables. The standard approach would strip the model of the gravitational character that should be at its core. That is because “the law of gravitation” would not be suitable to describe a simple econometric model of distance decay, which is closer to Tobler’s first law of geography (1970). Applying gravity to economics is much more complex and requires specific logics and econophysics structures, such as forces acting in space and principles of attraction as commercial and spatial concentration. Newton’s intuition of the existence of an invisible force capable of running long distances can be found in Stewart’s (1948) concept of attractivity of a populated area and Isard’s (1954) concept of business attractivity. This confirmed Isard’s idea to use this particular tool to move towards the formulation of a superior theory of international economics, which should merge the theories of commerce with those of localization in economic geography. Isard believed that, over time, this advanced model would have made it possible to reduce the gap between short- and long-run trade and location theories, once integrated into an interregional input-output framework.

3 A Critique of Previous Multilateral Theories and the Advantages of a Vector-based Approach

Before examining how multilateral economic theory has developed in line with Isard’s original econophysics approach (1954), it is important to briefly review the most significant attempts made by economists after Isard to develop a multilateral gravity model in economic literature. Particular attention will be given to the use of remoteness and multilateral resistance factors by Anderson and Van Wincoop. Even though in physics gravity as a multidimensional factor was suitable to describe multilateral trade, in economics, gravity began to be applied to the description of bilateral trade between two nations, in accordance with the standard formula developed by Tinbergen (1962). Thanks to Krugman’s (1991) New Economic Geography and McCallum’s (1995) Study of the Effects of Borders, the multilateral approach was later revived in international trade studies. However, these developments of the gravity model rather than being directly inspired by Isard’s original multilateral formula were a multilateral implementation of Tinbergen’s bilateral gravity formula.

Isard essentially considered the changes in distance and position as embedded in a comparative approach, where these factors are constantly evolving as a result of changing legislation, international agreements, etc. He argued that the concept of multilateralism in trade could not be separated from distance, and it had to be placed in a logic of comparative advantages due to the different locations. Every location in space has a different trade potential. For businesses proximity to large marketplaces is advantageous. The gravitational tool can be adapted to take these factors into account through a vector-type calculation logic where trade attraction forces represent vectors in a multilateral world, where they can be substituted, compounded, added to and subtracted from each other. High gravitational magnetism will correspond to high commercial potential.

At this point, I shall recall to the reader the main difference between the definition of comparative advantage, in particular Ricardo’s (1817), and Smith’s (1776) definition of absolute advantage. In the absolute advantage of Smith’s two-country model, there is already an implicit comparison on a logical and linguistic level. The comparative advantage introduced by Ricardo (1817) added a third comparison regarding the relative advantage among different productive sectors in the same country. Nevertheless, this has been applied to the H-O model too, but with a different connotation, namely as a comparison of relative factor endowments. Therefore, the concept of comparative advantage in economics has its own linguistic value that can be used beyond the historical and social boundaries of the Ricardian theory. It would be wrong to assume otherwise. This is further supported by the fact that "On the Principles of Political Economy and Taxation" (1817) was published more than a century before Ohlin (1933). Ricardo’s definition of comparative advantage might be considered as multilateral as it includes a third comparison, introduced by the scholar, concerning the relative advantage between the production of two different goods in the same country. Thus, it can be stated that it materializes on a multilateral level. Therefore, it makes sense to talk about comparative advantages when considering the trade advantages of two locations in relation to a third country or the rest of the world. In fact, multilateral models, unlike bilateral ones, simultaneously analyze a minimum of three countries. Looking at the logical meaning of such comparative approach, when considering a short distance as an advantage, there is a shift from a bilateral comparison (absolute advantage) to a multilateral one (comparative advantage). Consequently, when examining three countries rather than two in bilateral models, it is necessary to discuss comparative advantages in relation to multilateral gravity models. The relative advantages can thus be considered on a multilateral rather than a dual level, and a third relative element is added in the comparison. In this context, the gravity model could be adapted in order to analyze elements through a vector-based computing approach. This way, forces of commercial attraction and distances create a multilateral world of trade attraction vectors that can be substituted, added to or subtracted from each other. The outcomes of such measurements can be described through a series of arrows. The length of each arrow is proportional to the gravity force, and the orientation of the force matches that of the arrow. Thus, it is possible to draw a group of lines, called force lines, around the source-mass, so that the direction of the force in every point will be given by the direction of the force running through that point. The concept can be more easily understood looking at Figs. 1, 2, and 3, which show graphical representations of a vectorial field.

Fig. 3
figure 3

Representation of a vectorial field on a sphere. Source: personal elaboration

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Nijkamp and Ratajczak's (2021) analysis, which uses a vector technique similar to the one presented in Fig. 4, highlights the exports' macroscopic spatial structure of the Netherlands while also displaying the nations that the Netherlands sells the most to. The gradient configuration resembles the European "blue banana", which will be described later.

Fig. 4
figure 4

Gradient full vectors of Dutch exports potential model (forty-three European countries). Source: Nijkamp and Ratajczak's (2021)

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These vectors will be compared in the following paragraphs. This entails comparing different commercial potentialities (thus completely different advantages) that, in a multi-country model, depend on the position in space. This refers to the economic potential related to the geographical location. Since these vectors are a graphical and mathematical representation of commercial forces involved, when they are included in an economic gravitational field, they are the exact transposition of potentialities. It is possible to note a difference from Newton’s standard gravity formula, which describes a trade flow as a force at the econophysical level. On the other hand, a commercial field will inevitably represent the commercial potential as a function of space.

This type of advantage will be called herein the vector type advantage, which guarantees the quality of composability and decomposability (technically referred to as modularity). Trade advantages, considered and expressed as vectors and articulated in the concept of comparative advantage, are comparable among them given their quality using vectorial calculus. By expressing the gravitational advantage in vector form, it is possible to synthesize the gravity model with the "Intervening Opportunities" model proposed by Stouffer (1940) and Schneider (1960). This model suggests that the flow of goods and people is inversely related to the availability of alternative opportunities. By comparing these opportunities in vector form, it is possible to consider both the market size of the alternative destination and the gravitational distance. This can provide a more complete understanding of the factors that influence trading decisions.

4 Remoteness and Multilateral Factors of Resistance

Moving on to the remoteness variable, the first attempt among those mentioned above saw the use of the universal gravitational constant G, previously indicated as A in the trade model. Given its inherent coherence with physics and its multilateral character, when applied to economics, it should be used as a constant for a group of countries or for an entire market. This happens because the uniqueness and the invariability of the universal gravitational constant G do not exist in economics. Since the constant A depends on the two indexes i and j, it would be more accurate to refer to it as \({A}_{ij}\). The constant A will be determined by the specific system of countries taken into consideration. In its very formulation, the constant represents that value capable of converting the relationship between countries’ GDPs and their distances into a certain trade flow. Therefore, the impact of G should be general and consistent, contrary to the variability of economic distances and GDPs (for instance, the distance between two regions can vary following the construction of new railway junctions). In a way, it could only be affected by symmetrical variations with respect to the total number of countries, which, by definition, would be multilateral.

Deardorff (1998) argued that the volume of trade is influenced by the relative distance between business partners. Subsequent studies by Wei (1996), Helliwell (1997), Nitsch (2000) and Chen (2004) introduced a new variable that considered the impact of third countries, also known as the “third-country effect”, as they do not belong to the two trading partners in question. Among the authors who have extensively studied the subject, we find Bang (2006), Blonigen et al. (2007), and Baltagi (2008).

This development measured the isolation of a pair of countries, or as the variable termed it, their “remoteness”. According to the theory, trade between more isolated but close countries, such as New Zealand and Australia, is supposed to be larger in comparison to countries like Austria and Portugal. The latter pair consists of two countries close to each other, but contrary to the first couple, they are closer to a greater number of markets. According to this logic, the amount of trade exchanges between Austria and Portugal is lower than what it would be if the two countries were located in a remote area far from “third countries” (Italy, Spain, United Kingdom, etc.), as in the case of Australia and New Zealand.

