ï»¿ WPS6347
Policy Research Working Paper 6347
Integrating Gravity
The Role of Scale Invariance in Gravity Models
of Spatial Interactions and Trade
Jean-FranÃ§ois Arvis
The World Bank
Poverty Reduction and Economic Management Network
January 2013
Policy Research Working Paper 6347
Abstract
This paper revisits the ubiquitous bi-proportional is the only consistent maximum likelihood allocation of
gravity model and investigates the reasons why different a matrix of flows between origin and destination. The
theoretical frameworks may lead to the same empirical paper explores the feasibility of wider classes of non-scale
formula. The generic gravity equation possesses scale invariant gravity equations, where gravity is no longer bi-
invariance symmetries that constrain possible theoretical proportional by including nonlinear interactions between
explanations based on optimal allocation principles, trade costs and fundamental country factors such as
such as neoclassical or probabilistic frameworks. These economic size. It shows that such extensions are feasible
constraints imply that a representative consumerâ€™s but that they do not result in a significant improvement
utilities must be separable, and that an entropy model in the explanatory power of the empirical analysis.
This paper is a product of the International Trade Department, Poverty Reduction and Economic Management Network. It
is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development
policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.
org. The author may be contacted at jarvis1@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Integrating Gravity: The role of scale invariance in gravity models of
spatial interactions and trade
Jean-FranÃ§ois ARVIS 1
The World Bank
Keywords: Gravity models, international trade, trade costs
JEL Codes: C61, F12
EPOL, TRAN
1
Jarvis1@worldbank.org, International Trade Department
1 Introduction: Positioning the problem
The gravity model is the workhorse of much empirical work in major areas in the social
sciences, including economics, especially trade and transportation economics. It provides an
intuitive and very effective way to describe bilateral flows between a series of origins (exporters
in the case of international trade analysis) and destinations (importers). A flow between i and j
takes the form:
Flow i j = â€œsizeâ€? of i
x the â€œsizeâ€? of j
x function of variables measuring the separation between i and j,
such as geographical distance or the cost of transport
The initial models (e.g., Tinbergen) took a rather simple form, with explicit reference to the
Newtonian gravitational interaction proportional to each mass and inversely proportional to the
square of distance:
,
In basic economic applications this translates into:
, where the exponent is econometrically estimated.
Successor models have used more sophisticated approaches to estimate size and more
separation variables in addition to distance (see Section 2).
The tremendous empirical success of gravity modeling in economics has led many authors to
look for a theoretical foundation. Pioneers such as Zipf and Savage proposed heuristic
explanations, often based on probability theory (random matching between groups of exporters
and importers of various size), but could not properly account for the separation effects in terms
of cost or distance (Savage & Deutsch 1960); Deardorff 1998).
In the context of modeling transportation and mobility flows, Wilson (1970) proposed an
elegant and entirely consistent formulation based not on the paradigm of Newtonian gravity but
on the canonical ensemble of statistical physics. He postulated that the most probable flow
configuration maximizes entropy subject to budget constraints, and derived an implicit formula
for flows that includes separation costs (Wilson 1970) (Roy & Thill 2003).
2
How trade gravity modeling emerges from trade theory has been the object of intense
investigation following the influential work by Anderson and Wincoop (2003), itself following a
much earlier proposal by Anderson himself (Anderson 1979)and Bergstrand. As noted by
several observers, the problem is that there are almost too many successful explanations (De
Benedictis & Taglioni 2010)(James E. Anderson, 2011). However, as recently observed by Novy
(2012), at the core of most neoclassical explanations of trade gravity is the notion that a
representative consumer will source his or her imports according to destination, following the
Armington hypothesis, with his or her preferences derived from a CES utility.
This paper essentially reverses the problem. It does not try to explain the structure of the
gravity equation from an economic model but rather looks at how the formal structure of gravity
constrains the set of models that can be used to generate the gravity equation. The approach
followed amounts to an â€œintegrationâ€? of the gravity model, where the utility of a problem is
deduced or integrated from the specification of the demand functions.
Sections 2 and 3 discuss how neo-classical models or probabilistic derivations explain
gravity as an optimal allocation, the utility of a representative consumer, or the
likelihood/entropy of flows that are maximized under budget or market constraints. Section 4
describes how scale invariance of the generic bi-proportional model limits the functional form of
the objective function to be optimized to essentially the ones that are used in the literature (i.e.
