Leontief Meets Shannon - Measuring the Complexity of the Economic System

TL;DR

This paper introduces a novel complexity measure for economic systems based on Shannon entropy and Leontief's circular production model, quantifying the informational content of economic flows.

Contribution

It combines Leontief's production theory with Shannon entropy to create a new, quantifiable measure of economic complexity using Markov chains.

Findings

The measure is applied to data from seven countries over several decades.

Results show variation in economic complexity across nations and time periods.

The approach provides a new quantitative tool for analyzing economic structure.

Abstract

We develop a complexity measure for large-scale economic systems based on Shannon's concept of entropy. By adopting Leontief's perspective of the production process as a circular flow, we formulate the process as a Markov chain. Then we derive a measure of economic complexity as the average number of bits required to encode the flow of goods and services in the production process. We illustrate this measure using data from seven national economies, spanning several decades.

Tables

Table 1. Table 1 : Economic complexity H ​ ( 𝒮 ) 𝐻 𝒮 H(\mathcal{S}) , in bits and as a percentage of the maximum level log 2 ⁡ ( n ) subscript 2 𝑛 \log_{2}(n) .

	$n$	[bits]	[%]
AUS 1968	30	2.540	51.76
AUS 1974	30	2.509	51.14
AUS 1986	30	2.782	56.70
AUS 1989	30	2.863	58.34
CAN 1971	31	2.896	58.46
CAN 1976	31	2.863	57.78
CAN 1981	31	2.831	57.15
CAN 1990	31	2.751	55.52
CAN 1997	32	3.226	64.53
DEU 1978	29	2.844	58.53
DEU 1986	29	2.876	59.20
DEU 1988	29	2.890	59.48
DEU 1990	29	2.897	59.63
DNK 1972	28	2.898	60.29
DNK 1980	28	2.911	60.56
DNK 1985	28	2.965	61.67
DNK 1990	28	2.942	61.20
DNK 1997	36	3.310	64.02

	$n$	[bits]	[%]
AUS 1968	30	2.540	51.76
AUS 1974	30	2.509	51.14
AUS 1986	30	2.782	56.70
AUS 1989	30	2.863	58.34
CAN 1971	31	2.896	58.46
CAN 1976	31	2.863	57.78
CAN 1981	31	2.831	57.15
CAN 1990	31	2.751	55.52
CAN 1997	32	3.226	64.53
DEU 1978	29	2.844	58.53
DEU 1986	29	2.876	59.20
DEU 1988	29	2.890	59.48
DEU 1990	29	2.897	59.63
DNK 1972	28	2.898	60.29
DNK 1980	28	2.911	60.56
DNK 1985	28	2.965	61.67
DNK 1990	28	2.942	61.20
DNK 1997	36	3.310	64.02

	$n$	[bits]	[%]
FRA 1980	31	2.737	55.25
FRA 1985	31	2.773	55.98
FRA 1990	31	2.768	55.88
FRA 1995	37	3.033	58.23
GBR 1968	31	2.857	57.67
GBR 1979	31	3.027	61.10
GBR 1984	31	2.804	56.59
GBR 1990	31	3.027	61.10
GBR 1998	37	3.185	61.14
KOR 1975	346	1.858	22.03
KOR 1980	359	2.775	32.69
KOR 1985	363	2.996	35.23
KOR 1990	369	2.988	35.04

Equations

$$s_0\in \{1,\dots ,n\}$$

$$H(s_0)=-\sum _{i=1}^n{\pi }_i{\log }_2{\pi }_i\ge 0,$$

$$f_{ij}\ge 0,\forall i,j\in \{1,\dots ,n\}$$

$$p_{ij}\triangleq \frac{f_{ij}}{{\sum }_{j=1}^n{f}_{ij}}$$

$$H(s_1|s_0=i)=-\sum _{j=1}^n{p}_{ij}{\log }_2{p}_{ij}.$$

$$H(s_1|s_0)=\sum _{i=1}^n{\pi }_{i}H(s_1|s_0=i).$$

$$s_0\to s_1\to \cdots \to s_k,$$

$$H(s_k|s_{k-1},\dots ,s_0)$$

$$\begin{array}{rl}H(s_k|s_{k-1},\dots ,s_0)& \le H(s_k|s_{k-1},\dots ,s_1)\\ & =H(s_{k-1}|s_{k-2},\dots ,s_0).\end{array}$$

$$H(\mathcal{S})\triangleq \underset{k\to \infty }{\lim }H(s_k|s_{k-1},\dots ,s_0),$$

$$\boxed{H(\mathcal{S})=-\sum _{i=1}^n\sum _{j=1}^n{\pi }_i^{\star }{p}_{ij}{\log }_2{p}_{ij},}$$

$$0\le H(\mathcal{S})\le {\log }_{2}n,$$

Full text

1 Introduction

It appears self evident that an industrial economy is more complex than a pre-industrial one, but how are we to measure this complexity?