Therefore, the remoteness equation for a pair of countries i and j will be:

$${R}_{i}={\left( \sum \frac{{Y}_{z}}{{D}_{iz}} \right)}^{-1}$$
(4)

In the formula, z represents all trading partners different from j for country i, \({Y}_{z}\) is k’s GDP and \({D}_{iz}\) is the distance between country i and any trade partner other than j.

As anticipated, given its multilateral approach, from a theoretical and mathematical point of view, in the standard bilateral formula the remoteness replaces the constant A found in the Eq. (1) which is the transposition in econophysics of the gravitational constant G, so that:

$${F}_{ij}={R}_{i} \frac{{Y}_{i}{Y}_{J}}{{D}_{ij}}$$
(5)

Since the value of the gravity constant G remains stable and consistent throughout the universe, the above transposition is correct from a theoretical perspective. On the other hand, R can basically change but it is invariable with respect to a fixed set of countries at a specific time. If we consider this set of states as the universal set, R will be a constant too. It follows that R should be recalculated every time the spatial or temporal dimensions of our set of countries change. More specifically, the spatial dimension deals with the geographical extensions of the region under consideration, and therefore changes if the states taken into account are different or if a new state is conceived. When operating within a dynamic model in which the economic masses of countries can change through time, it is necessary to take into consideration the temporal dimension. Analogously, if we apply the same variable in economics, the influence on it would be exogenous since its value would be subject to the changes of the whole economic environment. A’s impact will always be multilateral, but its constant character is not fixed, even though it tends to be so in the short term. On the contrary, the distances between countries and GDPs will experience asymmetrical changes due, for example, to the stipulation of new bilateral agreements that would modify the distance between two countries or regions. Contrary to physics, in economics, symmetrical adjustments of the market under examination determine variations of the constant. Factors such as the use of digital currencies in the countries involved could have a positive impact on trade thanks to the rise of on-line purchases. A is assumed to be a constant because of its tendency not to change easily in the short term. Since this variable takes into consideration the general context of the market and has a universal character, economists often prefer to omit it.

On the other hand, when it is applied to a bilateral analysis and assumed as an element of multilateralism, other economists place great significance on it. They try to overcome the limits of the standard bilateral trade model, which for decades had been regarded as the standard gravity model in the study of international trade. I stress that this framework, even though indirectly, represents a revival of the model created by Isard (1954), both logically and practically. This claim is supported by the fact that by isolating the constant \(k\) as the unknown quantity in the Eq. (2), the outcome will be similar to the one found in formula (4), being \(k={iV\left(\sum_{j=1}^{n}\frac{{Y}_{j}}{{d}_{ij}^{a}}\right)}^{-1}\), and the substitution of remoteness in constant A, in the standard formula, is a clear attempt to introduce multilateralism in a bilateral trade model. Moreover, in these economic models, remoteness is arbitrary since it describes exclusively the effect of third countries and no other elements of general context. Within a commercial flow, there are a number of variables that affect the relation between the economic masses of the countries and the distance. Only considering the third-countries effect in the calculation of a constant that, by definition, is universal and defines the dimension of the interdependence between the aforementioned variables, would be reductive. Other variables, such as the countries’ internal distance, the geopolitical context, or technology, may influence this econophysical constant.

The model developed by Anderson and Van Wincoop (2003) outsets from McCallum’s (1995) considerations, which stated that borders pose an obstacle to international trade. McCallum found that bordering regions are close in terms of cultural, linguistic and cultural characteristics. However, national borders have, ceteris paribus, negative effects on their trade. This issue, known as the “national border effect” or the "McCallum Border Puzzle", is one of the most debated subjects in economic literature. McCallum (1995) used empirical evidence to support his research on commerce between the US and Canada and found that the inclination for trade is stronger between adjacent regions in the same country than between adjacent regions of different states. This claim was supported by data on interprovincial and international commerce in Canadian provinces between 1988 and 1990. Further research has confirmed that boundaries have a negative effect on trade. This was shown by a comparison between domestic and foreign trade using the gravity model in economics. Such a finding, though, shouldn’t come as a surprise for two main reasons. The first is the rise in border-related business expenses. In this case, factors like customs formalities, differences in monetary regimes, legal systems, languages, and other non-measurable elements contribute to difficult-to-identify transaction costs.

The same phenomenon can be observed in physics when two solid objects come into contact with each other: the pull of gravity that draws them together cannot be infinite because, according to the law of physics, the distance between two masses cannot be equal to zero. In a sense, this confirms McCallum’s assumption that for different countries the effect of borders’ distance decay will never be equal to zero. A second reason concerns the use of the gravity model, which in this context serves as an exemplification with respect to physics. In economics, Newton’s spherical shell theorem can be also applied to asymmetrical objects like countries. Given a group of countries, the force of trade attraction will be exerted outside national borders. To make it more complex, the attraction would be also exerted within the country itself, between its regions and cities, and even on individual economic subjects. According to the theorem, gravitational acceleration within the shell will be totally absent, and the mass is no longer treated as a set of smaller masses, but as a unique entity. The main advantage is that it allows to consider the internal distances among the different parts equal to zero. In this sense, states can be considered as unified economic entities, regardless of the parts they are made up of. Thus, borders are, by definition, economic distances and, for simplicity, include those distances that would otherwise have to be calculated within countries. Taking trade between Italy and France as an example, it would be assumed that the two countries are homogenous, single entities, considering as negative border effects both the trade flow of a border region in northern Italy and that of a more distant southern region. McCallum’s Border Puzzle could thus be solved by a vector-based gravity model that considers economic distances rather than geographical ones.

Anderson and Van Wincoop’s model added the so-called multilateral factors of resistance to the standard bilateral model, thus introducing a multilateral approach to the framework. However, it lacks the theoretical transposition of multilateral components of resistance to the force of gravity in physics, in contrast to the classical approach, which embraces the concept of gravity through the translation of masses and distances in physics into GDPs and distances in economics. By no longer referring to the force of gravity, there is no need to describe an econometric model as gravitational. I argue that most subsequent authors most likely used such redundant references to gravity in their works for historical reasons and due to the suggestive power of the word “gravity”, rather than to establish an actual logical link between the two disciplines. The reader might be wondering why not letting go of the model’s physical roots at this point. There is nothing wrong with fields diverging, but it is important to call them by their correct names. An econophysics model divergence would return economics to economics and physics to physics. The question here is whether it is still appropriate to discuss econophysics. In my opinion, this is not the case. A purely econometric methodology is insufficient for an econophysical approach. As econometrics applies mathematics and statistics to economics, econophysics applies physics and its logic to economics. To “render unto Caesar” is one of the objectives of this paper. Looking closely at Isard’s formula (2), we notice that a multilateral attribute is already inscribed on it, for it considers the effect of a group of countries on a country \(i\) rather than the effect of just two of them, as those conceived and created by Anderson and Van Wincoop (2003). I argue that given the existence of the first multilateral transposition of gravity in economics already given by Isard’s formulation, albeit theoretical, it would be improper to attempt to develop the model starting from Tinbergen’s 1962 standard bilateral formula. Trying to modify the standard bilateral formula considering an econometric approach and adapting it to a multilateral context would be pointless. In this way, there is the risk of omitting the reference to this multilateralism’s origins in Isard.

The goal of multilateral resistance is not to read more complex multilateral phenomena, but to provide a better estimate of the bilateral flow. In this sense, it aims at preserving the basis of Tinbergen’s analysis, considering at the same time the effect of the third country. Nevertheless, the multilateral factors of resistance appear to be inconsistent with the econophysical model as well as with the previous, simpler, and more consistent multilateralism of Isard. As a result, it becomes apparent that it is a stand-alone tool, disconnected from a broader gravitational framework and unable to be integrated into more complex multilateral economic models.

