Modelo Gravitacional

Gravity for Beginners

Keith Heady

October 22, 2000

Contents

1 The Basic Gravity Equation 2

1.1 Origins: Newton’s Apple . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Economists Discover Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Economic Explanations for Gravity . . . . . . . . . . . . . . . . . . . . . 3

2 Estimation of the Gravity Equation 4

2.1 Economic Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Remoteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 “Augmenting” the Gravity Equation 8

3.1 Income per Capita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Adjacency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Languages and Colonial Links . . . . . . . . . . . . . . . . . . . . . . . . 8

3.4 Border Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Evaluating Trade-Creating Policies 10

4.1 Free Trade Agreements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2 Monetary Agreements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

ƒMaterial presented at Rethinking the Line: The Canada-U.S. Border Conference, Vancouver, British Columbia, October 22, 2000.

yFaculty of Commerce, University of British Columbia, 2053 Main Mall, Vancouver, BC, V6T1Z2, Canada. Tel: (604)822-8492, Fax: (604)822-8477, Email:keith.head@ubc.ca

1 The Basic Gravity Equation

1.1 Origins: Newton’s Apple

In 1687, Newton proposed the “Law of Universal Gravitation.” It held that the attractive force between two objects i and j is given by

Fij = G MiMj ; (1)

Dij2

where notation is defined as follows

• Fij is the attractive force.

• Mi and Mj are the masses.

• Dij is the distance between the two objects.

• G is a gravitational constant depending on the units of measurement.

1.2 Economists Discover Gravity

In 1962 Jan Tinbergen proposed that roughly the same functional form could be applied to international trade flows. However, it has since been applied to a whole range of what we might call “social interactions” including migration, tourism, and foreign direct investment. This general gravity law for social interaction may be expressed in roughly the same notation:

Mi‹MjŒ

Fij = G Dij’ ; (2)

where notation is defined as follows

• Fij is the“flow” from origin i to destination j, or, in some cases, it represents total volume of interactions between i and j (i.e. the sum of the flows in both directions).

• Mi and Mj are the relevant economic sizes of the two locations.

– If F is measured as a monetary flow (e.g. export values), then M is usually the gross domestic product (GDP) of each location.

– For flows of people, it is more natural to measure M with the populations.

• Dij is the distance between the locations (usually measured center to center). Note that we return to Newton’s Law (equation 1) if ‹ = Œ = 1 and ’ = 2.

2

1.3 Economic Explanations for Gravity

Think of gravity as a kind of short-hand representation of supply and demand forces. If country i is the origin, then Mi represents the amount it is willing to supply. Meanwhile Mj represents the amount destination j demands. Finally distance acts as a sort of tax “wedge,” imposing trade costs, and resulting in lower equilibrium trade flows.

More formally: Let Mj be the amount of income country j spends on all goods from any source i. Let sij be the share of Mj that gets spent on goods from country i. Then Fij = sijMj. What do we know about sij?

1. It must lie between 0 and 1.

2. It should be increased if i produces goods in wide variety (n) and/or of high quality (–).

3. It should be decreased by trade barriers such as distance, Dij.

In light of these arguments we suggest

sij = g(–i; ni; Dij) ;

P` g(–`; n`; D`j)

where the g(•) function should be increasing in its first two arguments and decreasing in distance but never less than zero.

3

To move forward, we need a specific form for g(). One approach uses the Dixit and Stiglitz model of monopolistic competition between differentiated but symmetric firms. This model sets –i = 1 and makes ni proportional to Mi. A second approach assumes a single good from each country, ni = 1, but lets the preference parameter –i differ in such a way as to also be proportional to the size of the economy, Mi. Both let trade costs be a power function of distance.

Thus, we obtain

sij = MiDij€’Rj;

P

where Rj = 1=( ` M`D`j€’)). After substituting and rearranging we obtain a result that is very close to what we had sought for:

MiMj

Fij = Rj Dij’ : (3)

The main difference is that now the term Rj replaces the “gravitational constant,” G. We will discuss the interpretation of that term in the next section.

2 Estimation of the Gravity Equation

The multiplicative nature of the gravity equation means that we can take natural logs and obtain a linear relationship between log trade flows and the logged economy sizes and distances:

ln Fij = ‹ ln Mi + Œ ln Mj € ’ ln Dij + š ln Rj: + •ij: (4)

The inclusion of the error term •ij delivers an equation that can be estimated by ordinary least squares regression. If our derivations in the earlier section are correct, we would expect to estimate ‹ = Œ = š = 1.

2.1 Economic Mass

The economic sizes of the exporting and importing countries, Mi and Mj, are usually measured with gross domestic product. The estimated coefficients are usually close to the predicted value of one. However, it is not unusual to obtain values ranging anywhere between 0.7 and 1.1.

4

2.2 Distance

Distance is almost always measured using the “great circle” formula. This formula approximates the shape of the earth as a sphere and calculates the minimum distance along the surface.

Tip: To calculate great circle distances you need the longitude and latitude of the capitol or “economic center” of each economy in the study. The apply the following formula to obtain the distance measure in miles:

Dij = 3962:6 arccos([sin(Yi) • sin(Yj)] (5)

+ [cos(Yi) • cos(Yj) • cos(Xi € Xj)]);

where X is longitude in degrees multiplied by 57.3 to convert it to radians and Y is latitude multiplied by €57:3 (assuming it is measured in degrees West).

Even for air travel, great circle distances probably underestimate true distances since they do not take into account that most flights avoid the North Pole. For maritime travel, they do not take into account indirect routes mandated by land barriers. Furthermore international shipping cartels often set freight costs that bear little relationship to dis-tance traveled. Also, the costs of packaging, loading and unloading, seem to be primarily fixed costs that do not vary with distance. Taken together, these considerations suggest that distance should matter very little for trade.

While he have many ex-ante reasons to expect little relationship between trade and distance, the facts say that distance dramatically impedes trade. For this presentation, I averaged the distance coefficients from 62 regressions reported in eight fairly recent papers. The samples ranged from 1928 to 1995. The trading partners were mainly nations though some results for the trade of Canada’s provinces were included as well.

ˆ

The average distance effect turns out to be ’ = 1:01: This means that a doubling of distance will decrease trade by one half.

Leamer and Levinsohn’s (1994) survey of the empirical evidence on international trade offers the identification of distance effects on bilateral trade as one of the “clearest and most robust empirical findings in economics.”1

They asked “Why don’t trade economists ‘admit’ the effect of distance into their thinking? One [answer] is that human beings are not disposed toward processing num-bers, and empirical results will remain unpersuasive if not accompanied by a graph.” They showed Germany’s trade but—in the spirit of this conference—I will stay

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