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Quantitative Justification for the Gravity Model in Economics

Vladik Kreinovich1 and Songsak Sriboonchitta2

1University of Texas at El Paso, El Paso, Texas 79968, USA

vladik@utep.edu

2Faculty of Economics, Chiang Mai University, Thailand

songsakecon@gmail.com

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1. What Is Gravity Model

• It is known that, in general:

– neighboring countries trade more than distant ones, and – countries with larger Gross Domestic Product (GDP) g have a higher volume of trade.

• Thus, in general, the trade flow tij between the two

countries i and j: – increases when the GDPs gi and gj increase and – decreases with the distance rij increases.

• A qualitatively similar phenomenon occurs in physics:

the gravity force fij between the two bodies: – increases when their masses mi and mj increase and – decreases with the distance between then increases.

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2. What Is Gravity Model (cont-d)

• Similarly, the potential energy eij of the two bodies at

distance rij: – increases when the masses increase and – decreases when the distance rij increases.

• For the gravity force and for the potential energy, there

are simple formulas: fij = G · mi · mj r2

ij

; eij = G · mi · mj rij .

• Both these formulas are a particular case of a general

formula G · mi · mj rα

ij

.

• So, researchers use a similar formula to describe the

dependence of the trade flow tij on gi and rij: tij = G · gi · gj rα

ij

.

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3. Remaining Problem and What We Do in This Talk

• The formula tij = G · gi · gj

rα

ij

is known as the gravity model in economics.

• It has indeed been successfully used to describe the

trade flows between different countries.

• An analogy with gravity provides a qualitative expla-

nation for the gravity model.

• It is desirable to have a quantitative explanation as

well.

• Such an explanation is provided in this paper.

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4. Additivity

• We would like to come up with a function F(a, b, c) for

which tij = F(gi, gj, rij).

• At first glance, the notion of a country seems to be

very clear and well defined.

• However, there are many examples where this notion

is not that clear.

• Sometimes, a country becomes a loose confederation of

practically independent states.

• In other cases, several countries form such a close trade

union – from Benelux to European Union – that: – most trade is regulated by the super-national or- gans – and not by individual countries.

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5. Additivity (cont-d)

• So, we have several different entities i1, . . . , ik, . . . , iℓ

located nearby forming a single super-entity.

• If we apply our formula to each individual entity ik, we

get the expression tikj = F(gik, gj, rikj).

• Since all the entities ik are located close to each other,

we can assume that the distances rikj are all the same: rikj = rij hence tikj = F(gik, gj, rij).

• By adding these expressions, we get the trade flow be-

tween the super-entity i and the country j: tij =

ℓ

• k=1

tikj =

ℓ

• k=1

F(gik, gj, rij).

• Alternatively, we can treat the super-entity as a single

country with the overall GDP gi =

ℓ

• k=1

gik.

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6. Additivity (cont-d)

• In this case, by applying our formula, we get

tij = F(gi, gj, rij) = F

• ℓ
• k=1

gik, gj, rij

• .
• Our estimate should not depend on whether we treat

this loose confederation a single country: F(gi1, gj, rij)+. . .+F(giℓ, gj, rij) = F(gi1+. . .+giℓ, gj, rij).

• So, the function F(a, b, c) must be additive in a:

F(a, b, c) + . . . + F(a′, b, c) = F(a + . . . + a′, b, c).

• A similar argument can be make if we consider the case

when j is a loose confederation of states, then: F(a, b, c) + . . . + F(a, b′, c) = F(a, b + . . . + b′, c).

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7. Scale-Invariance

• The numerical value of the distance depends on what

unit we use for measuring distance.

• For example, the distance in miles in different from the

same distance in kilometers.

• If we replace the original unit with a λ times smaller
• ne, all numerical values multiply by λ:

r′

ij = λ · rij.

• It is reasonable to require that the estimates for the

trade flow should not depend on what unit we use.

