# A New and Stable Estimation Method of Country Economic Fitness and Product Complexity

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## Abstract

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## 1. Introduction

## 2. Metric Definition

#### 2.1. The Original Metric

- Convergence issues: As stated in a recent paper dealing with the stability of calculating this metric [12]:
- If the belly of the matrix [${M}_{cp}$] is outward, all the fitnesses and complexities converge to numbers greater than zero. If the belly is inward, some of the fitnesses will converge to zero.
- Since an inward belly is the rule rather than the exception, some countries will have zero fitness and as a result all the products exported by them get zero complexity (quality). This is mathematically acceptable, though it heavily underestimates the quality of such products: Even natural resources need the right know-how to be extracted so that their quality would be better represented by a positive quantity. To cure this issue one has to introduce the notion of “rank convergence” rather than absolute convergence, i.e., the fixed point is considered achieved when the ranking of countries stays unaltered step by step.

- Zero exports: The countries that do not export any good do have zero fitness independently from their finite capabilities.
- Specialized world: In an hypothetical world where each country would export only one product, different from all other products exported by other countries, this metric would assign a unity fitness and quality to all countries and products. Though mathematically acceptable, this solution does not take into account the intrinsic complexity of products.
- Equation symmetry: This is rather an aesthetic point, in that Equation (1) are not cast in a symmetric form.

#### 2.2. The New Metric

## 3. Results

#### 3.1. Dependence on the Non-Homogeneous Parameter

#### 3.2. Analytic Approximate Solution

#### 3.3. Country Inefficiency and Net-Efficiency

#### 3.4. Local Convergence

#### 3.5. Robustness to Noise

## 4. Discussion

## 5. Materials and Methods

#### 5.1. Construction of the **M** Matrix

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Second Order Expansion of Fitness and Qualities

## Appendix B. Important Quantities Defined Throughout the Text

$\mathcal{C}$, $\mathcal{P}$: | Total number of countries and products |

$\mathbf{M}$: | Binary matrix with element ${M}_{cp}=1$ if country c is a competitive country in exporting product p; ${M}_{cp}=0$ otherwise; export competitiveness is estimated by means of export volumes |

${F}_{c}$, ${Q}_{p}$: | Fitness of country c and quality (complexity) of product p at the fixed point |

${P}_{p}$: | Inverse of the quality of product p; it is a sort of product “simplicity” (${P}_{p}={\left({Q}_{p}\right)}^{-1}$) |

$\delta $: | Inhomogeneous parameter; this parameter is crucial in achieving a stable algorithm to evaluate the metrics; it will be eventually let go to 0 to get a parameter free metric |

${\tilde{F}}_{c},{\tilde{P}}_{p}$: | Rescaled versions of the corresponding un-tilded quantities: ${\tilde{F}}_{c}={F}_{c}\delta $, ${\tilde{P}}_{p}={P}_{p}/\delta $; these quantities do not depend on $\delta $ as soon as $\delta \to 0$ and are better suited to represent fitness and complexity rather than the un-tilded ones |

${\tilde{Q}}_{p}$: | Similar to the complexity ${Q}_{p}$ above, but for the new metrics calculated with the inhomogeneous algorithm: ${\tilde{Q}}_{p}={({\tilde{P}}_{p}-1)}^{-1}$ |

$\mathbf{K}$: | Coproduction matrix with element ${K}_{c{c}^{\prime}}$ equal to the number of the same products exported by countries c and ${c}^{\prime}$; $\mathbf{K}=\mathbf{M}{\mathbf{M}}^{T}$ |

${D}_{c}$: | Diversification of country c, i.e., the number of products the country c is competitive in exporting |

${I}_{c}$: | Inefficiency of country c defined as ${I}_{c}={D}_{c}-{\tilde{F}}_{c}$; it represents the fitness penalty resulting from exporting goods that are also exported by other countries |

${N}_{c}$: | Net-efficiency of country c; it is a de-trended version of the inefficiency; in the dataset considered ${N}_{c}\approx {D}_{c}^{0.75}-{I}_{c}$; it represents how effectively a country diversifies its exported goods by focusing on products not exported by others, which are usually among the most complex ones |

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**Figure 1.**Dependence on the non-homogeneous parameter: Dependence of fitness and quality at the fixed point on the parameter $\delta $. One country (Afghanistan) and one product (live horses) were chosen arbitrarily from the sample of year 2014.

