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

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