[Paper Review] Methods and Concepts in Economic Complexity

Andrés Gómez-Liévano|arXiv (Cornell University)|Sep 27, 2018

Economic and Technological InnovationEconomics, Econometrics and Finance11 references3 citations

TL;DR

This paper introduces a unified mathematical framework within the Theory of Economic Complexity (TEC) to quantify collective knowhow and predict economic development through the interplay of capabilities, products, and countries. It formalizes the Economic Complexity Index (ECI) as the dominant left-eigenvector of a stochastic matrix derived from trade data, linking ECI to national development and revealing its orthogonality to country diversity, thus providing a principled, non-arbitrary definition of complexity.

ABSTRACT

Knowhow in societies accumulates as it gets transmitted from group to group, and from generation to generation. However, we lack of a unified quantitative formalism that takes into account the structured process for how this accumulation occurs, and this has precluded the development of a unified view of human development in the past and in the present. Here, we summarize a paradigm to understand and model this process. The paradigm goes under the general name of the Theory of Economic Complexity (TEC). Based on it, we present a combination of analytical, numerical and empirical results that illustrate how to characterize the process of development, providing measurable quantities that can be used to predict future developments. The emphasis is the quantification of the collective knowhow an economy has accumulated, and what are the directions in which it is likely to expand. As a case study we consider data on trade, which provides consistent data on the technological diversification of 200 countries across more than 50 years. The paradigm represented by TEC should be relevant for anthropologists, sociologists, and economists interested in the role of collective knowhow as the main determinant of the success and welfare of a society.

Motivation & Objective

• To develop a unified, quantitative formalism for modeling the accumulation and coordination of collective knowhow across societies.
• To resolve ambiguities in the interpretation and uniqueness of the Economic Complexity Index (ECI) and Product Complexity Index (PCI).
• To formalize the relationship between national economic complexity, technological diversification, and long-term development using mathematical and empirical tools.
• To establish a principled, axiomatic basis for economic complexity that avoids circular definitions and flawed interpretations.
• To demonstrate how trade data can be systematically processed to extract measurable indicators of collective learning and development potential.

Proposed method

• Modeling economic complexity using a stochastic matrix C derived from country-product trade data, where C represents the probability of capability diffusion across countries.
• Defining the Economic Complexity Index (ECI) as the dominant left-eigenvector of the matrix C, ensuring it is uniquely determined by the Perron-Frobenius theorem.
• Using the matrix decomposition C = R L^T to express the stochastic process of capability diffusion, with R and L representing country and product transition probabilities.
• Applying diffusion maps and consensus dynamics to analyze clustering and similarity in product and country spaces, revealing structural patterns in economic development.
• Deriving the orthogonality between the country diversity vector d and the ECI vector, proving that d^T · ECI = 0 via eigenvector properties of C.
• Reinterpreting the conventional ECI calculation as a special case of eigenvector centrality in a bipartite network of countries and products, ensuring consistency and uniqueness.

Experimental results

Research questions

• RQ1 How can collective knowhow in economies be formally quantified using trade data and network structures?
• RQ2 What is the mathematical basis for the Economic Complexity Index (ECI), and why is it uniquely defined?
• RQ3 How does the structure of capability diffusion across countries relate to economic development and technological diversification?
• RQ4 Why is the ECI orthogonal to the country diversity vector, and what does this imply about the nature of complexity and diversity?
• RQ5 Can the iterative definition of product and country complexity be axiomatically justified, or is it inherently circular?

Key findings

• The Economic Complexity Index (ECI) is mathematically defined as the dominant left-eigenvector of the stochastic matrix C, ensuring its uniqueness and stability under the Perron-Frobenius theorem.
• The country diversity vector d is orthogonal to the ECI vector, as proven by d^T · ECI = 0, which arises because d is the dominant left-eigenvector of C and ECI is a right-eigenvector.
• The product space matrix P = R^T L has the ubiquity vector as its dominant right-eigenvector and the PCI as its sub-dominant left-eigenvector, mirroring the structure of the country matrix.
• The iterative definition of complexity—where country complexity is the average of its exported products’ complexity and product complexity is the average of the countries exporting it—does not uniquely define the vectors, as any right/left-eigenvector pair of C or P satisfies this condition.
• The ECI is positively correlated with national income levels and income growth, and its physical interpretation as a measure of collective knowhow is now grounded in eigenvector centrality and network dynamics.
• The framework resolves prior ambiguities in ECI interpretation by showing it is not a direct measure of knowhow but a derived index reflecting the structural complexity of a country’s productive capabilities.