Is the complexity a matter of the number of people involved? That probably has something to do with it since the population density in Europe today is well above that in Europe 500 years ago. But that scarcely seems enough. Clearly along with the increase in the number of people has gone an increase in the number of distinct products, and associated with that an increase in thenumber of trades or specialisations that people occupy. The classical economists viewed this social division of labour as a critical step in raising the productive capacity of an economy [1].

An initial approach might be to quantify complexity simply as the number of trades followed. But that does not seem enough either. Suppose we have two economies, each with one hundred distinct trades. In the first economy, 90% of the population are still farmers, with the special trades being distributed among the the 10% townsfolk. In the second economy, the great majority are urban with the population more evenly distributed between trades. It would be reasonable to say that the 90% rural country was economically simpler and less complex.

So intuitively, complexity ought to depend both on the number of trades and the distribution of the population into them. One way of approaching it would be to say that in a complex economy more information has to be provided to describe the average life outcome of an individual: will they be a peasant, a smith, a taylor, a wheelwright etc. This notion is possible to operationalize, if we view the economy as a process which randomly assigns individuals places in the social division of labour. The occupations each individual ends up in is then a random variable $s$ drawn from a set of distinct trades. Then we know from Shannon [2] that the average amount of information, (in bits) required to describe where the individuals end up is given by the entropy, denoted $H(s)$.

The entropy has several nice properties as a measure [3]. It has a lower bound 0 and an upper bound for a given a number of trades or economic sectors. It is also invariant to the order in which one lists economic sectors, assuming that the employment in each sector is given. However, $H(s)$ does not take into account that the economic system constitutes a process with interconnected sectors. Twelve complexity measures based on sectoral connectedness were surveyed in [4]. Some of them are bounded but provide no interpretable notion of complexity nor do they immediately relate to the economic production process. Alternative notions of complexity formulated from an algorithmic point of view do, however, provide an interpretable meaning. One notion is to measure the complexity of the interconnections of the economy in terms of the shortest algorithm that could generate the economic structure [5]. Another is to measure the number of steps required for an economy to solve its resource allocation problem [6]. These notions are dependent on how closely tied up the economic sectors are. The more they are interlinked, the greater the potential set of interactions, and the greater the algorithmic complexity. If there are $n$ distinct sectors, then the runtime complexity of solving for equilibrium prices in a fully connected economy is on the order of $n^{3}$. The fact that market-based economies with very large $n$ manage to coordinate production via a price mechanism, suggests that the complexity is significantly lower than this. One reason is that the interconnections between sectors in real economies are typically sparse, being more like Erdös-Rényi graphs than fully connected ones [7].

Leontief was one of the pioneers in developing input-output models of the entire economic production system, which are essential in the construction of national accounts data [8, 9, 10]. Starting from Leontief’s fundamental insights, we develop a complexity measure of an economic system using ideas originating from Markov [11] and Shannon [2]. The measure quantifies the average number of bits required to describe the flow of goods and services, from one sector to the next, when viewing production as a continual process. This provides both a practical interpretation, rooted in the engineering sciences, as well as a notion related to the production process, unlike the measures surveyed in [4]. The measure is applied to data from seven national economies, spanning several decades.

2 Economic complexity

From Leontief’s point of view, the economic production system has multiple inputs and multiple outputs of distinct goods and services [12, 13, 14]. Under capitalist institutions, it takes the form of production of commodities by means of commodities [15]. To begin the analysis of such a system, we partition the economy into sectors, each producing a distinct type of good or service. A fraction of output from sector $i$ is required for production in sector $j$. A certain fraction is also used up internally. Sectors with outputs that enter directly and indirectly in the production of all other sectors are denoted ‘basic’ [15, ch. 2]. The basic sectors are labeled $i=1,\dots,n$. Together they form an interconnected reproducing economic system, which we denote as $\mathcal{S}$ and corresponds to an irreducible and aperiodic graph.