So far, we have seen how gravitational attraction is expressed as a vector quantity in physics and is generated by the interaction between two or more bodies (altering their state of rest into one of motion). Similarly, in economics, forces moving from the gravity trade model act as vectors on two or more countries (multilateral model). Each vector of commercial attraction consists of a magnitude (or length) and a direction or versor (a unit vector indicating the direction of a directed axis). Therefore, it is not necessary to add multilateral factors external to the theorization of the gravity model itself. Thereby, there is no need to econometrically add a variable to the model to consider the effect of the third country on trade flows, given that it will be already included in the gravitational model as part of its conceptual framework.

Tobler (1970) reintroduced a concept similar to the one previously addressed. According to Claude Grasland (2010), the most significant of Tobler’s findings is the application of concepts like movement and accessibility to the symmetrical study of positions and interactions, rather than the effectiveness of the gravity model (which had previously been defined). In this view, the common distinction advanced by Castells (1996) between the “space of places” and the “space of flows” seems to lose its logic. Therefore, it is important to define the appropriate patterns for a universal description of locations and interactions. Tobler also claims that these concepts should not limit the analysis of the relations between flows and structure only to the Euclidean distance. This final point shows how Tobler’s assumptions (1970) are in line with the econophysics development presented in this article. A space of flows can be geometrically described only through a vector-based computation where positions and interactions can be analyzed symmetrically through the concepts of movement and accessibility. As for the previously discussed idea, knowing the vector of potential attraction for each country should allow us to calculate the bilateral or multilateral trade flow between each pair (or even more than two) of these countries. Considering several countries and their respective forces of commercial attraction, the driving force in a state, region, or point will result from the vector composition of the attraction forces considered (vector sum). Thus, in a multilateral model of countries, not only will there be the advantage of remaining in the conceptual domain of the gravity model, but it will also be possible to evaluate the effect provoked by a third country on a bilateral exchange in a more accurate manner. Multilateral factors of resistance (MTR) are static in theory and vary depending on the different combinations of countries. To vary their intensity, the researcher must arbitrarily measure their value. On the contrary, a vector sum has the advantage of always being valid/effective, as well as being composable, partitionable, and calculated without any external action. To analyze a bilateral flow in the presence of third countries, a scholar must calculate a vector sum or, more simply, calculate the vectors’ coefficients derived from the direction and magnitude of the third countries considered.

Moreover, literature claims that multilateral factors of resistance are influenced by endogeneity and reverse causality or, more accurately, they are exogenous factors added to the model, which still depend on the trade distances already included in the model itself. Therefore, it will be assumed that the MTR element in the right-hand side of the equations is correlated with the error term, given that it depends in part on the distances. This problem connected to the multilateral factors of resistance will be more apparent in cases where the commercial distance is not only geographical, but metaphorically representative of other variables. The so-called “third country effect”, would deviate from the bilateral attraction between a couple of countries, not only in relation to their geographical proximity (closeness), but also to various qualitative variables, such as language, distance and colonial heritage, among others. Since each vector is a structural component inherent to the entire system, such issue does not arise in an econophysical model of vector-based gravity. Therefore, what is known as multilateral resistance is just the outcome of the economic attraction or vector composition of the agents within the system. Single economic agents (states, regions, cities, etc.) only exert their respective forces of attraction among each other. When considering the action of such forces in a multilateral model, it is necessary to take account of their contrasting and competitive nature. In a pair of countries, the limiting effect (or opportunity cost) produced by a third country will be represented by a vector with an opposite direction (running on the same axis but in the opposite direction), or a partially opposite direction (different direction). It is reasonable to anticipate that the bilateral negative effect between the two countries will never be total (attraction will never be reduced to zero), but the greater the number of third countries involved, the greater the possibilities to trade with them, the greater the opportunity cost effect will be. As it will be explained in sections seven and eight, the domestic market exerts an attraction on businesses and other economic entities, which adds to the magnitude of the same elements within a country, since the domestic market also produces an attraction on companies and subjects of trade.

To simplify, usually, previous gravity models do not consider such internal attraction. In fact, states are seen as unitary economic entities (see the spheric shell theorem) without considering the fact that they are often made up of different regions and other separated economic subjects divided in space. Hence, it would be wiser to consider this internal distance, provided that the trade attraction is also directed towards the inside territory of a country in an autarchic way. A strong internal attraction may therefore discourage international trade flows. In addition, it embodies the degree of openness of a country to foreign trade; larger countries with considerable GDPs could trade less than expected due to a lower degree of openness.

5 Newton's Three Laws of Motion in Economics

Considering the econophysics framework discussed in the previous paragraphs, the elaboration in economics of a gravity model developed through the laws of physics will logically start from the transposition of Newton’s three laws of motion in order to achieve an advanced concept of economic field. The application of the physical theory of gravity and gravitational fields to the international economy will follow an approach consistent with Isard’s theoretical method. Such theoretic model will be developed through a logical and mathematical approach. Moreover, the transposition of actual physics concepts will result from the combination of an interdisciplinary method consistent with the concepts and characteristics of physics and economics, in line with the approaches adopted by Stewart (1950) and Isard (1954), whose research took Newton’s (1687) “Philosophiae Naturalis Principia Mathematica” as its starting point. The reader may think that the slavish transposition of certain laws of physics to economics is misleading and may lose sight of the heuristic character of the economic discipline’s specific tools. However, the methodological approach used in this study, based on a reasoned theoretical transposition, is the same as that used by Isard (1954) and Stewart (1941, 1947, 1948, 1950) before him. Considering the inspiration that the former had by translating the principles of the latter’s people flow theories into economics, it seems entirely logical and reasonable to complete this translation by also using the astrophysicist’s more refined concepts about gravitational fields. To do this, it is essential to understand Newton’s principles, which are part of the gravity model applied in economics. Moreover, it would be a mistake to think that this translation covers all of economics in a broad sense. Rather, it fits within the gravitational models in economics. Gravitational models, like any economic theory, have limitations and are true only under a well-defined number of assumptions and variables. Although considering their systematic placement in economics, in the broader sense of economic forces and entities, would be fascinating, doing so would require one to accept mechanism and determinism in economic sciences. If this is an essential premise for accepting gravitational reasoning in economics, it might be more difficult to generalize and accept it for all of economics. As Einstein reiterated to Niels Bohr, this is a problem of philosophy and opposing worldviews (Bohr-Einstein debates, 1925), to take up an important dispute in physics that has much in common with the problem being highlighted for economics alone. While it is true that the application of Newton’s three laws to physics follows a framework that is largely neoclassical in its assumptions and simplifications, it is also true that the slavish transposition of some of the laws of physics to economics could be misleading and could cause one to lose sight of the heuristic character of the specific tools of the economic discipline. As already noted, the general equilibrium is regarded as the neoclassical paradigm’s cornerstone. In order to provide a quantitative mathematical explanation for the notion of supply and demand equilibrium, Walras (1874) analogized Newtonian mechanisms. For instance, an equilibrium scenario would result from equal and opposite forces acting on a planet in a stationary orbit around the sun. However, they were firmly convinced that, much like the law of gravity remains universally valid and can be comprehensively understood through mechanical isolation from other factors such as air friction, wind currents, and so on, under a regime of completely free competition, the factors of production, products, and prices would naturally adjust to equilibrium. Walras set out to develop a quantitative mathematical representation for the idea of economic equilibrium in order to describe the general economic equilibrium in a market with the same accuracy and precision as Newton’s celestial clock. This is why neoclassicists had to include a significant amount of mathematics in their models. In mathematics, equilibrium is equivalent to finding the maximum of the applicable function.

  • The first principle of inertia stems from the studies of Galileo and Descartes, who believed that Aristotle’s claim that the natural state of objects is rest unless an external object exerts a motion on them was incorrect. The principle states that: “Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon”. Therefore, according to Aristotle, an object in a state of rest moves when external forces are applied to it. Aristotle observed that a moving body tends to gradually slow down, but he mistakenly believed that this was an absolute behavior of matter. Newton correctly adds that, even in the absence of such forces, the Greek philosopher’s principle holds true for objects in motion. Objects continue to move in a straight line at a constant speed (rectilinear uniform motion). In everyday life, the speed of an object in motion decreases due to the action of friction forces, as those exerted by air. Thus, in physics, we will have that if the vector sum of the external forces acting on an object is zero, the object’s state of motion will not change (that is, it will continue to move at the same speed if it is in motion or it will remain at rest if the object is not moving). Therefore, the natural transposition of the first law of dynamics in physics states that: “Every trade flow (or more generally, economic flow) in a market stays constant in time, unless modified by variations of the existing forces operating on that market (or unless new forces are produced).”