CES). Sections 5 and 6 look at how to construct alternative models where the scale invariance is
broken, and gravity is no longer strictly bi-proportional because the impact of trade costs on
flows is no longer independent from size.
2 The generic gravity equation and its scale invariances
The generic gravity model takes the general bi-proportional form describing flows between
origin i and destination j
(0)
where
is a factor that represents the push potential of origin i
is a factor that represents the pull potential of destination j
is the impedance representing the intensity of the interaction between i and j
The potentials A and B incorporate information about the "size" of origin and destination.
Potentials are either fixed effects or exogenously determined from country variable explaining
the â€œsizeâ€? of origin and destination.
3
The impedance K incorporates bilateral factors that explain the friction between i and j.
Following Anderson, this friction is referred to as trade costs.
The bilateral impedance is a parametric function of the otherwise exogenous bilateral trade
costs
(0)
where the impedance decreases with higher cost: f is a monotonous increasing function. The
parameter acts as a scale parameter for the trade costs in the impedance. is either
exogenously given or endogenously determined.
Typically the impedance is normalized to one for zero friction or bilateral cost and goes to
zero for very large costs. For instance, in Wilsonâ€™s model the dependence is exponential
(0)
In the CES based models, Ã la Anderson
, where is the Armington constant elasticity of substitution. The two
models essentially correspond through a logarithmic transformation , and
In what follows, except when specified otherwise, we shall refer to the Wilsonâ€™s exponential
dependence.
In early gravity proposals the impedance is simply the negative power of distance (Zipf
1946). More advanced specifications of bilateral trade costs can refer to time or transportation
costs and introduce dummy variables. A typical gravity model takes the expanded form:
.
(0)
4
Scale invariance
The gravity specification of bilateral flows is trivially invariant by the scaling
transformations.
or in the case of exponential dependence on costs
where the and are arbitrary real values.
The possibility of such a transformation is obviously consubstantial to the bi-proportional
standard gravity structure, and tied to the fact that trade costs and country size factors act
independently: trade costs have the same effects on bilateral flows independently irrespective of
the size of the partners. The scale invariance means that if a partner i is further away but bigger,
the corresponding trade flows are unchanged: twice as big potential yields the same bilateral
flows when the corresponding bilateral costs are increased by .
This phenomenon where rescaling of potential is compensated by a shift in trade costs is not
just an algebraic observation, but carries some economic significance. Obviously the inverse
problem (inverse gravity) of determining trade costs from trade flows is ambiguous and requires
additional assumptions to resolve the ambiguityâ€”e.g., that the diagonal trade costs with oneself
is zero or unit impedance or (Novy 2009), (Arvis & Shepherd, 2011). It means
also that trade costs are to some extent not directly observable, although individual components
of trade costs dependent on directly measurable factors such as distance or transportation cost
are.
In the context of trade economics, bilateral trade costs include two categories:
â€¢ Bilateral costs that depend on both origin and destination (group III in the above log-
linear gravity equation)
â€¢ Endogenous costs that represent the thickness of the borders at origin and destination and
do not depend on the country pair, typically represented in log-linear trade gravity
regressions by country specific dummies or non-dimensional variables where country
size does not intervene explicitly.
5
Endogenous costs are in the group II of variables in equation (4). However this group may
also explain the magnitude of the trade potential A, B, along with the size variable in group I.
The frontier between variables that explain endogenous trade costs or trade potential is a matter
of convention as some factors (labor costs, logistics performance/efficiency, cost of market
access) affect both production for export and domestic production, albeit not with the same
intensity.
3 Derivations of the bi-proportional gravity equation
Neo-classical models or probabilistic explanations of the gravity equation derive it as an
optimal allocation, under one or more budget or market clearance constraints, and depending
upon bilateral trade costs. The models found in the literature belong to one or the other category:
â€¢ One-dimensional allocation: an asymmetrical model where the allocation is determined
separately either for the origin or destination. In trade language, the model derives the
optimal allocation for representative importers and exporters for each origin destination
(partial equilibrium).
â€¢ Two-dimensional allocation: a symmetrical model where the full flow/trade matrix is
determined by the optimization problem, under marginal supply and demand constraints.
One-dimensional allocation: Representative trader
The reference asymmetrical model has been proposed by Anderson & Van Wincoop (2003).
