The gravity equation in international trade: an overview of the introduction of gravity to the study of economics and its systematic barriers

Abstract

This article explores the evolution of the gravity model in international trade studies, emphasising systematic barriers to its acceptance. Employing a chronological and argumentative approach, it examines the emergence of the gravity concept, comparing Isard’s original contribution with Tinbergen’s econometric approach. It discusses how the heuristic nature of economics and a perceived lack of theoretical foundation initially hindered the gravity equation’s use. The works of Tinbergen’s contemporaries, including Linnemann and Anderson, are introduced. Lastly, recent findings in research will be reviewed, such as Krugman’s New Economic Geography, which seeks to encompass Isard’s intention to combine international trade with localisation theories.

Keywords:

• Gravity model
• econophysics
• history of economics
• localisation theories
• new economic geography

JEL classification:

• A12
• B27
• F10

Acknowledgements

I thank Prof. Giancarlo Corò (University of Venice) and Prof. Elisa Barbieri (University of Venice) for their support and helpful discussions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 $A$ typically stands for a constant of proportionality (as in Krugman and Obstfeld 2005), but it may also denote the alternative in the rest of the market. The idea is that as the number of alternative trading partners increases, the trade volume with each of them decreases. In Krugman and Obstfeld (2005), A is a constant, whereas in Yotov et al. (2016, 17), $A$ denotes the inverse of world production (1/Y_world).

2 $Y_i$e $Y_i$ do not necessarily indicate a pair of countries, they may also refer to cities, regions, FTAs, single markets (e.g., the EU) and economic zones (e.g., South America or Southeast Asia). Some authors in the literature consider $Y_j$ as the total spending of a country rather than its GDP (ibid).

3 His extensive approach contains multiregional input-output frameworks, the use of the gravity model and potential models, game theory, interregional programming, industrial complex analysis, conflict resolution strategies, and environmental and ecological analysis (Isard 1960).

4 Van Bergeijk and Brakman (2010) built upon these theories.

5 Samuelson (1949) used constraints on economies of scale and a limited market, along with non-zero travel costs in a two-country model, to weight the income gap between otherwise equal countries.

6 The concept is based on Newton’s groundbreaking discoveries. It is an example of a quantitative expression that is capable of maintaining an economic system’s supply and demand equilibrium.

7 By acknowledging that transport inputs are not derived from a fixed quantity of resources, the author concedes to classify them as intermediary inputs instead of being considered a production factor.

8 Isard formulated a multilateral model to analyse the correlation between a group of countries’ wealth and its impact on the income of nation i. The model considers the friction caused by distances and includes several countries. Isard used a formula similar to that of a gravitational field in physics, as shown in Figure 1.

9 The fact that his model was published with no references in a little-known book’s appendix may be the reason why Isard did not mention it.

10 In economics, gravity started to be used to explain bilateral trade between two countries, following the conventional formula developed by Tinbergen (1962), even though gravity in physics was a multidimensional element ideal for describing multilateral trade. This represented a regression from Isard’s (1954) earlier promotion of a multilateral model. The gravity model faced resistance due to its perceived empirical nature and lack of theoretical foundation. Isard’s conceptual assumptions were disregarded in the literature, resulting in criticisms of the gravity model by the academic community. The model was assessed only based on Tinbergen’s empirical and practical approach because of its widespread diffusion. Probably due to its formal simplicity, and the similarity to Newton’s law, the model became the root of econophysics. If, on the one side, the pragmatic nature of the model and its adaptability were the major factors behind its extensive adoption, on the other side, the lack of sound theoretical foundations somehow restrained its success (Baier and Standaert 2020).

11 In “Demographic Gravitation, Evidence and Applications”, Stewart (1948) applied physics and gravity to demographic research and the study of human behaviors and interactions, finding that borders reduce the “mutual relationships” of flow in gravity models.

12 Understanding the First Law of Geography requires knowing the various methods by which distance can be estimated. Location-allocation models have existed for centuries both in Euclidean space and network space. Theoretically, human geography models are developed in continuous space. However, Hakimi (1964) tracked optimal facility location in discrete space, while Cooper (1972) developed his solution in Euclidean space. Moreover, the Dictionary of Distance, published by Deza and Deza (2006), distinguishes distance regardless of Euclidean or network distance. Geographers may benefit from geospatial distances, Voronoi diagrams, network diagrams, and Internet distances (defined by virtual connections). Deza Michel M. and Elena Deza spatialize non-spatial information, such as new stories, by equating distance with similarity in the Montello manner et al. (2003). Since analogous objects are located closer on a map, high correlation and dissimilarity values imply a close relationship between the two locations. Association and correlation metrics can determine the relations between variables.

13 Different studies tried to employ a gravity model that would gather position from data flows. For example, in a study published in 2021, Steegmans and De Bruin applied a gravity framework to user-generated data of a large housing market platform, showing that gravity describes the patterns of inflow and outflow of hits (i.e., mouse clicks) from one municipality to another.