• Of

course, we cannot simply require that F(gi, gj, rij) = F(gi, gj, λ · rij).

• This would mean that the trade flow does not depend
• n the distance at all.

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8. Scale-Invariance (cont-d)

• This is OK, since:

– the numerical value of the trade flow also depends

• n what units we use:

– we get different numbers if we use US dollars or Thai Bahts.

• It is therefore reasonable to require that:

– when we change the unit for measuring rij, – then after an appropriate change tij → t′

ij = µ · tij

we get the same formula.

• In other words, we require that for every λ > 0, there

exists a µ > 0 for which F(gi, gj, λ · rij) = µ · F(gi, gj, rij).

• In other words, we require that

F(a, b, λ · c) = µ · F(a, b, c).

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9. Third Natural Property: Monotonicity

• The final natural property is that:

– as the distance increases, – the trade flow should decrease.

• In other words, the function F(a, b, c) should be a de-

creasing function of c.

• Now, we are ready to formulate our main result.

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10. Definitions and the Main Result

• Definition 1.

– A non-negative function F(a, b, c) is called additive if the following two equalities hold: F(a, b, c) + . . . + F(a′, b, c) = F(a + . . . + a′, b, c); F(a, b, c) + . . . + F(a, b′, c) = F(a, b + . . . + b′, c). – A function F(a, b, c) is called scale-invariant if for every λ, there exists a µ for which, for all a, b, c: F(a, b, λ · c) = µ · F(a, b, c). – A function F(a, b, c) is called a trade function if it is additive, scale-invariant, and increasing in c.

• Proposition 1.

Every trade function has the form F(a, b, c) = G · a · b cα for some constants G and α.

• Thus, we have indeed justified the gravity model.

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11. Trade Flow May Depend on Other Character- istics

• So far, we assumed that the trade flow depends only
• n the GDPs and on the distance.
• The trade flow may also other depend on other char-

acteristics, such as the country’s population pi.

• Indeed, intuitively:

– the larger the population, the more it consumes, – so the larger its trade flow with other countries.

• Similar to GDP, population is an additive property, in

the sense that: – if two countries merge together, – their population adds up.

• How can we describe the dependence of the trade flow
• n two or more additive characteristics?

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12. Let Us Describe This Problem in Precise Terms

• Let us consider the case when each country is described

by several additive characteristics.

• Now, gi is now a vector consisting of several compo-

nents gi = (g1i, . . . , gmi).

• We are interested in the dependence tij = F(gi, gj, rij).
• Let us describe the reasonable properties of this depen-

dence.

• Similarly to the GDP-only case, we can conclude that

F(gi1+. . .+giℓ, gj, rij) = F(gi1, gj, rij)+. . .+F(giℓ, gj, rij); F(gi, gj1+. . .+gjℓ, rij) = F(gi, gj1, rij)+. . .+F(gi, gjℓ, rij).

• Also, it makes sense to require that F(a, b, c) is de-

creasing in c.

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13. Definitions and the Auxiliary Result

• Definition 2. Let m > 1.

– A non-negative function F(a, b, c) is called additive if the following two equalities hold: F(a, b, c) + . . . + F(a′, b, c) = F(a + . . . + a′, b, c); F(a, b, c) + . . . + F(a, b′, c) = F(a, b + . . . + b′, c). – A function F(a, b, c) is called scale-invariant if for every λ, there exists a µ for which, for all a, b, c: F(a, b, λ · c) = µ · F(a, b, c). – A function F(a, b, c) is called a trade function if it is additive, scale-invariant, and decreases in c.

• Proposition 2. Every trade function has the form

F(gi, gj, rij) =

• β
• γ

Gβγ · gβi · gγj rα

ij

for some Gβγ and α.

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14. Discussion

• Example. For the case of GDP gi and population pi,

we have tij = Ggg · gi · gj + Ggp · gi · pj + Gpg · pi · gj + Gpp · pi · pj rα

ij

.

