## Introduction

Trade flows are a fundamental part of any economy. Due to this, researchers remain keenly engaged in uncovering the intricate details as to what determines and shapes this critical aspect of the global economy. The gravity model of international trade is one of the leading tools for the empirical investigation of international trade flows. Jan Tinbergen (1962) formulated the traditional gravity equation over half a century ago in analogy to Newton’s law of gravitation. His equation is based on the intuitive relationship that: The bilateral trade flow *X*” *ij *between two countries i and j increases the larger the economies and the closer

(geographically) they are to each other

“*X**ij *= *G **Y**i**α**Y**j**β*

*dist**γ **ij*

Here *Y*” *i *and *Y*” *j *are the mass variables representing gross domestic production in countries i

and j. *d*” *ist**ij *denotes the distance. *α*” , *β*, *γ *are the elasticities to weight the factors. G is a

constant. This simple relationship is surprisingly stable, so that the strong predictive power of the gravity analysis has made it a standard tool in studying international trade and its determinants. However, Anderson and van Wincoop (2003) showed that a lack of theoretical foundation and an incorrect specification cause distorted estimates and lead to incorrect comparative-statics exercises often resulting into misleading conclusions and incorrect interpretations.

In this extract, I independently derive the structural gravity equation a’la Anderson and van Wincoop (2003). I explain its key aspects, its underlying assumptions and its’ essential difference to Tinbergen’s (1962) traditional gravity equation.

“1

The Gravity Equation in general

Head and Mayer (2014) present three definitions of a gravity equation.1 Their properties and structural differences are essential for understanding the derivation below, which is why they are presented at first:

**1. The General Gravity Equation**

“*X**ij *= *GS**i**M**j**φ**ij*The general gravity equation determines the bilateral trade flow

*X*”

*ij*by simply multiplying

the factors *S*” *i *and *M*” *j*. Both terms represent potentials of the exporting country *i*” and the importing country *j*” : They incorporate the whole range of options available for *i*” to export

and for *j*” to import goods. *φ*” *ij *is a dyadic term that affects both trading partners: it indicates the extent to which exporter *i*” has access to importer *j*” in order to deliver the goods. *φ*” *ij *can be

interpreted as trade costs that inhibit trade between the two countries, with 0″ ≤ *φ**ij *≤ 1. G is called the gravity constant. Head and Mayer (2014) particularly emphasize the multiplicative

form of the general gravity equation and its benefits for an estimation.2

**2. The Traditional Gravity Equation:**

“*X**ij *= *GY**i**a**Y**j**b**φ**ij*

The traditional gravity equation is a simple analogy to Newton’s law of gravity. It is not based on any economically theoretical foundation. The traditional gravity equation was also used by Tinbergen (1962) in his gravity analysis.3 Estimating using this equation has resulted in numerous distorted results and misinterpretations in the literature. Despite these shortcomings, it may be emphasized that it contains the significant empirical relationships that made gravity analysis so popular.

1 Head and Mayer (2014, pp. 11-12). 2 Head and Mayer (2014, p. 14).

3 Tinbergen (1962, p. 60).

“2

**3. The Structural Gravity Equation**

*X*” *i j *= ( *Y **i *) ( *E **j *) *φ **i j *Π*i **P**j*

The structural gravity equation differs from the general gravity equation in the degree of its structure: it is expanded by the multilateral resistance terms. The monadic terms *S*” *i *and *M*” *j*

are replaced by

” *S **i *= ( *Y **i *) a n d *M*” *j *= ( *E **j *)

Π*i **P**j*

In this form, the main difference to the traditional form can be seen: In the structural variant, the gravity constant G becomes a variable factor. Baldwin and Taglioni (2007) call it the “**Gravitational Un-constant**** “**:4

“*G *= 1 1 Π*i **P**j*

The termsΠ” *i *=∑*φ**il*⋅*E**l *and*P*” *j *=∑*φ**lj*⋅*Y**l *change with the trading partner and *l**P**l **l*Π*l*

over time. They also correlate with policy variables that influence trade.5 These important points are ignored in the traditional gravity equation, since G is kept constant there. Anderson and van Wincoop (2003) call Π” *i *and *P*” *j *the multilateral resistance terms. As the

multilateral resistance terms and the included cost terms change with their respective trading partners and over time, the structural gravity equation enables a more complete calculation of trade effects through the general equilibrium.6

Two crucial assumptions lead from the general to the structural gravity equation:

**Spatial allocation of expenditures of the importer ***j***! ****:****
**(3.2.1)*X*” *ij*=*π**ij*⋅*E**j*with*π*“*ij*≥0and”∑*π**ij*=1

*j*

4 Baldwin and Taglioni (2007, p. 4).

5 Cipollina et al. (2016, p. 2); De Benedictis and Taglioni (2012, p. 59). 6 Head and Mayer (2014, p. 44).

“3

This assumption states that the exports *X*” *ij *from country *i*” to country *j*” represent the share “*π**ij *of total expenditures from the country of destination *E*” *j *that country *j*” spends on the purchase of a single product from country “*i*.