For sake of illustration, suppose labour is required to initialize the production process and let the direct labour requirement in each sector be denoted as $\ell_{i}$. Consider a unit of the total social labour to be allocated to an initial sector

[TABLE]

randomly. Then $\pi_{i}=\ell_{i}/\sum_{j}\ell_{j}$ is the probability that a unit of labour is allocated to produce in sector $i$. That is, $\Pr\{s_{0}=i\}$ equals $\pi_{i}$. Using this formalism, we can study the economic requirements of production as a random process. The entropy of the random initial sector $s_{0}$ is then given by

[TABLE]

which gives the average number of bits required to encode the initial sector [2].222By convention, we have $0\log_{2}(0)=0$, which is justified by continuity [16]. This addresses the concern raised in [3] for sectors where $\pi_{i}=0$. To illustrate the economic meaning of this quantity in the case of labour, consider a pre-industrial economy. Here the great bulk of the working population are peasants and only a small proportion are employed in non-agricultural sectors. In consequence, the entropy $H(s_{0})$ in the pre-industrial economy is low. Industrialisation takes people from the countryside, and randomly casts them into a plethora of urban trades. This transition process will then increase the entropy $H(s_{0})$. The maximum entropy is attained if all sectors require an equal amount of labour. Then $H(s_{0})=\log_{2}(n)$.

Production in sector $s_{0}$ yields outputs that are requirements in a subsequent sector $s_{1}$. Analogous to the reasoning above, we consider this as a random transition $s_{0}\rightarrow s_{1}$, which enables a stochastic formulation of Leontief’s input-output model. Let

[TABLE]

denote the cross-sectoral requirements of output $i$ in sector $j$.

Definition 2.1**.**

An $n\times n$ matrix $\mathbf{P}=\{p_{ij}\}$, where

[TABLE]

is the fraction of sector $i$:s cross-sectoral output required for production in sector $j$.

The production requirements, in the form of forwarding cross-sectoral outputs from one sector to the next, can now be modeled as a Markov chain: Consider a unit of cross-sectoral output from $s_{0}$. It becomes an input in sector $j$ with a probability $p_{ij}$. Thus we consider the random transition $s_{0}\rightarrow s_{1}$ to occur with a probability $\Pr\{s_{1}=j\>|\>s_{0}=i\}=p_{ij}$. See Figure 1 for an illustration.

The entropy of the subsequent sector $s_{1}$ given the prior sector is

[TABLE]

To illustrate the economic meaning of this quantity, consider an industrial economy prior to electrification and suppose the output of sector $i$ is electricity generated by coal power plants. Initially this output enters into the production of only a few sectors. In consequence, the entropy $H(s_{1}|s_{0}=i)$ prior to electrification is low. With the introduction of transmission lines and power grids, electricity becomes widely used and $H(s_{1}|s_{0}=i)$ rises. The number of bits required to encode $s_{1}$ given a random initial state $s_{0}$ is obtained by simple averaging,

[TABLE]

Thus as the economy diversifies and more sectors require a complex mix of inputs, the conditional entropy $H(s_{1}|s_{0})$ will increase. Note that since the measure is logarithmic, an increment of one full bit corresponds to a doubling of complexity.

We can now extend above approach to trace a sequence of unit outputs through subsequent sectors in the production process. That is, the sequence of $(k+1)$ random variables,

[TABLE]

that represent the $k$th order production requirements of the economy in a forward direction. The conditional entropy of the current sector $s_{k}$ in the sequence is denoted

[TABLE]

and can be derived by extension of the first-order conditional entropy (3). As Leontief realized, the production process is a ‘circular flow’ of requirements [8]. The sequence (4) is a therefore a representation of the entire process when $k$ tends to infinity. Given the the properties of the sectors in $\mathcal{S}$, the distribution of $s_{k}$ will then tend to a unique stationary distribution [16, ch. 4] and under stationarity, the conditional entropy of the sequence is a nonincreasing quantity, i.e.,

[TABLE]

In other words, while the production process is conceptualized as an unceasing circular flow, it is possible to define a finite limit of its conditional entropy.

Definition 2.2** (Economic complexity).**

The entropy of the economic system $\mathcal{S}$ is defined as

[TABLE]

and represents the number of bits required to encode $s_{k}$ in the limit.

The entropy $H(\mathcal{S})$ is the average code length required to encode the destination of an output in the production process. Thus it is a natural measure of complexity of $\mathcal{S}$ in logarithmic scale. For any given entropy, the economic system must effectively discriminate between $2^{H(\mathcal{S})}$ output destinations at each step of the process. The entropy of two systems, $\mathcal{S}$ and $\mathcal{S}^{\prime}$, enable a comparison across time and space. Suppose $\mathcal{S}$ and $\mathcal{S}^{\prime}$ both produce basic four-wheeled automobiles, and that its production in $\mathcal{S}$ requires a complex mix of inputs, each of which in turn require another a complex mix to produce. If $\mathcal{S}^{\prime}$ produces automobiles in a simpler manner, then $H(\mathcal{S})$ will be greater than $H(\mathcal{S}^{\prime})$. Since the relative difference in the effective number of output destinations is $2^{H(\mathcal{S})-H(\mathcal{S^{\prime}})}$, each additional bit to $\mathcal{S}$ represents a doubling of the complexity of the economy $\mathcal{S}$ relative to $\mathcal{S}^{\prime}$.

