# On the Structure of the World Economy: An Absorbing Markov Chain Approach

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. World Input–Output Database

- $n\times n$ matrix $\mathbf{Z}=\left[{z}_{ij}\right]$ with $i=(\widehat{i},r)$ and $j=(\widehat{j},s)$ and $n=JS$, so that $ij$ element of the matrix describes the sales of intermediates from country—industry pair i to country—industry pair j$$\begin{array}{cc}\hfill \mathbf{Z}& =\left[\begin{array}{ccc}{z}_{11}& \dots & {z}_{1n}\\ \vdots & \ddots & \vdots \\ {z}_{n1}& \dots & {z}_{nn}\end{array}\right].\hfill \end{array}$$
- n dimensional final demand vector $\mathbf{f}={\left(\right)}^{{f}_{1}}T$ with the i-th entry describing the sales from country—industry pair i to final users.
- n dimensional gross-output vector $\mathbf{x}={\left(\right)}^{{x}_{1}}T$ where ${x}_{i}={\sum}_{j}{z}_{ij}+{f}_{i}.$
- n dimensional value-added vector $\mathbf{w}={\left(\right)}^{{w}_{1}}T$ where ${w}_{i}={x}_{i}-{\sum}_{j}{z}_{ji}.$

#### 2.2. World-Input and World-Output Networks

#### 2.3. Absorbing Markov Chains

## 3. Results and Discussion

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Proof of the Theorem 1

#### Appendix A.1. Perron–Frobenius Theorem for Positive Matrices

- There is a positive real number ${\lambda}_{1}$, (called the Perron root, the Perron–Frobenius eigenvalue, the leading eigenvalue or the dominant eigenvalue), such that ${\lambda}_{1}$ is an eigenvalue of $\mathbf{M}$ and any other eigenvalue (possibly, complex) in absolute value is strictly smaller than ${\lambda}_{1}$,$$\mid {\lambda}_{i}\mid <{\lambda}_{1},$$
- ${\lambda}_{1}$ is a simple root of the characteristic polynomial of $\mathbf{M}$.
- There exists a right eigenvector ${\mathit{\rho}}^{r}={[{\rho}_{1}^{r},\dots ,{\rho}_{N}^{r}]}^{T}$ of $\mathbf{M}$ with eigenvalue ${\lambda}_{1}$ such that $\mathbf{M}{\mathit{\rho}}^{r}={\lambda}_{1}{\mathit{\rho}}^{r}$, ${\rho}_{i}^{r}>0$ for $i=1,\dots ,N$. Respectively, there exists a positive left eigenvector ${\mathit{\rho}}^{l}={\left(\right)}^{{\rho}_{1}^{l}}T$ such that ${\left(\right)}^{{\mathit{\rho}}^{l}}T$ and ${\rho}_{i}^{l}>0$ for all i.

#### Appendix A.2. Spectral Theorem for Diagonalizable Matrices

#### Appendix A.3. Functions of Matrices

#### Appendix A.4. Proof of the Theorem

## Appendix B. Proof of Theorem 2

## Appendix C. Proof of the Theorem 3

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**Figure 1.**A simple example with three countries ${C}_{1},{C}_{2},{C}_{3}$ and one industry I. The world economy includes three pairs: $1\equiv ({C}_{1},I)$, $2\equiv ({C}_{2},I)$ and $3\equiv ({C}_{3},I)$. The first pair produces intermediate products in amount of p percentage of the total output for the second pair, and final products in amount of ${a}_{1}$ percentage of the total output for the country ${C}_{1}$. The second pair produces intermediate products in amount of q percentage of the total output for the first pair, intermediate products in amount of p percentage of the total output for the third pair and final products in amount of ${a}_{2}$ percentage of the total output for the country ${C}_{2}$. The third pair produces intermediate products in amount of q percentage of the total output for the second pair and final products in amount of ${a}_{3}$ percentage of the total output for the country ${C}_{3}$. (

**a**) The world economy is represented as an absorbing Markov chain with one absorbing state. In this case $\gamma ={[{a}_{1},{a}_{2},{a}_{3}]}^{T}$. (

**b**) The world economy is represented as an absorbing Markov chain with three absorbing states. In this case, ${\mathbf{D}}_{\eta}$, Equation (18), is a $3\times 3$ diagonal matrix with elements ${a}_{1}$, ${a}_{2}$, and ${a}_{3}$. (

**c**) The Ghosh-inverse matrix $\mathbf{L}={(\mathbf{I}-\mathbf{B})}^{-1}=\mathcal{G}$, Equation (11), the output upstreamness, $\mathbf{g}$, Equation (13) and the matrix $\mathbf{M}$. The largest eigenvalue of the matrix $\mathbf{B}$ is $\lambda =\sqrt{2pq}$. The left and right eigenvectors are also shown with $A=p+q+\lambda $. For this example, the product distribution does not depend on p and q and is equal to $\{1/4,1/2,1/4\}$.