**General equilibrium – The market clearing condition:****
***Y***! ***i *= ∑ *X **i j*

*j*

This assumption states that the entire production of country i (numeraire good) is being exported and therefore traded (including with itself).

In the following, a possible derivation of the structural gravity equation a’la Anderson and van Wincoop (2003) will be shown. Because of its simplicity, it is predestined for an introduction to the theory of gravity analysis. After the derivation of the gravity equation I will highlight the importance of the multilateral resistance terms for a sound gravity analysis.

The Armington model

The model by Anderson and van Wincoop (2003) (also called the Armington model) was presented in 2003 in their well-known work *Gravity with Gravitas*. It was one of the first models that fits into the definition of a structural gravity equation. The Armington model is based on the work of Armington (1969) and Anderson (1979). According to the Armington assumption each country produces its own variety. Their distinction is modeled by their properties – here in the form of quality. Their distinctness also implies that each good can be clearly differentiated by its country of origin. It thus marks the country of origin in which it is produced and from which it is exported.

*A**i*7 denotes the quality index. *c**ij *is the amount of goods consumed by the inhabitants of the importing nation “*j *from the exporting country “*i *import.

7 Anderson and van Wincoop (2003) use the inverse simulating a distribution. In Head and Mayer (2014) A is interpreted as a quality factor. In this work I follow this convention. See Head and Mayer (2014, p. 15) for more details on that.

“4

**Assumptions of the Armington Model**

On the consumption side, the consumers in the importing country *j*” maximize their utility under the budget constraint. The utility function is modeled with constant elasticity of substitution (CES) so that the preferences are homothetic and identical across all countries. On the cost side, the Armington model is based on the concept of iceberg transport costs: The costs *t*” *ij *are measured in such a way that a share of *c*” *ij *from country *i*” to country *j*” is

considered as a loss. In order to make sure that one unit of a good arrives, *t*” *ij *units of the good need to be exported. Under perfect competition, this also implies *p*” *ij *= *p**i *⋅ *t**ij *, since

there is no surplus.8 Anderson and van Wincoop (2003) also assume that the bilateral trading costs are symmetrical *t*” *ij *= *t**ji *and that there are no costs for domestic trade, which leads to

“*t**ii *= *t**jj *= 1.

**Derivation of the structural gravity equation**

First, the nominal demand function has to be derived. For this, the consumers within the importing nation “*j *maximize their utility:

*σ*

*i*

*U*” = (*Ac*)*σ*−1 *σ*−1 *j iij**σ*

under the constraint of the budget restriction *E*” *j *= ∑ *p **i j **c **i j *.

*i*

The Lagrangian function to be maximized in order to determine the optimal amount of consumption is therefore:

[∑ consumer in country*j*” are:

” *dL *= *dL *= 0 *dc**ij **d**λ*

*σ *]*σ *− 1

*p**ij**c**ij*)

The first-order conditions for determining the optimal amount of consumption for a

+ *λ*(*E**j *− *ii*

*σ *“*L *= (*A**i**c**ij*)*σ *− 1

∑

8 De Benedictis and Taglioni (2011, p. 65).

“5

The solution for ” *dL *= 0 is: *dc**ij*

*i*

*σ*][∑

*σ*−1]

*σ*−1 [

*σ*−1]

(*A**i**c**ij*) *σ **σ*−1 ⋅ *σ*

*σ*−1

⋅(*A**i**c**ij*) *σ *−1⋅*A**i *=*λ*⋅*p**ij*

The solution for ” *d L *= 0 is the equivalent to the budget restriction: *d**λ*

“⇔*E**j *=∑*p**ij**c**ij i*

Next, equation (3.3.1) needs to be rearranged in a way so that the budget restriction can be inserted. This is done by summing over all i:

