GAINS FROM TRADE: THE COSTA RICAN CASE

INTRODUCTION
One of the oldest and most interesting questions in the economic literature is how to quantify the gains from trade (2). Recent work by Costinot & RodriguezClare (2014) (CRC) described how the results of a wide array of trade models developed in the last two decades can provide parsimonious measures of the gains from trade. Those include, for example, one sector models, multiple sector models, and models with intermediate goods. Different structures for how competition works in those markets are also considered, such as perfect, Bertrand, and monopolistic competition (3).
The results presented in CRC (4) are useful for evaluating the effects of globalization and the differences that arise for different countries depending on the level of integration to the rest of the world. The authors use the World Input Output Database (WIOD) constructed by Dietzenbacher, Los, Stehrer, Timmer & de Vries (2013) for computing the gains from trade. However, this database does not include Costa Rica as an individual country, it is included as part of the "Rest of the World".
For a small, open economy such as Costa Rica it is of particular interest to quantify how much does the country gain from having its economy open to trade with the rest of the world. The two main contributions of this paper are that, first, it updates the results from CRC to the 2011 data, which changed quantitatively after the trade collapse that followed the Great Recession and, second, it computes the gains from trade for Costa Rica using this version of the WIOD.
The results are, in general, consistent with the gains from trade from similar small open economies. The gains from the current situation are above the average of the rest of the world, while increasing dramatically when the assumptions allow for multiple sectors in perfect competition.

THE NEW DATABASE
Costa Rica did not update its own Input Output Matrix (IOM) for many decades. Leiva & Vargas (2014) mention that before 2014 there had been only two matrices in the history of the country, one from 1969 (Modelo InsumoProducto para Costa Rica1969: Un ensayo de Economia Interindustrial), and the 2011 version developed by the Banco Central de Costa Rica (BCCR). There have been other approximations in between, such as the matrix from 1991, which had been the most widely used before the new publication. Even though in February 2016 the newest version of the IOM (with data for 2012) was published by BCCR along with a new set of data for the national accounts, this matrix has not been included in a newer version of the WIOD.
The 2011 version which was published in 2014 was constructed using the most recently available information in accordance with the best practices recommended by the United Nations Statistical Commision. In Bullon, Mena, Meng, Sanchez, Vargas & Inomata (2015) the authors document how this IOM was embedded into the World Input Output Database. Their work allowed for the publication of a domestic version of the WIOD that includes Costa Rica as a single country in this database, and not part of the Rest of the World. The authors of Bullon et al. (2015), the Ministry of Foreign Trade (COMEX) and the Central Bank (BCCR) deserve recognition for preparing this database for external use. There is a significant amount of work that can be done thanks to this effort, and the trade and industrial organization literature of Costa Rica can expand much more thanks to this accomplishment.
The Costa Rican 2011 IOM has 76 products, which were aggregated into 35 industries to match the international version. The results shown in this paper are not exactly the same as those presented in Costinot & RodriguezClare (2014) because the version into which the Costa Rican IOM was embedded was the 2011, whereas the authors use the 2008 version. It is also the case that this database shows trade data after the 20082009 crisis that caused a collapse of the quantity of international trade in the following years, which affects the magnitude of the gains from trade.
One relevant difference from CRC is that for the calculations presented in this paper, 16 sectors are used for the aggregation levels, instead of the 31 sectors used in the original paper (5). The reason for this is that the Costa Rican IOM lacks data on some of the sectors, and makes the inversion of the matrices required for the computation impossible without some additional aggregation.