**Figure 2.**Mean values for the off-diagonal blocks of $\mathbf{A}$ and $\mathbf{B}$ (in blue). Mean values for the diagonal blocks of $\mathbf{A}$ and $\mathbf{B}$ (in orange).

**Figure 4.**Spearman’s rank correlation for the World Input–Output Database (WIOD), year 2014. Upstreamness $\mathbf{u}\left(\kappa \right)$ and downstreamness $\mathbf{d}\left(\kappa \right)$ are computed as $\mathbf{u}\left(\kappa \right)={(\mathbf{I}-\kappa \mathbf{B})}^{-1}\mathbf{1}$ and $\mathbf{d}\left(\kappa \right)={(\mathbf{I}-\kappa {\mathbf{A}}^{T})}^{-1}\mathbf{1}$, respectively, where $\kappa $ is a parameter. $\kappa =1$ corresponds to the values of the $\mathbf{u}$ and $\mathbf{d}$ of the world networks and the year 2014. The limits $\kappa \to {0}^{+}$ and $\kappa \to {(1/\lambda )}^{-}$ correspond to $\lambda \to {0}^{+}$ and $\lambda \to {1}^{-}$, respectively (see Appendix A).

**Figure 5.**Comparative boxplots of $\mathbf{g}$ and $\mathbf{h}$. Expectation, panels (

**a**) and (

**b**), and standard deviation, panels (

**c**) and (

**d**) of the random variable time to absorption for both the world-input and world-output networks.

**Figure 6.**Histograms (WIOD, year 2014) of (

**a**) the product distribution, (

**b**) the output upstreamness and (

**c**) the input downstreamness. The product distribution does not depend on the type of network (input/output).

**Figure 7.**Global patterns of final demand for the warehousing and support activities for transportation in 2014.

Quantity | Computed with | Equation | |
---|---|---|---|

$\mathbf{g}\left(\mathbf{B}\right)$ | n-dim vector | $\mathbf{B}$ | (13) |

$\mathbf{g}\left(\mathbf{A}\right)$ | n-dim vector | $\mathbf{A}$ | (13) |

$\mathbf{h}\left(\mathbf{B}\right)$ | n-dim vector | $\mathbf{B}$ | (14) |

$\mathbf{h}\left(\mathbf{A}\right)$ | n-dim vector | $\mathbf{A}$ | (14) |

${\mathit{\rho}}_{prod}$ | n-dim vector | $\mathbf{A}$ or $\mathbf{B}$ | (17) |

${\mathbf{m}}_{i}$ | J-dim vector | $\mathbf{B}$ and ${\mathbf{D}}_{\eta}$ | (27) |

${\zeta}_{i}$ | J-dim vector | $\mathbf{A}$ and $\delta $ | (29) |

$\mathbf{PP}$ | $J\times J$ matrix | $\mathbf{B}$ and ${\mathbf{D}}_{\eta}$ | (28) |

$\mathbf{WP}$ | $J\times J$ matrix | $\mathbf{A}$ and $\delta $ | (30) |

GDVA | scalar | $\mathbf{A}$ and $\delta $ | (31) |

GIEVA | scalar | $\mathbf{A}$ and $\delta $ | (32) |

GDFU | scalar | $\mathbf{B}$ and ${\mathbf{D}}_{\eta}$ | (33) |

GIEFU | scalar | $\mathbf{B}$ and ${\mathbf{D}}_{\eta}$ | (34) |

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**MDPI and ACS Style**

Kostoska, O.; Stojkoski, V.; Kocarev, L.
On the Structure of the World Economy: An Absorbing Markov Chain Approach. *Entropy* **2020**, *22*, 482.
https://doi.org/10.3390/e22040482

**AMA Style**

Kostoska O, Stojkoski V, Kocarev L.
On the Structure of the World Economy: An Absorbing Markov Chain Approach. *Entropy*. 2020; 22(4):482.
https://doi.org/10.3390/e22040482

**Chicago/Turabian Style**

Kostoska, Olivera, Viktor Stojkoski, and Ljupco Kocarev.
2020. "On the Structure of the World Economy: An Absorbing Markov Chain Approach" *Entropy* 22, no. 4: 482.
https://doi.org/10.3390/e22040482