“⇔ *σ*−1

*i*

(3.3.1)”⇔ (*Ac *)*σ*−1 *σ*−1 ⋅(*Ac *)*σ*−1 =*λ*⋅*p c*

*i ij **σ **i ij **σ **ij ij*

1

“⇔[∑(*Ac *)*σ*−1]*σ*−1 ⋅∑(*Ac *)*σ*−1 =*λ*⋅∑*p c*

*i ij **σ **i ij **σ **ij ij iii*

Inserting into (3.3.1) gives

*σ*−1[]1 “⇔[∑*Ac**σ*−1] ⋅∑(*Ac *)*σ*−1 = *i **i ij i ij*

*σ**σ*1 ∑

*Ac*

*σ*−1

*σ*−1⋅(

*Ac*)

*σ*−1

⋅∑*p c*

*i **ij**σ **i ij **σ **p**ij*⋅*c**ij **iii*

*ij ij*

Inserting *X*” *ij *= *p**ij**c**ij *and solving for *X*” *ij *reveals the nominal import demand function depending on the consumption “*c**ij*:

(*Ac *)*σ*−1 (3.3.2)”⇔*X**ij*= *i ij **σ *⋅*E**j*

∑ (*A c *)*σ *− 1 *i **iij **σ*

In order to take into account the first assumption of a structural gravitation equation – the spatial allocation, see equation (3.2.1) – the price index *P*” *j *for the consumers in country j has

to be taken into account.

Anderson and van Wincoop (2003, p. 8) use the Dixit-Stiglitz price index:

“6

*P*” *j *= *p **i j *1 − *σ *[∑( )1−*σ*] 1

*i **A**i*

!⇔*P*1−*σ *=∑(*p**ij*)1−*σ **j **i **A**i*

For this, equation (3.3.2) is transformed as follows:

“⇔*p**ij*= *p*

(*Ac *)*σ*−1*i ij **σ *⋅*E**j*

*c *∑ (*A c *)*σ *− 1 *ij **i **iij **σ*

(*Ac *)−1*i ij **σ *⋅*E**j*

“⇔ *ij *=*A **σ*−1

*i *∑*i*(*A**i**c**ij*) *σ *“⇔(*p**ij*)−*σ *= [ *A**i**c**ij*

⋅*E*−*σ **j*

⋅*E*−*σ **j*

*i **iij **σ *“⇔∑(*p**ij*)1−*σ*=[ ∑*i**c**ij**p**ij *] ⋅*E*−*σ*

*i **A**i *∑(*Ac*)*σ*−1 −*σ **j **i **iij **σ*

Inserting the budget restriction ” ∑ *c**ij **p**ij *= *E**j *and the price index yields *i*

*A**i *∑ (*A c *)*σ *− 1 *i **iij **σ*

−*σ *]

“⇔(*p**ij*)1−*σ *= [ *A**i*

*p**ij**c**ij *] ∑ (*A c *)*σ *− 1 −*σ*

∑(*p *)1−*σ *” ⇔ *P*1−*σ *= *ij*

*E*1−*σ*

*j*]

= [

To derive the gravity equation, it is further transformed:

*j **i **A**i*

∑(*Ac*)*σ*−1 −*σ **i **iij **σ*

“⇔[∑(*Ac *)*σ*−1]−*σ *=(*P**E**j*)1−*σ **iij **σ*

*i**j*

“7

[∑ ] (*E*)

*σ*−1 and inserted into the equation (3.3.2):

(*Ac *)*σ*−1*i ij **σ *⋅*E**j*

“⇔ (*Ac*)*σ*−1 = *j **σ **i**iij**σ **P**j*

“⇔*X**ij*=

( *j*)*σ*−1 With

*E**j **σ **P*

*X*

(3.3.3) *X*” *ij *= ( *p**ij *)1−*σ**E**j *with “*σ *> 1. *A**i **P**j*

(*AX *)*σ*−1 *i ij **σ*

“*X*=*pc*⇔*c*= *ij*⇔(*Ac*)*σ*−1= *ijijijij**p**ij**iij**σ **p**ij*

above equation results in:

As can be seen, the nominal demand *X*” *ij *dependents on the relative (quality scaled) price ( *p**ij *) (*p**ij*)

” *A**i**P**j *, which corresponds to the relationship between the bilateral price ” *A**i *and the ideal (CES) price index *P**j *.9 The term ( *p**ij *)1−*σ *= *π**ij *denotes the share of expenditures that

*A**i**P**j*

country *j*” spends on a produced variety from country *i*” . Equation (3.3.3) also implies that bilateral trade depends on relative trade costs, because of *p**ij *= *p**i**t**ij*.10