    This logical transposition is rationally consistent with both the gravity model and the basic principles of economy. Similarly, in a broader sense, Dornbusch and Fischer (1994), in their Macroeconomics manual, explain that the concept of economic equilibrium of a model derives in part from physics and refers to a state of rest of the endogenous variables that can only be modified by a change of exogenous variables or parameters (exogenous constants). As can be seen, the concepts are similar to each other.

  • Newton’s second law, also known as the law of acceleration (or real law of motion), states that: “The alteration of motion is ever proportional to the motive the force impressed; and is made in the direction of the right line in which that force is impressed. […] If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force is impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subducted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined when they are oblique, so as to produce a new motion compounded from the determination of both”. It follows that the force exerted by a body will be equal to the vector sum of the forces applied to it. Thus, there will be a change in the momentum in relation to the direction or magnitude of the velocity vector in time, resulting in a deceleration or an acceleration. In physics, “momentum” refers to the product of the mass of a particle and its velocity. It is a vector quantity, i.e., it has both magnitude and a direction.

    This principle is expressed in the formula:

    $$\left(F=m a\to a=\frac{F}{m}\right)$$
    (6)

    F (force) and a (acceleration) are vectors, m is the inertial mass, that is the resistance a body opposes to acceleration. A force exerted on a mass will produce an acceleration that will be greater the smaller the mass of the body. This second law, combined with Galileo’s discovery that the acceleration of an object does not depend on its mass but that objects of different masses accelerate at the same rate under gravity, leads us to the conclusion that an object's weight must be proportional to its mass multiplied by the acceleration of gravity. The transposition of the second law of motion in economics, with specific regards to the gravitational model and not economics in a broad sense, would be as follows: "the resulting trade flow (or the general economic one), is equal to the vectors sum of all the attractive gravitational trade forces applied in a certain market (in relation to an economic agent)." Mathematically, this principle is crucial to the thesis of this article because it explains how the gravity model acts in practice, that is, across a vector sum, in a multilateral trade framework. The reader might object that, for example, the simple concept of increasing economies of scale appears to contradict the principle of proportionality implicit in Newton’s second law. However, as previously explained, we confined the theoretical law to the econophysical model of gravity. Moreover, as with any economic theory, we will have a simplification of reality and it will be true all other things being equal. The concept of increasing economies of scale might seem to challenge both theoretically and empirically Ricardo’s model of trade specialization, but it actually enriches it or is simply considering a more specific case than a model with different premises and objectives. The same would happen considering classical mechanism and the principle of proportionality of the gravity model in economics versus the economics of increasing scale.

  • The third law of motion, also known as the law of action and reaction, is clearly explained by Newton as follows: “To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal and directed to contrary parts. […] If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn backwards to the stone: for the distended rope, by the same endeavor to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinge upon another, and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes in the velocities made towards contrary parts are reciprocally proportional to the bodies”.

    In mathematics we will have that:

    $${F}_{AB}={- F}_{BA}$$
    (7)

    The vector force that operates from A towards B, is equal to the vector force acting at the same time on A and B but in an opposite direction.

    The transposition of the third law of motion in economics states that: “The trade flow (or, generally economic), which is created among a country A, towards a country B (or between a subject A towards B), is equal to an opposite flow originated from country B towards A.”

    The importance of this law lies in the fact that it shows how the gravitational trade model is never unidirectional (more properly, in econophysics, we should refer to it as direction or versor). The trade flow of imports and exports between a large country and a smaller one will be the same in both directions, so that country A’s exports to country B will be country B’s imports to country A, and vice versa.

6 Theory of Commercial Gravitational Fields in Economics

Given the logical and coherent transposition of Newton’s three laws in the gravitational trade model, the only coherent approach to analyze the latter from a multilateral perspective is through a vector-based calculation. In this section, I will try to adopt an econophysics approach to evaluate global trade and develop a mapping method of the gravity models’ attraction fields in relation to a group of countries. Since Isard’s transposition of gravity into commerce (1954) is based on the larger set of transpositions previously made by Stewart in his studies about migration (1947), the most logical and coherent way to proceed for the development of the former is to broaden the commerce theory using the physical theory of gravitational fields, as the latter has done.

Starting from the vector equation of gravity in trade, and given the conformability of Newton’s three laws of motion to the commercial gravity model, we will have:

$${\overrightarrow{F}}_{ij}=- A \frac{{M}_{i}{M}_{j}}{ {\left|{\overrightarrow{D}}_{ij}\right|}^{2 }} \frac{{\overrightarrow{D}}_{ij}}{\left|{\overrightarrow{D}}_{ij}\right|}$$
(8)

\({\overrightarrow{F}}_{ij}\) is a vector and the attractive force exerted by country i on country j and \(M_i\) and \(M_j\) are the two GDPs; \(\left|{D}_{ij} \right|=\left|{D}_{j}-{D}_{i}\right|\) is the distance between the two countries, \(A\) is a constant that substitutes the gravitational constant G found in the gravity model’s standard formula, which allows the transposition of the ratio between the product of economic dimensions and the distance in a given set of countries \(. \frac{{\overrightarrow{D}}_{ij}}{\left|{\overrightarrow{D}}_{ij}\right|}= \frac{{D}_{j }-{D}_{i}}{\left|{D}_{j}-{D}_{i}\right|}\) is the unit vector from country \(i\) to attractor country \(j\). In the vector form, we will have that \({F}_{ij}={-F}_{ji}\).

As in physics, in economics we can define the gravitational economic field of an economic entity \(i\) (such as a state, region, city, firm, etc.) as a vector field expressing the force of commercial attraction that the economic entity considered would apply in any point in space. The formula will be:

$$g =A\frac{{M}_{1}}{{r}_{ij}^{2}} \to g\ state=A \frac{{M}_{1}}{{D}_{ij}}$$
(9)

To determine the attraction force exerted by other states at any point, considering trade attraction as a vector, it is suggested in this paper to sum the vectors of every state-generated field in question at any point (vector composition), as already seen in the formula (9).

The economic fields resulting from the different states will form the overall economic field, just as the trading force produced at a point will be obtained through the vector sum of the trading forces involved at that point.

In formula:

$$\begin{array}{l}Ftot={F}_{1}+{F}_{2}+\left[\dots \right]{+ F}_{n}\\\quad {g}_{tot }={g}_{country\ 1}+{g}_{country\ 2}+[\dots ]{+ g}_{country\ n}\end{array}$$
(10)

By measuring the force operating on a small observer (testing mass), it is possible to draw a map of the gravity field around a given source State \(M\). Each economic field is the result of the mass attraction at any point. Therefore, in order to measure several economic fields, it is necessary to calculate a vector sum. Thus, the commercial gravitational field is an econophysical and graphic representation of the attractive commercial forces in a market. When that market consists of a group of countries, the gravitational field will be the result of the sum of the single commercial gravitational field of those countries. Proceeding in accordance with an inverse theoretical reasoning, the existence of an important gravitational field indicates the presence of a well-defined market, such as the European market, the US market, Southeast Asia’s market, et cetera.

Moreover, in physics, the concept of gravitational field is expressed as a potential in a mobile position in space, and not as a force in act. In this sense, commercial potentials are considered through the commercial gravitational field formula. They represent a variety of commercial advantages that can be estimated through vector calculation. These potentials can be easily compared to each other through vector calculus and their modularity qualities. This will result in comparative gains-of-trade advantages.