**Figure 2.**Comparison between the original and the revised metric: Differences in country fitness (left panel) and product complexity (right panel) calculated in the original metric of Ref. [1] (vertical axes) and new metric (horizontal axes) as referred to year 2007. The green line in the left panel is the best least square approximation of power-law type (correlation coefficient 0.989) with exponent ca. 1.53. The dark line in the right panel is the best power-law approximation (correlation coefficient 0.971) resulting with an exponent of ca. 1.38. The year 2007 was chosen randomly. Similar results apply to all the years considered. In particular, the correlation coefficient and the exponent of the green line in the left panel lie between 0.987 and 0.990, and 1.48 and 1.61 respectively throughout the years. For the black line in the right panel we find a correlation coefficient between 0.950 and 0.979, and an exponent between 1.34 and 1.47.

**Figure 3.**Numerical vs Analytic relative error: The histogram of the relative difference $({\tilde{F}}_{c}^{\left(\mathrm{fixed}\phantom{\rule{3.33333pt}{0ex}}\mathrm{point}\right)}-{\tilde{F}}_{c}^{\left(\mathrm{approximated}\right)})/{\tilde{F}}_{c}^{\left(\mathrm{fixed}\phantom{\rule{3.33333pt}{0ex}}\mathrm{point}\right)}$ is plotted with the number of countries on the vertical axis. The approximated values are calculated using Equation (10).

**Figure 4.**Role of diversification: The country inefficiency ${I}_{c}={D}_{c}-{\tilde{F}}_{c}$ is plotted vs. the diversification ${D}_{c}$ with the black line representing the power-law relation ${I}_{c}\approx {D}_{c}^{\phantom{\rule{0.166667em}{0ex}}0.75}$ (linear regression with correlation coefficient 0.994). In the inset the net efficiency ${N}_{c}$, defined as the difference between the black line and the inefficiency of the main graph, is shown. Plotted data pertain to year 2007. We find a similar behaviour for all the years considered with the exponent of ${D}_{c}$ between 0.73 and 0.76, and the correlation coefficient between 0.993 and 0.995.

**Figure 5.**

**Inefficiency cartoon:**Large ovals represent three countries, while small circles represent products. In this simple example, the inefficiency ${I}_{1}$ of country 1 is ${I}_{1}={K}_{12}/{D}_{2}+{K}_{13}/{D}_{3}$. From the figure we get ${K}_{12}=2$ and ${K}_{13}=4$, i.e., the number of products exported by both countries (the cardinality of the intersection sets), and the diversifications ${D}_{1}=17$, ${D}_{2}=5$, ${D}_{3}=20$. Thus, ${I}_{1}=2/5+4/20=0.6$ and the approximated fitness ${\tilde{F}}_{1}\approx 16.4$.

**Figure 6.**Time evolution of fitness and net-efficiency: (Left panel) Country fitness yearly evolution as estimated by the new metric. (Right panel) yearly time evolution of country net efficiency. The net efficiency is a detrended version of the inefficiency defined in the text and displayed in the inset of Figure 4 for the year 2007. Curves were artificially smoothed by a cubic spline for a better visual representation.

**Figure 7.**Noise robustness: Spearman correlation between the ranking of countries based on fitness at zero noise and at different noise levels $\eta $ (see Section 3.5 in the main text). The performance of the two metrics is practically indistinguishable. Note that at $\eta =1$ all the elements of matrix $\mathbf{M}$ are flipped so that the perturbed system is perfectly anti-correlated with the original one.

**Figure 8.**Net-efficiency vs fitness: each line corresponds to the time evolution of the connection between the fitness of a country and its net-efficiency in the period between 1995 and 2014. This figure connects the quantities on the vertical axes of the plots displayed in Figure 6. Lines start from a large circle (year 1995) and end with a small one (year 2014).

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**MDPI and ACS Style**

Servedio, V.D.P.; Buttà, P.; Mazzilli, D.; Tacchella, A.; Pietronero, L. A New and Stable Estimation Method of Country Economic Fitness and Product Complexity. *Entropy* **2018**, *20*, 783.
https://doi.org/10.3390/e20100783

**AMA Style**

Servedio VDP, Buttà P, Mazzilli D, Tacchella A, Pietronero L. A New and Stable Estimation Method of Country Economic Fitness and Product Complexity. *Entropy*. 2018; 20(10):783.
https://doi.org/10.3390/e20100783

**Chicago/Turabian Style**

Servedio, Vito D. P., Paolo Buttà, Dario Mazzilli, Andrea Tacchella, and Luciano Pietronero. 2018. "A New and Stable Estimation Method of Country Economic Fitness and Product Complexity" *Entropy* 20, no. 10: 783.
https://doi.org/10.3390/e20100783