COMPUTING GAINS FROM TRADE
As it was mentioned, the goal of this paper is to apply the same methodology of CRC for the version of the WIOD that includes Costa Rica as a separate country (CRWIOD henceforth), which was developed by COMEX (see Bullon et al., 2015). Even though there is no new theory developed throughout this paper, some basic elements of how the gains from trade are computed in each version will be described explicitly.
I present the main elements that help to understand how the model described works: the preferences, the price index that corresponds to those preferences, the price of each good according to the assumption made regarding the competition of the economy, and the gravity equation that results from the solution of the model. While I give general notions of the relevant elements of each model, the complete description of the derivation can be found in CRC. The discussion of the many caveats that should be considered when analyzing each type of model are also found in that paper.
Armington Model
The simplest multicountry gravity model used in international trade which can match trade patterns across countries is an Armington model, which assumes an endowment economy. This setup can serve as a benchmark for comparison for the rest of the models and assumptions presented in the rest of the paper. In each of the i = 1, ..., n countries there is an endowment of a unique domestic good Q. The preferences take the form
[mathematical expression not reproducible] (3.1)
with [C.sup.ij] is the quantity consumed of a good exported by the country denoted with the first subscript, to the country denoted with the second subscript. In the notation of this paper, i is the country of origin of the goods, where they are exported from. In contrast, j is the country where goods are received; hence, measures like welfare have to be analyzed in country j. The parameters [[psi].sub.ij] > 0 are exogenous preference parameters, and [sigma] > 1 is the elasticity of substitution between the goods. There is a corresponding price index associated to preferences 3.1 with the goods consumed in each destination country j:
[mathematical expression not reproducible] (3.2)
where [P.sub.ij] is the price of the good produced in country i (also called "good i" because it is endowed to that country) exported to country j. The trade costs [[tau].sub.ij] of a good being exported from country i to country j are assumed to take an iceberg form, hence [P.sub.ij] = [[tau].sub.ij][P.sub.ii]. Given that the price index [P.sub.ii] can be expressed as a function of the county i's total income, [Y.sub.i], and the endowment [Q.sub.i], [P.sub.ii] = [Y.sub.i]/[Q.sub.i], we can get the following expression for the price of each good exported from country i to country J.
[P.sub.ij] = [Y.sub.i][[tau].sub.ij]/[Q.sub.i] (3.3)
This simple economic environment results in a gravity equation that describes the trade flows between each pair of countries, [X.sub.ij]. This gravity equation relates the preference parameters and [psi], country j's total expenditure [E.sub.j] [equivalent to] [[summation].sup.n.sub.i=1] [X.sub.ij], the countries' total endowment [Q.sub.i] and income [Y.sub.i], and the transport costs. This expression takes the form:
[X.sub.ij] = [([Y.sub.i][[tau].sub.ij]).sup.[epsilon]][X.sub.ij]/ [([[summation].sup.2.sub.l=1]([Y.sub.l][[tau].sub.lj]).sup.[epsilon]][X.sub.lj] [E.sub.j] (3.4)
where [X.sub.ij][equivalent to] [([Q.sub.i]/[[psi].sub.ij]).sup.[sigma]1]]), and [epsilon] is the trade elasticity:
[epsilon] [equivalent to] [sigma] 1 = [partial derivative]ln([X.sub.ij]/[X.sub.jj])/[partial derivative]ln[[tau].sub.ij]
In general equilibrium, two conditions must hold: [Y.sub.i] = [E.sub.i] and [Y.sub.i] = [[summation].sup.n.sub.i=1] [X.sub.ij]. Hence, the gravity equation can be written
[Y.sub.i] = [[summation].sup.n.sub.j=1] [([Y.sub.i][[tau].sub.ij]).sup. [epsilon]][X.sub.ij]/[[summation].sup.n.sub.l=1] [([Y.sub.l][[tau].sub.lj]).sup.[epsilon]][X.sub.lj] (3.5)
This system of n equations can be used to compute an equilibrium with n unknowns, which can then in turn be used to compute the levels of expenditure and bilateral trade flows using the budget constraints and the gravity equation 3.4. This system of equations can be used to develop counterfactual welfare exercises, given changes in the trade costs that countries face.
The relevant measure for welfare is real consumption, which can be defined as [C.sub.j][equivalent to][E.sub.j]/[P.sub.j] Arkolakis, Costinot & RodriguezClare (2012) show that for a wide variety of trade models it is possible to compute the changes in real consumption when comparing steady state equilibriums (6) from two sufficient statistics, namely the elasticity of imports with respect to the variable trade costs, e, and the share of expenditure in domestic goods. The variable [[lambda].sub.jj]. is formally defined as the share of expenditure on goods from the same country, [[lambda].sub.jj] = X.sub.jj]/[E.sub.j].
[[lambda].sub.jj] = [X.sub.jj]/[E.sub.j] = 1  [[summation].sub.i[not equal to]j] [X.sub.ij]/[[summation].sup.n.sub.i=1][X.sub.ij] (3.6)
For these types of models, welfare changes are defined as changes in real consumption given a foreign shock. In the case of the Armington model, the welfare consequences of changes in trade costs from [tau] to [[tau].sup.0] can be computed simply as:
[mathematical expression not reproducible] (3.7)
where, for any variable X, [??] =...