The trade flow then becomes: “⇔*X *=( *p**i**t**ij *)1−*σ**E*

*ij *(*i*)*j*

*A**i*∑ *p**i**t**ij **iA*

9 Baldwin and Taglioni (2007, p. 3).

10 Due to the assumption of perfect competition (Anderson and van Wincoop, 2003, p. 7).

“8

This equation states that the trade between two countries depends on their respective trade costs in relation to the average of the trade costs of both countries.11

In order to derive the structural gravity equation, the condition for a general equilibrium – market clearing – must apply:

*Y*” *i *= ∑*X**ij *= ∑( *p**ij *)1−*σ**E**j *= (*p**i *)1−*σ*∑(*t**ij *)1−*σ**E**j jj**A**i**P**j **A**i**j**P**j*

After solving for the quality scaled price

( *p**i *)1−*σ **E**j *“*A**i *=(*t**j*)1−*σ*

and insertion into the equation (3.3.3) we get the bilateral trade flows depending on bilateral trade costs:

∑*ij **E**j **jP*

*X*“=*Y**i**E**j*( *t**t**ij *)1−*σ **ij**Y**w *(*P*1−*σ**E*)

*j*

In the last step, the world production is being added by expanding the equation with the term ” 1 / 1 .

This leads to the structural gravity equation a’la Anderson and van Wincoop (2003): *X*” *i j *= *Y **i **E **j *( *t **i j *) 1 − *σ*

*Y**w *Π*i**P**j*

The bilateral trade flow therefore depends on income shares from the importing country i, the exporting country j, and the bilateral trade costs in relation to the multilateral resistance (price indices). Anderson and van Wincoop (2003) define Π” *i *and *P*” *j *as the inward and

∑ *ij j *⋅*P **j*1−*σ**Y**w **j*

*Y**w **Y**w*

outward multilateral resistance:

Π”*i*= *t**ij **E**j*1−*σ*

*σ*

*w*]1

*j*

*P*

*j*

*Y*

11 Anderson and van Wincoop (2003, p. 10).

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*P*“*j*= *t**ij **Y**i*1−*σ *[∑( )1−*σ **w*]1

*i *Π*i **Y*

Under the assumption of symmetric costs *t**ij *= *t**ji*12 and balanced trade *X**ij *= *X**ji *both terms become equal: “Π*i *= *P**i*:

*X*! *i j *= *Y **i **E **j *( *t **i j *) 1 − *σ **Y**w **P**i**P**j*

Anderson and van Wincoop (2003, p. 5) make the additional assumption of Π” = *P*1−*σ*. *ii*

However, the model can then no longer be used for panel data.13 The modeling leads to the result:*α*” =*β*=1.Sothemodelpredictsthattheelasticitiesofthemassvariablesareequal

to one.14 Due to the assumption of constant elasticity the Armington equation can be (dis)aggregated and can therefore be used for estimation at the aggregate level and at firm or sectoral level.15

Key aspects of the structural gravity equation

The derivation above shows two key aspects of the traditional gravity equation:

**1. The „Multilateral Dimension“ **16

The structural gravity equation shows that the bilateral trade flow depends on the bilateral resistance term relative to the two multilateral resistance terms Π” *i *and *P*” *j*. Trade is therefore

determined by the relative barriers to trade.17 The multilateral resistance terms capture the effects of the general equilibrium enabling a correct comparative-static analysis. They

12 This assumption is violated when it comes to analyse the effects of preferential trade agreements (De Benedictis and Taglioni, 2011, p. 68).

13 Baldwin and Taglioni (2007, p. 5).

14 Chaney (2011, p. 2).

15 Anderson (2011, p. 5).

16 De Benedictis and Taglioni (2011, p. 64). 17 Anderson and van Wincoop (2003, p. 10).

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measure the aggregated barriers that each country suffers from this trade with its respective trading partners.18

**2. Omitted Variables Bias**

The modeling has shown that the two multilateral resistance terms are a crucial component of the structural gravity equation. Omitting them in the estimation equation therefore leads to an Omitted Variables Bias. Anderson and van Wincoop (2003, p. 4) demonstrate this using McCallum’s (1995) estimation equation:

“*ln*(*X**ij*) = *α*0 + *a*1*lnY**i *+ *a*2*lnY**j *+ *a*3*lnd**ij *+ *a*4*δ**ij *+ *ε**ij*.

