Measuring the Intangibles: A Metrics for the Economic Complexity of Countries and Products

We investigate a recent methodology we have proposed to extract valuable information on the competitiveness of countries and complexity of products from trade data. Standard economic theories predict a high level of specialization of countries in specific industrial sectors. However, a direct analysis of the official databases of exported products by all countries shows that the actual situation is very different. Countries commonly considered as developed ones are extremely diversified, exporting a large variety of products from very simple to very complex. At the same time countries generally considered as less developed export only the products also exported by the majority of countries. This situation calls for the introduction of a non-monetary and non-income-based measure for country economy complexity which uncovers the hidden potential for development and growth. The statistical approach we present here consists of coupled non-linear maps relating the competitiveness/fitness of countries to the complexity of their products. The fixed point of this transformation defines a metrics for the fitness of countries and the complexity of products. We argue that the key point to properly extract the economic information is the non-linearity of the map which is necessary to bound the complexity of products by the fitness of the less competitive countries exporting them. We present a detailed comparison of the results of this approach directly with those of the Method of Reflections by Hidalgo and Hausmann, showing the better performance of our method and a more solid economic, scientific and consistent foundation.


Revealed Comparative Advantage
The Revealed Comparative Advantage [1] is defined by the ratio between the export share in the considered year of product p in country c and the share of product p in the world market: where X cp represents the dollar exports of country c in product p.

Further considerations on the weighted matrix M
The two most important ways of generating a country-product matrix M starting from Eq. (1) have been presented in the main text (see Eqs. (1) and (2)). Other possible choices for the weights are in principle possible, however the one discussed in the paper seems to be the most appropriate to describe the effect of the differences of volumes of export for a given product by different countries for fundamental reasons explained in Sect. 4.1 of the main text. We have seen that the analysis of the global export taking into accounts these weights, from one side confirms qualitatively the results of the simplest case without weights, on the other displays additional important economic statistics on the exports as the Pareto-Zipflike distribution of the ranking of "weighted" fitnesses of countries in strict analogy with the behavior of the total GDP distribution of countries.
3 Further considerations on the triangularity of the countryproduct matrix Poorly diversified countries export almost exclusively ubiquitous products, which are presumably of low complexity value as widely diffused on the market. In other words there is a systematic relationship between the diversification of countries and the ubiquity of the products they make and export: poorly diversified countries have a Revealed Comparative Advantage almost exclusively in ubiquitous products, whereas the most diversified countries appear to be the only ones with RCAs in the less ubiquitous products which in general are of higher value on the market. This means that the wealthiest countries should produce only few products with a high degree of specialization. As wealth spreads in the market, other countries specialize in different (and restricted) sets of products. If we represent all the trading relations in a matrixM describing the bipartite network of countries and products, it should be therefore possible to rearrange rows and columns of the matrix so that a mostly block diagonal matrix appears. This looks instead impossible with the actual matrixM in Fig. 1 of the main text.
4 Algebraic approach to the Method of Reflections and convergence to trivial fixed points As shown in [2], Eqs. (11) of the main text can be rewritten in the vectorial form, p , and where we have calledĴ A =ĈM andĴ B =PM t (the suffix t standing for "transpose"), withĈ andP respectively the N c × N c and N p × N p square diagonal matrices defined by C cc = k −1 c δ cc and P pp = k −1 p δ pp . The second one of Eqs. (2) can be substituted into the first one so that to directly relate only even iterations k In the same way we can write the two-steps equations for odd iterations k (2n+1) c . In an analogous way one can show that for products the following vectorial equation is valid: As shown in [2] the matrixĤ in Eq. (3) is a Markov transition operator, in the sense that H cc ≥ 0 and c H cc = 1 .
This matrix in principle defines a Markov chain C with n−order iterative probability distribution p (n) ≡ {p Note the difference with Eq. (3) in which the operatorĤ appears transposed with respect to Eq. (5). Therefore the MR does not define a Markov chain, but the evolution in the adjoint vectorial space to the one of the Markov measure. Moreover, for the given matrixM the stochastic operatorĤ t is ergodic [3], so that there is a finite order s such that at all s > s all matrix elements [Ĥ] s cc > 0 for each (c, c ). Under these hypotheseŝ H satisfies the Perron-Frobenius theorem stating that the matrixĤ is diagonalizable with a unique eigenvalue λ 1 = 1 and all the others have positive real part and are such that |λ i | < 1 for 2 ≤ i ≤ N c . This implies that Eq. (5) has a unique stationary probability distribution. This coincides with the asymptotic state of the same equation, given by the left eigenvector of the matrixĤ corresponding to the maximal eigenvalue λ 1 . The speed of convergence to this stationary/asymptotic state is substantially determined by the second eigenvalue λ 2 , i.e. of the eigenvalue with maximal real part below λ 1 = 1. For what concerns the MR, it is important to note that, given the structure of the eigenvalues of the matrix H, Eq. (3) converges exponentially in n to the right eigenvector ofĤ corresponding to the eigenvalue λ 1 = 1 with a rate determined by λ 2 . It is simple to show, as a general property of all ergodic Markov chains, that such an eigenvector is uniform, i.e., with all components k * c = k * independent of c. Therefore the MR makes all k . Therefore, at sufficiently large n, This means that in practice in [4] the authors check the correlations of the second right eigenvector v i and the vector whose c th component is the log of the GDP per capita of country c. This explains also why in [4] they stop the iterations to n = 18. This is due to the fact beyond this iterations the width of the distribution k (2n) c measured by Re[λ 2 ] n and for a such large n it is already at the computational limits of resolution.

Comparison of Fitness with the Global Competitiveness Index
A widely known indicator of competitiveness is the Global Competitiveness Index (GCI) issued every year by the International Monetary Fund. Citing the Global Competitiveness Report: The GCI is a comprehensive index that takes into account 12 pillars, or drivers, of competitiveness: institutions, infrastructure, macroeconomic environment, health and primary education, higher education and training, goods market efficiency, labor market efficiency, financial market development, technological readiness, market size, business sophistication, and innovation, and its value is determined mostly in relation to a survey conducted among a pool of Executives of each country.
This measure is then strongly related to the perception of efficiency and the quality of life that a country can offer. This is of course tightly related to the wealth of a nation. As show in Fig. 1 the correlation coefficient between GCI in 2010 and GDP per capita is very high, around 0.82. Conversely the aim of Fitness is to quantify intangible assets and capabilities that possibly are not yet "revealed" in terms of wealth and quality of life, but that are somehow inferable by the complexity of the productive system. For this reason, as shown in Fig. 1 fitness much less correlated with income. We expect to be able to extract valuable information about growth potential by the comparison of fitness and realized income. This aspect will be expanded in an upcoming work.