Following Armington (1969), for each importing country the representative consumer has
differentiated preferences according to the country of origin of the goods and according to a CES
utility function. The representative consumer in j sources goods from i at price , which include
the fob price plus trade costs
In anticipation of the developments of the next section, we use the language of the indirect
utility function. The latter is homogenous function U of degree zero of price and budget. Hence
U can be written as
,
where m is the budget of the buyer:
6
By Roy's theorem the quantity of goods bought by j from i is given by
Following Anderson, most authors posit a CES form
where is the constant elasticity of substitution, and the a_{i} are share
parameters independent of the destination ij.
Applying the previous formula for yields:
, independently of i
which corresponds to a standard, generic, bi-proportional gravity equation. We refer the reader to
De Benetis (2012) for some alternative representative traders explanation of gravity.
Two-dimensional allocation
Two-dimensional allocations are a one-step derivation of the gravity equation with an
entirely symmetrical treatment of origin and destination. Wilson (1970) proposed half a century
ago a most elegant solution. Wilson determines the flow according to the most likely bilateral
values of discrete consignment or movement (of people or vehicles) between origin i and
destination j, subject to constraints of marginal total in row and column:
. is an integer but expected to be large.
The probability or likelihood of a given configuration is given by
Making use of the Stirling's approximation, the log-likelihood is
The most likely configuration maximizes L under line and column constraints as well as a
budget constraint (Wilson 1970):
Using Lagrange's method, the problem yields an optimal solution which has the standard log
gravity form
7
where a and b are fixed effects and beta are Lagrange's multipliers which are determined by the
constraints of marginal total in row and column.
In the original Wilson model, the gravity exponent is endogenous to the model and hence
not necessarily constant. However the gravity formula can be also deduced by looking for the
maximum of the following Lagrangian derived from the previous log-likelihood
under supply and demand line/column constraints, and with the gravity exponent exogenously
given.
Wilsonâ€™s approach borrows the concept of entropy and draws an analogy from physics of the
canonical derivation of the partition function in statistical physics (Kubo 1967), where the most
likely probability allocation of states, or partition function, correspond to the most probable
configuration (or maximum entropy) for a given average energy (in place of budget constraint).
The Lagrange multiplier , known as the Boltzmanâ€™s factor, is inversely proportional to the
absolute temperature.
Thus, taking also the results of the previous section, it seems that in the context of economic
behavior, the equivalent of the temperature would be the inverse of the elasticity of substitution.
In the same way that the temperature is a scale for the distribution of energy level, the inverse of
(or of ) is a scale of the distribution of trade costs. However the analogy may not be
appropriate in other respects. For instance, in a transposition of the model to canonical or grand
canonical ensemble is not obvious and may not make sense, what would be the equivalent
concept in economics of a thermal reservoir?
The global budget constraint for all origins and destinations mirrors the conservation of
energy. However, the significance is not entirely obvious. The notion of a global arbitrage
between variety of trade linkages and trade costs supported by the trading community is expected
(as in the Lagrangian L1), but why would total trade costs be conserved? It amounts to imagining
one actor, like a multinational corporation, optimizing a variety of trade between locations under
its own budget constraints.
4 Compatible models: One-dimensional allocation case
8
The scale invariance of the generic bi-proportional gravity models constrains the possible
optimization problems that can produce it. The following proposition holds.
Proposition: The class of models compatible with bi-proportional gravity models are
restricted to the following:
1. For one-dimensional allocation (representative traders), the corresponding objective
function (e.g. utility) must be separable
2. For two-dimensional allocation, the corresponding objective function is the entropy
formula (Wilsonâ€™s model)
For the two-dimensional model, the fact that the entropy model is the only one consistent
with the row and column constraints is consistent with the findings by Arvis & Shepherd (2013)
that the Poisson Quasi Maximum Likelihood is the only one preserving marginal totals between
original and predicted value. Indeed the Poisson QML is essentially an entropy formula.
Compatible models: One-dimensional allocation model
The scaling invariance in the generic gravity model is related to some form of independence
of irrelevant opportunities of substitution, where the ratio of substitution between two possible
allocations are independent of changes affecting other cases of allocation. Take the case of
representative buyer k, then the ratio of allocation of supplies from i and j.
will be independent of changes in supplies from other origins or changes in the substitution ratios
for
The generic indirect utility for trader k takes the form of a function
homogeneous of degree zero. Hence it can be written as
where is the price of variety y and m the expenditure of the buyer, hereafter treated as
exogenous.