Here *Y*” *i *and *Y*” *j *represent the GDPs of both countries *i*” and *j*” . *d*” *ij *measures the distance

between *i*” and *j*” . *δ*” *ij *is a dummy variable. It indicates whether inter-provincial trade or state-

province trade takes place being 1 for the first and 0 for the latter. *ε*” *ij *is the error term and

measures the discrepancies between predicted and actual trade. Due to the modeling the

multilateral resistance terms are now added. The log-linearized form is now *t*” *ij *= *b**ij**d**ρ *(with *ij*

k as a constant and “*t**ij *= *b**ij**d**ρ *): *ij*

“*ln*(*X**ij*) = *k *+ *lnY**i *+ *lnE**j *+ (1 − *σ*)*ρ**lnd**ij *+ (1 − *σ*)*lnb**ij *− (1 − *σ*)*ln*Π*i *− (1 − *σ*)*lnP**j*

The crucial difference between the traditional form of McCallum’s (1995) equation and the structural gravity equation a’la Anderson and van Wincoop (2003) are the two added multilateral resistance terms. These terms are the essential part of the structural gravity equation. Omitting them in the estimation equation leads to a distorted result by omitted variables bias in McCallum (1995), since the bilateral trading costs appear in the multilateral resistance terms and do correlate with them. However, the multilateral resistance terms are theoretical constructs that are not observable.19 Taking them into account leads to a number of challenges when estimating.

18 Cipollina et al. (2016, p. 2).

19 Piermartini and Yotov (2016, p. 6).

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Different models lead to the (structural) gravity

An important finding through modeling is that it is not only the Armington model that leads to the structural gravity equation.20 It can also be derived from other models.21 The gravity model does not depend on specific assumptions such as constant elasticity of substitution. The model by Eaton and Kortum (2002) for example is based on Ricardo’s comparative advantage.22 The Melitz (2003) model explains international trade through monopolistic competition and free market entry. Chaney (2008) extends this including firm heterogeneity. The numerous models differ in their assumptions and the underlying explanations for international trade. As they are based on different assumptions and restrictions, each of them explains only a part of the bilateral trade flow. This aspect is crucial because the interpretation of the importer and exporter terms depends on the underlying model.23 Some models even compete with each other.24 25 Nevertheless, they all meet the two central assumptions of the structural gravity equation. The only requirements are that the importer and exporter terms are multiplicatively linked to the trading costs, and the mass variables are defined consistently.26 In the Armington model, for example, the importer term “*S**i *stands for the quality-based price to the power of 1″ − *σ*. In the case of the Eaton and Kortum model, it represents the product of the average level of productivity in a country and the marginal production costs.27

20 Head and Mayer (2014, p. 44); Santos Silva and Tenreyro (2006, p. 642). 21 Fally, (2015, p. 4).

22 Eaton und Kortum (2002, p. 1775).

23 Baltagi et al. (2014, p. 6).

24 De Benedictis and Taglioni (2011, p. 64).

25 Bergstrand (1990) constructed 1990 a model based on monopolistic competition. Deardorff (1998) shows, that the gravity equation can be derived from the „old” and „new” trade theory, see Gomez-Herrera (2013), p. 3.

26 Fally (2015, p. 5).

27 Baltagi et al. (2014, p. 4).

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Figure 3 shows different theoretical foundations leading all to the gravity equation:

Figure 3: The different theoretical foundations of a gravity equation, source: Larch et al. (2016, p. 12)

Head and Mayer (2014, p. 22) list different foundations for the exporter and importer terms of a structural gravity equation (“*S**i*, *M*” *j*, *φ*” *ij*) (here: *β*” = price elasticity of demand, *γ*” =

income elasticity of demand):

**Table 1: Different models determining bilateral flow**

Traditional Armington model

Krugman, monopolistic competition

*S**i*

*M**j **φ**i j*

ad hoc

*t*1−*σ **ij*

*t*1−*σ **ij*

Yiα | Yjβ |

(wi)1−σ Ai | Ej Pj |

Nw1−σ ii | Ej Pj |

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*S**i **M**j*

*φ**i j*(1 +

*γ*)(1 −

*σ*)

(1 − σ)γ Liwi(σ + γ) | Ej Pj |

Aiθ Ni wiθ | Ej Pj |

!Tiw−βθΠ1−β ii | Ej Pj |

Nα̂−θw−θ−μ[ θ −1] iiiσ−1 | Ej Pj |

CES-CET

Heterogenous consumer

Eaton and Kortum

heterogenous firm model

heterogenous firm model (log- concave)

*t**ij **σ*+*γ *( *a**ij *)*θ*

*t**i j*

!*t*−*θ **ij*

*N**i **E**j*

(*α*̂*i**w**i*)*θ **P **j*

!*t*−*θ **ij*

*ξ **θ *−1 *σ*−1

! *ij*

*t*

*θ*

*ij*

Source: Own presentation based on Head and Mayer (2014, p. 22).

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