I will now present a method for mapping the commercial gravitational field obtained. First, if we consider economic distances instead of geographic ones, representing this field on a political or geographical map will not be easy. As previously explained, gravity models use various measures of distance. Based on the approach created by Mayer and Zignago (2005), the distance between countries is measured in kilometers and reflects the distance between two cities, weighted by the proportion of the country’s population that resides in each city. A similar elaboration of the concept of field can be found in Nijkamp and Ratajczak (2021), as depicted in the following Fig. 5, which shows an export potential field map for the Netherlands that uses distances between capital cities. The gravitational potential surface's illustration highlights an important region of the European continent that is actively involved in trade with the Netherlands. It also demonstrates Germany's dominant position and a crescent-shaped ridge that extends across Italy and Great Britain. These regions are thought to have the easiest access to Dutch exports. Nijkamp and Ratajczak’s model aligns with our theoretical framework, although it starts from different premises. As discussed, the author believes that the original models proposed by Isard and Stewart already serve as the basis for potential analysis and that it is necessary to develop an econophysical framework around them in order to fully understand and analyze these models. That’s what we attempted to do in this work. Nijkamp and Ratajczak’s model has the merit of bringing the discussion about the gravitational model back to a more economic and physical plane. While their analysis presents theoretical limitations, particularly in its interpretation of the gravitational field of the Netherlands as if it were a single mass or country, it is important to recognize that the gravitational field for a set of masses and countries actually constitutes a resultant field and represents a market, i.e., a set of several countries, as discussed in this paper. Since markets may be evaluated, the perspective of the study of the gravitational field should be more ambitious.

Fig. 5
figure 5

Potential map of Dutch exports to forty-three European countries. Source: Nijkamp and Ratajczak's (2021)

Full size image

Geographical distance is used as a starting point and it is enhanced by several factors intended to account for more economic, cultural, social, and linguistic characteristics. The decision to consider a straight-line distance or, for example, the use of an alternative mode of transportation significantly impacts the measurement of geographic distance, much like it does with other variables related to the origin and destination locations. This consideration becomes particularly important when taking into account the potentially extensive physical regions that some nations can occupy, along with the fact that commerce and economic activities can occur across their borders. The most popular databases, CEPII and USITIC, take geographic distance into account. This includes population-weighted distances between countries and distances that represent the spatial distribution of activities inside a country. Other proxies for cultural proximity such as language, religion, legal system origins, colonial ties, etc. are historically linked to the fact that Tinbergen’s study group was the first to use colonial, postcolonial, religious, and cultural parameters in a trade model. Tinbergen (1962) developed the most popular and used formula for the gravity model, later formalized by his student Linnemann (1966) in his thesis. This is aimed at measuring the distance that international trade must cover in a more precise way. The estimated coefficient for the distance variable is significant and has a negative sign. In line with Herrera’s conclusion (Gómez-Herrera, 2013), it depicted any transportation costs or trade obstacles that might hinder trade flows. Therefore, if we build a gravitational field across this non-geographic distance, it will spread across these same economic distances, i.e., in a virtual and not real space. In these cases, the term “distance” does not refer to an actual distance measured in meters but to a dimensionless quantity (also known as a “pure quantity”) that measures the resistance of distance in the gravitational model of trade. Or, rather, it should be measured in economic distance meters. Therefore, in these empirical studies and in some gravitational model’s databases, we are not considering the absolute distance between countries expressed in meters, but a relative economic distance whose value depends on the comparison between the distances of other countries. The use of a weighted or network graph with vertices and edges that considers the economic distances between states or regions would be more appropriate. This could be an effective solution, given that economists are more interested in state-to-state or state-to-region distances than in the distance of any other point in space. On the other hand, the exclusive use of geographical distance would allow for an easier representation on a bi-dimensional map. For instance, in the map shown in Fig. 1, Stewart uses a geographical distance, even if it is a simplification of reality. However, thanks to its usefulness, it is still considered valid. An expedient could be considering the use of economic distances in the measurement of gravity fields and, for simplification, accepting a geographical propagation of the field. Despite the fact that it is estimated using economic distances, the force of this field does not necessarily propagate across those distances. Thus, the same field \(g= A \frac{{M}_{1}}{{D}_{ij}}\) constructed across an economic distance in the denominator can be propagated across a geographic distance through a simplification, which is necessary to represent that field on a geographic or political map. However, since the points of origin and destination in a Cartesian space are not that clear, geographic distances could also raise issues similar to those already addressed. A possible solution could be to use the distance between capitals. Nevertheless, from a realistic point of view, it is still a simplification, since states have different shapes, and their regions are differently inhabited or have different concentrations of productive activities. When using a geographical map, this leads to obvious complications and the subsequent need for simplifications. In social sciences, the distance variable, no matter its method of construction, will always be, to some extent, an arbitrary choice. The application of gravity in social sciences describes an attraction dynamic based on variables of dimension and resistance. The latter also considers the friction caused by distance. Although gravity models are empirically effective in economics, their use will never be as precise as in physics. Therefore, even if imperfect, geographical distance is a simplification that provides a valuable basis for future and more complex research. In this sense, this simplification, i.e., the use of a geographic distance to propagate a field of commercial attractiveness calculated with economic distances, might be necessary for a realistic geographic representation. The latter would be useful for a regional analysis from a strictly visual perspective, since the commercial field of a state would expand outwards (and not inwards), with isotropic values and following the shape of the state’s borders, defining different levels of spatial magnetism. This is because in physics, the gravity force of bodies with an asymmetric spherical extension is the sum of the inputs of their point particles.

An alternative is to consider a point (for example, a city) as a source of propagation from that state. However, this would also be an arbitrary choice. In physics, regions of space with high and low magnetism can be identified by looking at the map of a gravitational trade field obtained from the economic attractiveness of several states. Regions with a high magnetism are characterized by more intense trade exchanges and economic attractiveness. Furthermore, in regions with high levels of magnetism, economic distances are shorter among states with larger economies. Thus, the representation of the resulting commercial gravity field describes the market that these different regions and their respective countries form. That is, an econophysical representation of the flows and exchanges that take place in that region or market. The concept of market here outlined concerns stylized geographical regions in which trade takes place. Therefore, a portion of countries with a certain degree of political or geographical homogeneity or other similar characteristics is considered. We can conclude that we are only taking into account different levels of disaggregation and, as a result, the “market” here considered is an efficient econophysical representation of the flows and trade of a certain area. Moreover, the vector principle we just analyzed allows us to consider these forces of trade attraction, which are modular and can be spatially divided into larger or smaller portions. The regional markets considered by the gravity model, contrary to the global market, could have been identified through a series of commercial treaties or common characteristics of a region’s territories. In this respect, the gravity model of trade is a further specification which focuses on the region’s magnetism. Thus, the effectiveness and the attractiveness of a market can be measured and represented through the degree of observable magnetism.

When discussing gravitational mapping methods, Tobler’s studies (1970) might be useful. Tobler formulated the first law of geography, stating that: “everything is related to everything else, but closer things are more closely related than distant things”. This law had a strong affinity with the gravity model and, nowadays, it can also be described through the concept of spatial autocorrelation, also known as the Moran’s Index (1950). Spatial autocorrelation consists of a unit similar to space parameters, and it can be either negative or positive. When the parameters are spatially concentrated, there will be a positive spatial correlation; in the opposite case, it will be negative (also known as spatial heterogeneity). Although this is a purely spatial concept, it presents many similarities with the representation of a magnetic field. This means that large, geographically close countries with low magnetism represent an underperforming market with little trade integration, and they probably indicate the existence of a conflict situation. The ideas mentioned above are not yet in their most useful form for international trade analysis. Further empirical research is required. The logic of the vectors is a determinant of the resulting flow or attraction. For instance, the attraction exerted by a third country can limit or strengthen the attraction among a pair of countries. This happens because the third country has a limiting or reinforcing effect, which is an opportunity-creative effect. A limiting effect will be represented, for instance, by a simple vector with the same directed axis but an opposite direction, or versor; naturally, the effect is never absolute (the attraction between the two countries is never reduced to zero), but the larger the third countries and their number (the trade possibilities), the stronger this effect will be.