9
Then by Roy's identity the ratios of allocation are
, where is the derivative of U in the i-th variable
Furthermore, for the one dimension allocation problem of a representative trader to be
invariant to irrelevant opportunities of substitution, it is necessary and sufficient that the utility
function is separable and additive (see Annex 1): .
If additional requirements are made that allocation should depend only on price ratios, then
the are restricted to power functions with the same exponent , yielding the CES
utility. However, this is not the only solution. For instance the logit discrete choice equation is
derived from the choice (Anderson et al. 1992). The later choice corresponds
to a case where allocation depends only on price differences.
Compatible models: Two-dimensional symmetric optimization model
In this case we are looking into the maximization of a Lagrangian L of the bilateral flows
under the constraints of conservation of total in line and column,
, and
and a budget constraint. .
The multiplier method applied to the Lagrangian L yields for each pair or origin destination
{a,i} the partial derivative of L is the sum of line and column constants plus costs:
If the solution of the problem is a generic, scale invariant bi-proportional gravity, then up to a
multiplicative constant the cost can be replaced by the logarithm of the flows
. Hence:
10
And by identifying the dependence in X, and integrating
, which is the
expected entropy like formula for the Lagrangian.
5 Modified gravity and breaking of scale invariance
Departing from the classical explanations of gravity implies a break in the scale invariance.
The easiest and most natural way to do this is to supplement the utility function and Lagrangian
with non-linear terms that explicitly break the symmetry of the model. As apparent in the
following, the models include at least one additional parameter beyond the scale giving and
become substantially more complex.
One-dimensional allocation: Modified translog gravity
In the representative consumer representation, breaking the symmetry means making the
indirect utility function explicitly non-separable by introducing multiplicative interaction
between the trade costs. For instance, Novy (Novy 2009) recently tried to describe trade flows
starting with a translog utility instead of a CES. Indeed, a translog utility is one of the simplest
explicit ways to introduce interactions and break the scale invariance.
Bilateral flows are in the form
where
As apparent in this structure of the "impedance" K1, this extension of gravity is no longer bi-
proportional and breaks a number of symmetries of the traditional equation:
11
â€¢ The bilateral impedance depends not only on the bilateral trade costs but also of other
trade costs of the importer.
â€¢ The size of the exporting economy matters: K1 is sensitive to the share coefficient. of
the exporter, with the direction effect depending on the sign of : if the latter is positive,
the "impedance" is higher for smaller exporter.
â€¢ The translog model breaks the formal symmetry between origin and destination.
â€¢ Transslog is essentially a nonlinear extension of the Cobb-Douglass utility, which
corresponds to a CES of one, while In the context of trade the typical observed CES is
much larger (seven).
We propose below a simpler alternative symmetry-breaking extension of gravity, which departs
more smoothly from standard gravity.
Two-dimensional allocation: Modified gravity equation
In the case of two-dimensional symmetric gravity, the simplest approach to symmetry-
breaking "gravity" consists in adding interaction terms between costs and flows in the
Lagrangian yielding the gravity model, so that the scale invariance does not hold. Indeed, the
original Wilsonâ€™s Lagrangian
also reads as a weighted average of a linear combination of and the trade costs (in the
following angle brackets stands for weighted averages )
,
which is obviously invariant by the scaling transformation
, and
Therefore, a minimalist way to break the scaling symmetry is to add to the Lagrangian a non-
linear multiplicative second-order interaction between trade costs and logarithm of flows, the
most natural choice being their covariance (which is invariant by translation of the log flows and
costs):
12
,
The optimization of this Lagrangian under row and column constraints yields
,
The previous equations yields an explicit solution for log-linearized flows as
,
where
This modified gravity equation
â€¢ breaks scale invariance, a positive increases the effect of trade costs for large flows
(larger than the ), while a negative Î³ further suppresses small flows for the same
trade costs.
â€¢ keeps symmetry between origin and destination.
6 Implementation
The modified gravity equation is applied to the World Trade matrix (the dataset is the same
as the one used in Arvis & Shepherd (2013), zero omitted, using the logarithm of distance
as a proxy for the trade cost. The regression takes the form:
, (5)
where t is the covariance variable
The following table includes the results of estimation of (5) using:
13
1. Poisson regression for the model without interaction ( )
2. Poisson regression for the model with interaction ( )
3. OLS for the model without interaction ( )
4. OLS for the model with interaction ( )
The graphs in Annex 2 plot for Poisson and OLS the predicted values for both standards and
modified gravity.