14 The supply curve differs from the neoclassical view, which is based on maximizing profit and suggests an alternative curve shape. Similarly, the demand curve does not conform precisely to the expectations of the neoclassical paradigm (Ippoliti 2014a)

15 Science history reflects that many scientific theories and laws, including Newton’s law of gravity, were historically regarded as deterministic, meaning events were caused by well-defined factors. However, with the advent of quantum physics and chaos theory, it has been understood that some situations can exhibit non-deterministic behaviour, even though Newton’s law of gravity remains valid for most everyday applications. Therefore, the phrase alludes to this historical evolution in scientific perspective.

16 Economies of scale increase production returns by exploiting the division of labour. They represent the cost advantages that businesses gain because of their size of operation, and they cause incomes to rise disproportionately to the factors of production employed (O’Sullivan and Sheffrin 2003).

17 OLS denotes the Ordinary Least Square estimation technique (Olanike Kareem and Kareem 2014).

18 Figure 2 displays a range of models upon which the gravitational model is based, including monopolistic competition (refer to Anderson 1979; Anderson and van Wincoop 2003), the Heckscher-Ohlin model (Bergstrand 1985; Deardorff 1998), the Ricardian model (refer to Ricardo 1817; Eaton and Kortum 2002), and the entry of heterogeneous firms and market selection (Chaney 2008; Helpman, Melitz, and Rubinstein 2008); sectoral Armington’s CES model (Anderson and Yotov 2016); sectoral Ricardian model (Costinot, Donaldson, and Komunjer 2012; Chor 2010); an input-output sectoral model that connects the gravity model based on Eaton and Kortum (2002) with Caliendo and Parro (2015) model, integrating it with a dynamic framework featuring asset accumulation (Olivero and Yotov 2012; Eaton et al. 2016).

19 If, from a physics perspective, neglecting the role of distance entails a limit case like a point, in economic terms, it transposes in perfect distribution and the absence of concentration. It follows that if the inexistence of distance is implied, the gravity model would be unenforceable.

20 Despite several excellent contributions to gravity theory, among which Krugman (1980), Bergstrand (1985, 1989, 1990), and Deardorff (1998), the gravity model was just considered an example of “social physics” and some of the theoretical rationale had been disregarded in most of the empirical literature until the early 2000s (Yotov 2022), when researches started focusing on a solution to the model’s identification problem (Evenett and Keller 2002).

21 Sometimes countries choose certain trading partners because of ‘‘home bias’’, which can stem from various circumstances, such as historical colonial ties or a shared language. They influence the impact of trade costs, and the term is used with the assumption that they may be reflected. However, as pointed out by Eichengreen and Irwin (1998), such biases, in particular historical ones, are not considered by the traditional gravity model. Consequently, it is difficult to determine the impact of history and other factors on existing trade arrangements.

22 Linnemann (1966) found that the elasticity of trade flows to distance was different from 2 (De Benedictis and Taglioni 2011).

23 Linnemann’s dataset included information on 80 countries and 6300 bilateral trade flows.

24 The model offers an equilibrium analysis of international trade that includes factor movements, spatial dimension, and location issues. These factors also impact on other analysis fields, such as the housing market. In a study published in 2016, McAllister and Nanda investigate the scale and determinants of foreign investments flows between national real estate markets, demonstrating that foreign direct and portfolio investment variables, such as size and distance, have significant effects on foreign real estate investment flows.

25 Similarly, Samy and Dehejia (2011) employ a gravity model to analyze trends and labor standards in the European Union, demonstrating that labor-standard improvements are associated with improved export performance.

26 Studies of the latter were based on bilateral comparisons using the gravity equation to estimate the importance of national borders.

27 Among the many authors that studied the topic see: Baltagi (2008), Bang (2006), and Blonigen et al. (2007).

28 They identified three errors in the gravitational model’s economic formulation and ranked them according to their severity: the golden error, concerning the inaccurate commercial cost estimation; the silver error, where the direction/timing of trade flows must be considered when analyzing trade flows; and the bronze error, which indicates that measuring trade using US average prices leads to deflated estimates.

29 For instance, Flowerdew and Salt (1979) conducted a study on migration between labour market areas in Great Britain. They examined the effects of size and distance and suggested explanations for the geographical patterns identified, which indicated that the gravity model allows for a thorough description of the nature of the effect of size and distance on migration flows.

30 Vozna’s (2016) study investigates wealth determinants by using Wilson’s model of entropy. The findings suggest a plausible similarity between profit and wealth dispersion, while also highlighting how the concept of equality applies to both economic actors and individuals across different countries. The model indicates that greater entropy results in more competitive markets, as equality and dispersion are prevalent in distance absence.

31 It is worth mentioning the works of Batty (1970), Berry and Schwind (1969), Bussière and Snickars (1970), Cesario (1973), Champernowne and Barker (1972) Charnes, Raike, and Bettinger (1972), Openshaw (1976), Scott (1971), Sheppard (1976).

32 About transportation, Bevis (1959) describes the traffic model developed by the Chicago Area Transportation Study to predict interzonal traffic volume, which is a gravity model combined with linear programming techniques. Also, Wiant (1961) analyses the simplified gravity model procedures used to predict traffic volumes in seven Iowa cities in 1980.