A further clarification of a subject that might be up for debate among econophysicists is required. In its basic formulation, the gravitational model of trade is correctly intended as gravitational and not as magnetic. Indeed, a gravity field can only attract, while a magnetic field also produces a repulsive force. Logically, the mass of a country (GDP) attracts trade and cannot reject it. In this sense, the fact that a third country is capable of deviating trade flows towards itself, shouldn’t be considered as a repulsive force in relation to a pair of countries, but rather an opportunity cost. As with gravity in physics, every commercial attraction will continue to exist and will not be nullified, but the trade flow will simply aim at one country rather than another. The attraction or magnitude of the country’s internal components must be added to the third country’s opportunity-cost effect because its internal market can produce a force of attraction on firms and other economic entities. As aforementioned, usually, this aspect is not directly considered because it is included in the assessment of distance. If considered, it should describe the degree of openness of the country to foreign trade. Large countries, with cities and inhabitants well distributed in space, with a dense or concentric territorial extension and high-developed GDPs, could trade less than expected due to a low degree of openness to foreign trade and to the specific location of population and production centers. It is important to stress that the scarce degree of openness discussed above should be understood as the result of the relevant regions and masses’ conformation, rather than a politically-motivated shift towards autarchy. The countries taken as examples will be characterized by free trade. Moreover, a low degree of openness could be caused by geographical isolation, and other spatial and cultural characteristics. In our case, it is important to understand how internal commercial attraction among the regions of a country can have a negative influence on the openness to foreign trade of the country itself. Given a fixed area, the circle is the geometric figure whose sum of distances from its center mass to its borders at each point is as small as possible. If we apply this reasoning to a state and consider the force of attraction between the different regions of the country, we will have that, ceteris paribus, the internal commercial gravitational pull will be stronger in more compact states where bordering regions are not far from the central ones. In addition, the economic weight of every single region should be considered in the gravitational calculus, as well as the surrounding foreign countries’ position and economic weight, and the geographical distribution of populations. For example, in geometry, a square is more similar to a circle than to a rectangle. In a rectangle, the shorter sides will be geometrically more distant from the barycenter of the rectangle itself, and therefore less influenced by it. The same could occur with asymmetrical shapes, but with different degrees of openness to the foreign market within a state's sub-units. This isolation of the peripheral regions could favor the national market over the foreign one. The internal gravitational pull will have a negative impact on trade with foreign countries since trade between nations will be concentrated within each country. The concept is that domestic trade attraction naturally takes away some of the foreign-directed flows on an econophysical level as well. However, a country with a high level of domestic trade may paradoxically attract smaller countries with a higher degree of openness under its economic influence, which may result in those nations falling under the country’s sphere of economic influence. From an econophysics perspective, one could speak of smaller countries entering the economic orbit of larger countries. Such theory is indirectly and empirically proved by the studies of Tinbergen (1964), Anderson and Yotov (2010), Alesina et al. (2005) and Rose (2006). The scholars found out that, contrary to what they expected, larger and wealthier countries traded “less than normal”, that is, below the regression line generated by the gravity model equation. On the other hand, smaller countries appeared to be more dependent on foreign exchanges. This outcome may appear to be inconsistent with the accuracy of the gravity equation in economics, but, in fact, it can be easily interpreted in an econophysics model of trade. As previously stated, gravitational trade forces exist not only between countries, but also within them, just as in physics, gravity occurs both outside and inside the masses, which, in turn, consist of an ensemble of units with their gravitational force. Therefore, smaller states will end up commercially gravitating toward or orbiting within the sphere of economic influence of larger countries, just as smaller masses gravitate within the sphere of influence of larger ones in physics. The accuracy of the gravity equation in economics is yet again empirically demonstrated only through a vector-based logic of analysis.

One of the most discussed issues surrounding the study of gravitational fields in international trade might be whether to talk about different markets, and thus different fields, or to consider a single global market and, consequently, a single trade field. When calculating flows or trade attractiveness, it is useful to consider (single) portions of individual nations. Despite the theoretical convenience of this operation, the researcher should be aware that, in a globalized and extremely connected world like ours, omitting a portion of third countries, even if quite distant, is a simplification. For instance, new trade agreements between New Zealand and Australia may have an impact, albeit a small one, on trade between Spain and Portugal. This problem is not new to economics, and it affects the boundaries of its theories, laws, or analysis in systems that can either be individual areas or geographical sections. However, for practical reasons, this can be divided into different markets and gravitational fields that are connected to each other and represent more or less homogeneous sections from a geographical, political, or social perspective. Moreover, a market can be divided into more markets depending on the economist’s interest, which can range from general to specific. Thus, it makes sense to refer to larger regional markets, such as the European, Southeast Asian, Mediterranean, and Middle Eastern markets, but also to single national markets (for example, the Italian or Spanish market). Depending on the goal of the economist, the analysis can go further into detail to regional or provincial markets, as well as to markets made up of single individuals. Surprisingly, the same thing happens in physics. Bodies can have different spatial extensions, and the gravity force among them is calculated by summing up the “points mass” inputs that make up the bodies. Likewise, the gravity field of masses separated from each other will result from the vector integration of the single fields. This observation is significant from a theoretical point of view. Just, as in physics, a change in mass, even if peripheral, affects the gravitational field of the entire universe, in economics a variation of commercial forces in a market can influence (even in the slightest way) the forces acting in global trade. The more interconnected the economies, the more the global trade will change. This parallelism between physics and social sciences can be clearly seen in the works of Isard (1954) and Carey (1858:42), who saw humans as molecules in a society. The advantage of considering gravitational forces in economics tackles precisely these issues. According to the reductionist logical properties of mechanical physics, it is possible to apply the same rules of trade attraction on larger markets as well as on smaller ones through the principle of superposition, and composition and decompositions of vector forces. Although this is the result of significant economic simplifications, as mentioned, they are the same as those underlying econophysics (with the exception of applications of quantum physics) and, to some extent, the neoclassical paradigm. Trade agreements and the creation of single markets, considered as multilateral agreements between states aiming at removing trade barriers, influence and modify commercial gravity fields. Nevertheless, proceeding with a reverse logical reasoning could be helpful to define markets through the gravity model theory. It would be possible to derive markets according to a logic of distance reduction. Each geographical region of the same market would share one or more variables in all its spaces, which in turn would reduce the distance between the countries that are part of it. For example, in the eurozone, besides geographical proximity, a single currency would reduce distances by facilitating exchanges. Another instance involves South and North America, where a reduction of distance will be brought about by shared languages and similar economic and social traits. The stipulation of trade agreements like Mercosur and NAFTA will further contribute to favoring exchanges between the two geographical areas. Having calculated all economic distances, which implies taking into account all conceivable factors, it appears that it would be possible to provide a map of the forces of commercial attraction and the commercial gravity fields generated using a graph, or, to simplify, through a geographical expansion propagation. Therefore, areas with a strong magnitude between different states indicate the existence of important trade agreements between those states. In fact, high gravitational attraction is a sign of large masses and low trade barriers.