• An interesting property of this example is that:

– in contrast to the GDP-only case, when we always had tij = tji, – we can have “asymmetric” trade flows for which tij = tji.

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15. Acknowledgments

• We acknowledge the support of the Center of Excel-

lence in Econometrics, Chiang Mai Univ., Thailand.

• This work was also supported in part:

– by the National Science Foundation grants HRD- 0734825, HRD-1242122, and DUE-0926721, and – by an award from Prudential Foundation.

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16. Proof of Proposition 1

• Let us first use the additivity property.
• For every b and c, we can consider an auxiliary function

fbc(a)

def

= F(a, b, c).

• In terms of this function, the first additivity property

takes the form fbc(a + . . . + a′) = fbc(a) + . . . + fbc(a′).

• Functions of one variable that satisfy this property are

known as additive.

• It is known that every non-negative additive function

has the form f(a) = k · a.

• Thus, F(a, b, c) = fbc(a) = a · k(b, c) for some function

k(b, c).

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17. Proof of Proposition 1 (cont-d)

• Substituting F(a, b, c) = fbc(a) = a · k(b, c) into the

second additivity requirement, we get: a · k(b + . . . + b′, c) = a · k(b, c) + . . . + a · k(b′, c).

• Dividing both sides of this equality by a, we get:

k(b + . . . + b′, c) = k(b, c) + . . . + k(b′, c).

• Thus, the function kc(b)

def

= k(b, c) is also additive.

• Hence, k(b, c) = kc(b) = b · q(c) for some constant q(c)

depending on c.

• Substituting this expression for k(b, c) into F(a, b, c) =

a · k(b, c), we get F(a, b, c) = a · b · q(c).

• Hence, to complete the proof, it is sufficient to find the

function q(c).

• For a = b = 1, we have F(a, b, c) = q(c).

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18. Proof of Proposition 1 (cont-d)

• We know that F(a, b, c) is a decreasing function of c.
• Thus, q(c) = F(1, 1, c) is also decreasing in c.
• To find the function q(c), let us now use scale invariance

F(a, b, λ · c) = µ(λ) · F(a, b, c).

• Substituting F(a, b, c) = a · b · q(c) and dividing both

sides by a · b, we get q(λ · c) = µ(λ) · q(c).

• We have q((λ1 · λ2) · c) = µ(λ1 · λ2) · q(c).
• On the other hand, q(λ2 · c) = µ(λ2) · q(c) so:

q(λ1 · (λ2 · c)) = µ(λ1) · q(λ2 · c) = µ(λ1) · µ(λ2) · q(c).

• By equating these two expressions for the same quan-

tity q(λ1 · λ2 · c), we conclude that µ(λ1 · λ2) · q(c) = µ(λ1) · µ(λ2) · q(c).

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19. Proof of Proposition 1 (cont-d)

• Dividing both sides by q(c), we get

µ(λ1 · λ2) = µ(λ1) · µ(λ2).

• Functions µ(λ) with this property are known as multi-

plicative.

• Here, for every c, we have µ(λ) = q(λ · c)

q(c) .

• In particular, for c = 1, we get µ(λ) = q(λ)

q(1).

• Since q(c) is an decreasing function, we conclude that

µ(λ) is also an decreasing function.

• It is known that every monotonic multiplicative func-

tion has the form µ(λ) = λ−α for some α > 0.

• From q(λ) = µ(λ) · q(1), we can conclude that q(c) =

G · c−α, where we denoted G

def

= q(1). Q.E.D.

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20. Proof of Proposition 2

• This proof is similar to the proof of Proposition 1.
• First additivity requirement implies that F(a, b, c) is

linear in a.

• Then we show that F(a, b, c) is linear in b, so it is

bilinear in a and b.

• Now, scale-invariance implies that all the coefficients
• f this bilinear dependence are ∼ r−α

ij

for some α > 0.