Table 1 results
20710 observations 1 2 3 4
R2/ pseudo R2 nd nd 0.749 0.820
Log distance coeff ( âˆ’ Î² ) -0.837 -0.802 -1.739 0.837
Wald Chi2/ F 68035 55339 175 266
,, z statistics -34.0 -35.9 -77.4 24.2
,, standardized â€œbetaâ€?
coefficient -1.008 -0.965 -2.093 1.008
Interaction coeff ( âˆ’Î³ ) 0.114 0.298
,, z statistics 16.1 89.4
,, standardized â€œbetaâ€?
coefficient 0.013 0.034
The regression results suggests the following conclusions
1. As expected the Poisson regression has the most consistent result with trade spatial
interaction data (Silva & Tenreyro 2006)(Arvis & Shepherd 2013)
2. The improvement in fit provided by the model (R2 or pseudo-R2) is relatively small in
both models (also visually apparent in the shape of distributions in Annex 2).
3. In Poisson, the coefficient :
â€¢ has a small standardized (â€œbetaâ€?) value meaning that the effect of the non-linear
covariance term is small as compared with the classical log-linear impact of distance
â€¢ is negative and significant, which means a relatively small effect of stronger
suppression of small flows.
â€¢ the predicted values for the trade flows are relatively closed in each models
â€¢ the distance coefficients are close in both Poisson model
4. The OLS regressions has observed by various authors are less robust than the Poisson
ones
14
5. The OLS results (coefficient predicted value) are less consistent between standard and
modified gravity model:
â€¢ Loose fit between predicted and actual value of trade
â€¢ The coefficient of log-distance in the modified gravity turns positive with a relatively
higher (than in Poisson) coefficient of interaction, which suggests an issue of multi-
colinearity better addressed in the Poisson estimate.
7 Conclusions
The use of symmetry principles is not very popular in economics, although the quest for
symmetry is central to scientific understanding in other discipline (physics, chemistry). Hence
rather than a restriction they are a desirable property that make the description of a phenomenon
consistent, less dependent upon extra assumption or parameters, and likely more analytically
tractable (the parsimony principle). Scale invariance in gravity constrains the possible
explanation to essentially the known one. This should be considered more as a desirable property
than a strong restriction.
Possible scale symmetry breaking alternatives are indeed feasible, where trade costs and size
of trade flows interact, instead of the standard bi-proportional structure. However even with the
simplest modification, modified gravity equations add parameters and complexity including
nonlinear equations to estimate econometrically. They do not seem to provide a major
improvement over standard gravity. Thus, given the high quality of fit of modern gravity models,
especially when using Poisson Quasi Maximum Likelihood estimation, there does not appear to
be a very strong case for using an alternative to the bi-proportional standard gravity equation.
This result provides additional support for ongoing and recent efforts to better understand and
measure the origin and nature of bilateral impedances or trade costs within the standard model.
15
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17
Annex 1
Lemma: The property of independence of ratios of substitution holds if and only if the
indirect utility function is separable.
Let be the vector of the variables to be bought from various sources at prices and
the indirect Utility (without loss of generality the budget is set to one). Then the
ratio of substitution between i and j is by Roy's identity
(0), where is the derivative of U in the i-th variable
The independence property means that that this ratio does not depend of for
A separable utility would be of the form
(0)
That the condition is sufficient is immediate as
(0)
, which is independent of for .
To prove the necessary condition, let look at the ratio
, for all or in log
(0)
taking the partial derivative in implies that
(0)
for all and hence
18
must be a function of only for all i, which means by integration that the substitution ratios
are necessarily of the form
(0), where the are function of one variable.
Let be functions of one variable such that the log of their derivative is . Then
(0)
For (11) to hold for every i, j it is necessary that
, where G is the same for all i. Let G be expressed as a function of
the t then the equality
means that
, or
, for all i, j. This is possible only and only if G is a function of only the sum of
the
. Finally, taking U as a primitive of F yields the expected result
Q.E.D.
19
Annex 2
Log of Trade vs. predicted value for standard and modified gravity (Poisson Regression).
1
Log of Trade vs. predicted value for standard and modified gravity (OLS)
2