Theorem 2.1**.**

The entropy can be computed as

[TABLE]

where the elements $\{\pi^{\star}_{i}\}$ are given by the eigenvector $1\boldsymbol{\pi}^{\star}=\mathbf{P}\boldsymbol{\pi}^{\star}$ with eigenvalue that equals 1. That is, it is a nontrivial solution to $(\mathbf{I}-\mathbf{P})\boldsymbol{\pi}^{\star}=\mathbf{0}$. The entropy is bounded by

[TABLE]

and attains its maximum value when the transitions to each sector are equiprobable.

Proof.

See [16, thm. 4.2.4]. ∎

The result (6) provides an operational measure that can readily be applied to existing input-output data collected by national statistics services. Note that the $H(\mathcal{S})$ is invariant to the selection of the initial sector $s_{0}$.

Remark 1*.*

Remark: The economic system $\mathcal{S}$ can also be extended to include the reproduction of its workforce. This is achieved by including the households as a sector that outputs units of labour, with inputs in the form of the real wage vector of the productive workforce, cf. [17].

3 Numerical examples

In this section, we apply the $H(\mathcal{S})$ in (6) to input-output data from seven national economies found in [18, 19]. The entropy measure quantifies the complexity of an economy, and as it evolves over time, $H(\mathcal{S})$ also registers the degree of structural economic change.

One limitation of the data is that the sectoral classification of the economy is not consistent across all datasets. However, within each national economy the classifications are sufficiently similar to provide meaningful comparisons across time. Moreover, the number of basic sectors $n$ also vary depending on classification level. To mitigate these effects we also consider $H(\mathcal{S})$ as a percentage of its maximum value. The results are presented in Table 1 and Figure 2.

First, we observe that the complexity of each economy is low relative to its maximum possible value. That is to say, relative to the complexity that would prevail were all sectors equally likely as destinations of the cross-sectoral outputs. For small input-output tables, the complexity is about half of the maximum level. Across all of the datasets, the complexity is sufficiently low so as to discriminate between at most 10 effective output destinations (3.3 bits) at most, and at least 4 destinations (1.9 bits).

Second, note that while the South Korean dataset (KOR) specifies the economy at a much finer resolution than the other sets, the complexity is still comparable to them. This is consistent with the results presented in [7], using disagregated tables from the US economy. In that analysis it was found that the number of intersectoral links grew at a rate below $\log(n)$. If we express that structure as a directed graph with the nonzero elements having equal weights then we would expect the $H(\mathcal{S})$ to be of order $\log_{2}(\log_{2}(n))$. Based on this result, an advanced economy $\mathcal{S}$ with $n=369$ sectors would be expected to have an entropy less than 3.1 bits, which is what we observe here.

Third, $H(\mathcal{S})$ also registers structural economic change. This is most dramatic in the South Korean economy. Between 1975 and 1980, the entropy increased by nearly one bit which corresponds to a doubling of the complexity in five years. This is reflective of its rapid state-led industrialization process. By contrast, while economic complexity of the United Kingdom (GBR) increases between 1968 and 1979, it exhibits a notable drop by the mid-1980s, corresponding to a fifteen percent reduction of complexity. This would reflect the significant decline of the manufacturing industry, while the restructuring of the UK economy is followed by a subsequent rise in complexity. The German economy (DEU) shows a stable level of complexity in the period prior to unification. The trend in the Canadian economy (CAN) was similar, until the mid-1990s.

4 Conclusions

Starting from an input-output model of an economic system, we have developed a measure of its complexity based on Shannon entropy. Following Leontief’s perspective of the production process as a circular flow, we formuled the production process as a Markov chain. This enabled us to derive an operational measure of economic complexity as the average code length required to encode the destination of an output in the production process.

We applied the measure to real data from seven national economies. It was observed that the complexity of each economy is substantially below the maximum possible value. The results for the larger data sets were also consistent with scaling laws observed in other economies. In addition, the measure was found to register structural economic changes of industrialization and deindustrialization.

These results suggest that $H(\mathcal{S})$ is reasonably similar among advanced industrialized economies (approximately 3 bits in the 1990s), and therefore related to productivity levels and other economic development indicators. While there is no clear relation between small variations in the entropy and growth rates of outputs, the rapid structural changes registered in $H(\mathcal{S})$ do appear to be associated with changes in output trajectory of the system, as in the case of South Korea and the United Kingdom mentioned above. Further research using updated input-output data from the original national statistics bureaus would be needed to assess this as well as the possible noise fluctations in the estimated entropies. Such data would need to extend the time-series into the last two decades. It would then be possible to quantify the structural effects of the financial crisis of 2007-2008 and its repercussions.