7 Empirical Analysis of the Theory of Gravitational Field in Economics

The following Fig. 8 shows a simulation in Europe, i.e., an imaginary closed set that includes the five countries with the highest GDP (Italy, Spain, United Kingdom, France, and Germany). For each of them, annual GDP values in dollars for the year 2019, obtained from World Bank databases, were taken into consideration. This particular year was chosen because it provides more stable data than the following period of crisis caused by the global SARS-CoV-2 pandemic and the geopolitical instability generated by the war in Ukraine. For each of the five countries, a square tile grid map was created, with a single tile of the same size and shape serving as its representation. Each section of the map is then divided into equal squares by abstraction. Albeit somewhat arbitrary, the cell number is meant to fit the clusters and accurately reflect how they are identified by statistical research. It clearly represents a simplified checkerboard pattern, which can be expanded and made more complex by taking into consideration a higher number of countries. However, as our theoretical goals are to validate the reasoning process, we only need a simple checkerboard representation. For the purpose of the analysis, I will utilize a collection of Dynamic Gravity data collected by USITC (The United States International Trade Commission), which is almost identical to the one provided by CEPII (Centre d’Etudes Prospectives et d’Informations Internationales). According to the USITC website, the CEPII dataset presents several limitations, including a static set of countries that does not reflect the rise and fall of various states. These omissions may lead to bias in the assessments of the effects of certain gravity measures on trade flows. Moreover, many variables used by CEPII do not show significant variations over time or in relation to the magnitude of the conflict under consideration. In other words, these degrees of magnitude are not accurately represented in the CEPII dataset. To address these limitations, the USITC dataset considers the variables used by previous datasets while also considering the time and size variations of conflicts in order to improve the transparency of estimations. Essentially, there are three main barriers. States’ imprecise and asymmetrical shapes limit their inclusion within a square. The second one is that the estimated distance in the USITC dataset is a population-weighted distance between pairs of nations. Finally, given the size and irregularity of the states’ geographic boundaries, it is difficult to graphically represent the source of an economic gravity field on a political map. In order to create a Square Tile Grid Map that addresses all three issues, we opted for considering spatial population density, which is more concentrated. In particular, we originally used the spatial visualizations found in Professor Alasdair Rae’s (2018) in-depth study. His map is shown in the Fig. 6. Based on geospatial data from Eurostat, these graphics illustrate the population density across Europe in detail.

Fig. 6
figure 6

European population density. Data: Eurostat. Source: Alasdair Rae (2018)

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It was possible to graphically represent the gravity field with a Square Tile Grid Map by using population density, which is consistent with the employment of population distances found in the same USITIC gravity database. Albeit through simplifications, a more circumscribed spatial area can be identified as the source of each respective economic field. This zone, which is located within a cell of our square tile grid map, depicts the most urbanized and inhabited region of each state. The masses and the distances belong to the gravitational model data from 2015–2019. These are, therefore, normalized quantitative variables for which the measurement scale has been eliminated in order to obtain pure numbers within a certain range of values. We will have masses from value 2 to 4 and distance from value 1 to 3. To represent the field on a checkerboard with unit magnetic-field decay, it is necessary to consider masses with a minimum magnetism of two, otherwise, the field cannot be represented. In accordance with the previous range, for distances, one must consider a normalization between 1 and 3, since a unit decay outside the single country might result in a level of magnetism ranging from 1 to 3. The normalization formula from mathematics, here recalled in an interval other than [0–1], is \({z}_{i}=\frac{{x}_{i}-min}{max-min}+min\).

This normalization formula (Fig. 7) refers to the reduction of the dataset so that the normalized data fall between 0 and 1, eliminating the effects of variation in the dataset’s scale. The equation for normalization is derived by first subtracting the minimum value from the variable to be normalized, then subtracting the minimum value from the maximum value, and finally dividing the first result by the second.

Fig. 7
figure 7

GDP normalization. Source: Own elaboration

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Considering a distance decay effect of minus one, the unit of mass magnetism for each square of distance is added for horizontal, vertical, or diagonal movements. The numbers in the cells are simplified representations of the field’s commercial attractiveness potential, and they have the same colors as the country they refer to (Fig. 8). As mentioned, the commercial potential generated by the economic mass will decrease by 1 unit for every box, as it moves away from its country. In the same way, the lines of force decrease the farther they are from the field’s masses. We must keep in mind that when we consider the gravitational commercial field, we are referring to the potential commercial attraction of, for example, a mobile enterprise in space and not to the effective trade flow between states. In other words, we could consider the sum of the gravitational attraction potential generated by the various states in this precise point of space for each box. The resulting gravitational potential represents the trade opportunities for an enterprise at that specific point in space: the stronger the magnetic field in a given position, the greater the trade flow. Due to the properties of vectors, the overall field will be equal to the sum of all individual fields generated by each country. Therefore, the commercial potential of a box will correspond to the sum of the single potentials in this point for each nation. According to this imaginary data, the yellow squares represent the four regions of space where magnetism should be maximum.115 Since in the simplified example the masses of the four countries are given and the mass of the observer would be totally negligible, the only not given value would be the position of the observer. Hence, magnetism (income potential) is a function of the position \({D}_{ij}\) only.

Fig. 8
figure 8

Stylized example of localization theory through commercial gravity fields. Source: Own elaboration

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For example, in this political map of Europe (Fig. 9), it is possible to observe the so-called “Blue Banana”. The “Blue Banana” is a homogeneous portion of European space characterized by high levels of development and urbanization, which are among the highest in the world. The similarity with the previous figure is evident.

Knowing the basic economic, political and commercial information about Fig. 9, it is clear that the Blue Banana represents the high magnetism region of Europe, that is, where the vector sum of the individual fields is maximum, i.e., where the products of the masses (GDP) are at their maximum, while the distances at their minimum. In another sense, this localization is where the commercial gravity fields of Europe have the highest magnitude, as a result of the interaction between the attraction force of masses through the distances of the different countries. This principle is partially related to Smith’s (1776) idea that “the division of labour is limited to extension of the market”. In fact, larger markets, that is markets made up by several countries and characterized by greater trade integration, will allow for a higher level of magnetism and concentration. Hypothetically, this will therefore enable an increasing number of specialized industries to meet the demand of a growing number of close consumers.

Fig. 9
figure 9

Blue Banana, source: public domain

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As regards this idea, an economic agent at a point within the Blue Banana will have a greater comparative advantage in terms of production and transportation than one located outside of the Blue Banana. This is since input and output factors at the service of each European market, other factors being equal, are more easily available for activities located within this area of ideal localization. Obviously, an alteration of the distances considered in this case, caused by new trade agreements or distancing measures such as Brexit, will have an impact on the magnetism of the commercial gravity field. After considering this assumption, the position of the commercial center of gravity would be different. For example, assuming a closure of the United Kingdom from the European market, the ideal localization for enterprises, other factors being equal, could be shifted southward towards the other four major member States. The UK would most likely be absorbed by the orbit of different, large commercial gravity fields, for instance, to the benefit of the United States.

This suggests that trade agreements should never be evaluated solely in terms of absolute advantages, but rather considering the comparative advantages deriving from the treaty. In this paper, the term “comparative advantage” is used for its semantic and conceptual value, which goes beyond Ricardo’s two-country international trade model. Ricardo Crosswise deserves credit not only for the breadth of his model, but also for introducing a new concept in the study of economics. The concept of comparative advantage implies that one should focus more on the relative comparison of advantages between each other than to just absolute advantages. Certainly, a trade agreement may provide various commercial advantages to different sectors for the countries parties to the treaty, but these advantages may differ significantly among member states. In a competitive environment, when compared to economic competitors in other countries, a lower cost than in the past but one that represents a higher opportunity cost compared to other countries could potentially lead to the failure of those mentioned sectors in the long run. A theory of commercial gravity fields, as here suggested, using vector calculus, can achieve a more complete analysis of absolute and comparative advantages deriving from trade agreements. Clearly, the latter have a more direct influence on a country or a market’s gravity field, having moved the gravitational barycenter in a more or less advantageous way for a single country. Thus, it would make sense to speak of “compared vector-based advantages”, which refer to a comparison made through a simple vector calculus and the consequential use of commercial magnetism resulting in a whole market.

The European market was considered using the same USITC gravity data. In this regard, a mapping representation of the respective gravity fields for all EU countries was developed. Annual GDP dollar values for the year 2019 from World Bank databases were used for each of them.

