# Various Gravity Models Implemented in Practice

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Gravity is distinguished by its parsimonious and tractable representation of economic interaction in a many country world. Most international economic theory is concentrated on two country cases, occasionally extended to three country cases with special features. The tractability of gravity in the many country case is due to its modularity: the distribution of goods or factors across space is determined by gravity forces conditional on the size of economic activities at each location. Modularity readily allows for disaggregation by goods or regions at any scale and permits inference about trade costs not dependent on any particular model of production and market structure in full general equilibrium. The modularity theme recurs often below, but is missing from some other prominent treatments of gravity in the literature.

The story begins with setting out the traditional gravity model and noting clues to uniting it with economic theory. The traditional gravity model drew on analogy with Newton's Law of Gravitation. A mass of goods or labor or other factors of production supplied at origin i, Yi, is attracted to a mass of demand for goods or labor at destination j, Ej, but the potential ow is reduced by distance between them, dij. Strictly applying the analogy, gives the predicted movement of goods or labor between i and j, Xij. Ravenstein pioneered the use of gravity for migration patterns in the 19th century UK (Ravenstein, 1889). Tinber-gen (1962) was the rst to use gravity to explain trade ows. Departing from strict analogy, aditional gravity allowed the coe cients of 1 applied to the mass variables and of 2 ap-plied to bilateral distance to be generated by data to t a statistically inferred relationship between data on ows and the mass variables and distance. Generally, across many applications, the estimated coe cients on the mass variables cluster close to 1 and the distance coefficients cluster close to 1 while the estimated equation ts the data very well: most data points cluster close to the tted line in the sense that 80 90% of the variation in the ows is captured by the tted relationship. The t of traditional gravity improved when supplemented with other proxies for trade frictions, such as the e ect of political borders, common language and the like.

Notice that bilateral frictions alone would appear to be inadequate to fully explain the e ects of trade frictions on bilateral trade, because the sale from i to j is in uenced by the resistance to movement on i's other alternative destinations and by the resistance on move-ment to j from j's alternative sources of supply. Prodded by this intuition the traditional gravity literature recently developed remoteness indexes of each country's `average' e ective. Each sale has multiple possible destinations and each purchase has multiple possible origins: any bilateral sale interacts with all others and involves all other bilateral frictions. This general equilibrium problem is neatly solved with structural gravity models. For expositional ease, the discussion will focus on goods movements from now on except when migration or investment are specially treated.

## Frictionless Gravity Lessons

Taking a step toward structure, an intuitively appealing starting point is the description of a completely smooth homogeneous world in which all frictions disappear. Developing the implications of this structure yields a number of useful insights about the pattern of world trade. A frictionless world implies that each good has the same price everywhere. In a homo-geneous world, economic agents everywhere might be predicted to purchase goods in the same proportions when faced with the same prices. In the next section the assumptions on preferences and/or technology that justify this plausible prediction will be the focus, but here the focus is on what the implications are for trade patterns. In a completely friction-less and homogeneous world, the natural benchmark prediction is that Xij=Ej = Yi=Y, the proportion of spending by j on goods from i is equal to the global proportion of spending on goods from i, where Y denotes world spending. Any theory must impose adding up constraints, which for goods requires that the sum of sales to all destinations must equal Yi, the total sales by origin i, and the sum of purchases from all origins must equal Ej, the total expenditure for each destination j. Total sales and expenditures must be equal: i.e.

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Thus far, the treatment of trade ows has been of a generic good which most of the literature has implemented as an aggregate: the value of aggregate bilateral trade in goods for example. But the model applies more naturally to disaggregated goods (and factors) because the frictions to be analyzed below are likely to di er markedly by product characteristics.

## Structural Gravity

Modeling economies with trade costs works best if it moves backward from the end user. Start by evaluating all goods at user prices, applying demand side structure to determine the allocation of demand at those prices. Treat all costs incurred between production and end use as being incurred by the supply side of the market, even though there are often signi cant costs directly paid by the user. What matters economically in the end is the full cost between production and end use, and the incidence of that cost on producer and end user. Many of these costs are not directly observable, and the empirical gravity literature indicates the total is well in excess of the transportation and insurance costs that are observable (see Anderson and van Wincoop, 2004, for a survey of trade costs).

The supply side of the market under this approach both produces and distributes the delivered goods, incurring resource costs that are paid by end users. The factor markets for those resources must clear at equilibrium factor prices, determining costs that link to end user prices. Budget constraints require national factor incomes to pay for national expenditures plus net lending or transfers including remittances. Below the national accounts, individual economic agents also meet budget constraints. Goods markets clear when prices are found such that demand is equal to supply for each good. The full general equilibrium requires a set of bilateral factor prices and bilateral goods prices such that all markets clear and all budget constraints are met. This is a great advantage for two reasons. First, it simpli es the inference task enormously. Second, the inferences about the distribution of goods or factors is consistent with a great many plausible general equilibrium models of national (or regional) production and consumption.

## Gravity and Factor Flows

Gravity has long been applied to empirically model factor movements. As with trade ows, the model always ts well. But, in contrast to the recent development of an economic structural gravity model of trade, there has been little progress in building a theoretical foundation. This section sets out a structural model of migration, reviews promising steps toward a structural model of Foreign Direct Investment (FDI) and closes by pointing to the unsolved puzzle of modeling international portfolio capital movements.

## Integrated Superstructure

The gravity model nests inside a general equilibrium superstructure. As pointed out in Anderson and van Wincoop (2004), modularity implies that the problem of resource and expenditure allocation across sectors in the general equilibrium superstructure can be treated separably from the gravity module problem of distribution within sectors to destinations or from origins. Consistency between the two levels of the problem requires xed point calculations in general, but the economy of thought and computation due to separability is extremely useful, and in particular makes it possible to integrate gravity with a wide class of general equilibrium production models. So far, only very simple production models have been used for full general equilibrium comparative statics, but I anticipate that this situation will change.

Another attractive candidate is the Ricardian production model. Eaton and Kortum (2002) nest gravity inside a Ricardian model of production, a choice followed by a host of subsequent researchers such as Arkolakis (2008). An important feature of these models is the action on the extensive margin, as industries arise or disappear. In the Eaton-Kortum model of 2002, the extensive margin is the only margin. Arkolakis and others have variants in which both extensive and intensive margins are active. This is an important feature because disaggregated trade data and especially rm level data indicate that both margins are active.

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