Following an econophysical approach to economic analysis, analysis software typically used in the natural sciences was employed. There are many ways to represent a field on a political map since the concept of vector field in physics is not limited only to the study of gravity but is used in magnetism and electromagnetism, meteorology, acoustics, etc. Thus, in practice, a stylized commercial gravitational field can also be represented by using software for analyzing terrestrial magnetism, geopotentials, acoustic or thermal field, etc. For the analysis, a data visualization approach known as a spatial heat map (Microsoft Power BI), which uses color to depict the magnitude of a phenomenon that varies continuously in two dimensions, was chosen. In Fig. 10, the variation in color, by hue or intensity, provides hints about how occurrence is concentrated or fluctuates over space. Limits are several and well known. In particular, they have already been discussed in relation to the differences with the magnetic field. First, the need to use both geographical and non-economic distances if the phenomenon is to be represented on a political map. Secondly, there is two-dimensionality, where points or regions are considered the source of the field. The heat map addresses most of these needs and is capable of aggregating data points together by adjusting the width and dimensions of the map and the “radius”. It is also capable of including the vector principle of superposition previously discussed by aggregating intensity values in a two-dimensional space. However, most of the software has a limited value of radius, and this naturally puts a limit when applying the concept of economic field to markets and countries that are not close to each other in terms of space. However, this problem does not arise when considering the European market. In this sense, the representation below can serve as a useful representation of the trade field mutually generated by EU countries. The resulting gravitational field of that market is the sum of the gravitational fields of individual states. Similar measures could be achieved by changing the parameter for the mass with different locations or with a greater number of countries, or different disaggregations. For instance, one could look at the population and its distribution, at the GDP, at purchasing power parity, at the level of capital in each country, etc. The map clearly shows the continental position of the commercial center of gravity of the market made up of EU countries, as well as the peripheral condition of countries like Greece. Considering the previous theory, the conclusions regarding the interaction of attractive trade forces, the benefits of international trade, and the findings about business location described by this model, which have been extensively discussed earlier, are clear, assuming all other factors remain constant.

Fig. 10
figure 10

Empirical elaboration of the economic gravitational field of EU countries. Source: Own elaboration

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On the left and right of Fig. 11, a similar representation for European countries and eurozone’s markets is respectively shown. When enlarging the database and considering a greater number of countries, the shape becomes more similar to the “Blue Banana”. On the other hand, when the number of countries taken into account decreases, the high-magnetism area is altered and the peripherality of some countries, like Greece, increases.

Fig. 11
figure 11

Empirical elaboration of the economic gravitational field of European countries and eurozone countries. Source: Own elaboration

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By considering this model from a dynamic perspective and considering the economic distances, it is possible to draw interesting conclusions from an economic point of view. The field is constantly changing, first by looking at GDP through variations in the production function, which leads to theories of economic growth, but also by looking at variations in distance. For example, a trade agreement reduces the distance between partner countries by increasing their magnetism. Although this cannot be represented on a political map with geographical distances, it is easy to understand, and the positive effect that shortening the distance would have is easy to imagine. It would indeed favor a transition from more peripheral areas to more magnetic areas of the field. However, it is also evident how negative and conflicting factors can, on the contrary, increase the economic distance between countries by destabilizing and deforming the gravitational field of certain markets. Finally, in terms of comparative advantages, trade agreements between groups of countries may be advantageous in absolute terms for all members by reducing the distances between them. However, relocating the economic gravitational center of a market to a more northern, southern, eastern, or western position could offer significant relative advantages to certain countries. Evident economic and physical considerations would therefore derive from physics. For example, these are considered in the evaluation of the effects of a new trade agreement between Europe and the United States or between China and Russia. Moreover, the enforcement of the Brexit agreement could move Europe’s gravitational and financial center of gravity further south, shifting the United Kingdom economically into the orbit of another important gravitational field, that of North America. On the one hand, this analysis method could be read econophysically through the “Three-body problem” studied by Lagrange in 1772, i.e., an equilibrium problem. On the other hand, it could be read though a return to a bilateral perspective on the equilibrium between the force of gravity and other forces. In physics, an example could be considering both the gravitational and centrifugal forces. In economics, this reminds us of the relationship between a small country that ends up revolving around the orbit of a country with a stronger economic influence. Despite appearing more like an entropic (dispersion) force in economics, this centrifugal force also recalls Krugman's centrifugal force. The key technical point is that gravitational force has an infinite radius of application. However, in the natural sciences, it is difficult to find software that can account for a geographical map and a radius long enough to include all countries. In any case, this is a technical problem that does not affect the conclusions already drawn. In a larger market, the effects would be the same as in a smaller market. The economic gravitational field will simply change with the vector composition principle. The resulting field will therefore be the sum of the individual fields. As already seen, a peculiar characteristic of gravity is that it is a purely attractive and additive force, meaning that the gravitational fields involved always add up. There is no cancellation of gravity like that between positive and negative electric charges. One remark, in particular, concerns distance, which, as mentioned, in this map is purely geographical. However, theoretically, the principles will not change. The resulting field of the entire market changes when a geographic or economic distance is reduced, and the commercial center of gravity shifts proportionally to the mass and distance considered. Therefore, the theory of gravitational fields provides a new conceptual model with different capabilities, fully integrating theories of trade and theories of localization, in a real econophysical framework, in the wake of the founding father Isard (1954).

8 Conclusions

Drawing inspiration from Isard and Stewart’s studies, this work presents an alternative to multilateral developments of gravitational models of international trade. Therefore, the scholars’ theories were thoroughly analyzed, not only within the international economics framework, but also in relation to regional sciences and location theories (see Isard’s superior theory). The objective was to demonstrate that, by focusing on the spatial aspects of economy, trade theory can always be improved. Sometimes, looking back is necessary to move forward. In compliance with this tautology, I discussed the limitations of subsequent multilateral studies conducted by authors who succeeded Walter Isard (1954) and are distant from his econophysics approach. Therefore, the study emphasized the limitations of the multilateral gravitational models that followed Isard and dismissed them. The Anderson and Van Wincoop (2003) model was specifically cited to demonstrate how this is an econometric model that no longer refers to econophysics and, hence, gravity.

Moreover, the concept of remoteness was taken into consideration. This is a variable intended to introduce the effect that third countries have on the trade volume of a pair of countries. Nevertheless, such attempt tries to force multilateralism into a model that still maintains its bilateral structure. Furthermore, I have thoroughly analyzed gravity in physics and built an econophysics theoretical framework necessary to embrace the development of an innovative model of gravity fields in trade. To do this, an econophysical model that combines the theories of international trade and localization was outlined. It was demonstrated how knowledge of gravitational fields is essential to modify, accordingly to an econophysics approach, the development of the gravitational law in international economics. In economics, this vector-based multilateral model responds to the same logics of attraction and concentration typical of the gravity force in physics. Therefore, it is possible to elaborate a broader economic theory of gravitational fields, able to describe a multilateral market model that includes many countries. To simplify, this model was intended as a spatial bi-dimensional model, capable of illustrating commercial magnetism on a geographical or political map using fields and equipotential lines. More specifically, the author revises Isard's idea of combining localization choices analysis and trade flows observation into a single analytical approach, while also proposing a vector analysis as a tool for mapping gravitational fields in which the economic forces generated by interactions between and within countries unfold. With respect to prior dominant models, far from an econophysics approach, the advantages of this new vectorial model are obvious. Moreover, this model is based on the notion of vectorial composition and on a direct comparison of the vectors, as can be seen in the gravitational field map of the European states. It particularly provides an econophysical model that takes into account the shifting vector field and includes both the static and dynamic markets. Future research in this area could pursue several goals. One possibility is to extend the application of the model by considering different field origins and more thoroughly evaluating the activity distribution within each nation's regional space. Another limitation of the current model is the reliance on metric distance. This issue could be addressed by developing new techniques for integrating political maps with economic space, which consists of more than just physical distances. In conclusion, Isard’s theoretical analysis and the issues he raised are still relevant and have not yet been fully addressed. From what is highlighted in the paper, it is evident that, the current literature on the gravitational model is capable of neither successfully synthesizing economic geography and international trade nor creating a “superior theory of international trade”, as Isard sought to do. This study suggests that the gravitational field model here presented is headed towards the integration of trade theories and regional sciences by using a physics instrument, the field, which could pave the way for the development of authentic econophysics models.