Sustainable Development and Spatial Inhomogeneities

Abstract

Historical data, theory and computer simulations support a connection between growth and economic inequality. Our present world with large regional differences in economic activity is a result of fast economic growth during the last two centuries. Because of limits to growth we might expect a future world to develop differently with far less growth. The question that we here address is: “Would a world with a sustainable economy be less unequal?” We then develop integrated spatial economic models based on limited resources consumption and technical knowledge accumulation and study them by the way of computer simulations. When the only coupling between world regions is diffusion we do not observe any spatial unequality. By contrast, highly localized economic activities are maintained by global market mechanisms. Structures sizes are determined by transportation costs. Wide distributions of capital and production are also predicted in this regime.

Appendix

We here report the results of numerical tests to check the meta-stability of the observed patterns, possible scaling effects and the role of initial configurations.

1.1 5.1 Scaling

The system does no display noticeable side effects. We tested the number of active cells with A>100 for varying lattice sizes after 10000 iteration steps. The fraction of active sites was respectively 19.8 % for a 30×30 cell lattice, 19.1 % for 50×50 cells and 19.6 % for 100×100 cells. We found that the observed difference in percentage to be not significative.

1.2 5.2 Patterns Stability

Because the system is noisy we used the Pearson correlation coefficient to measure the correlation between a pattern of capital K taken at time 5000 and patterns observed at further times until t=10,000. We use a 50×50 network. The Pearson coefficient is measured by:

$$ r(\tau)=\frac{\sum_i (x_i(t)-\overline{x(t)})(x_i(t+\tau)-\overline{x(t+\tau)})}{ \sqrt{\sum_i (x_i(t)-\overline{x(t)})^2}\sqrt{\sum_i (x_i(t+\tau)-\overline{x(t+\tau)})^2}} $$

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The Pearson coefficient decreases from 1 to 0.99 in 5000 time steps which demonstrates a long term stability after quasi equilibrium has been reached. We also checked the stability of the correlations between one cell and its neighbours at varying distances over 5000 time steps and similarly checked stability (Fig. 8).

Fig. 8

Time evolution of capital spatial correlations at varying distances r(1), r(2), r(3) and r(4)

1.3 5.3 Influence of Initial Conditions

A choice of initial conditions is arbitrary and we reported in Sect. 3 results obtained from random initial conditions without any spatial correlation. On the other hand, since the present state of the world is already structured in industrial regions, the possible importance of initial structures is worthwhile to study. We then ran simulations with initial sinusoidal patterns of A and K such that:

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with several wave vectors k=2nπ/L, n=2, 5, 10 and lattice size L=50 (Fig. 9). Resource initial distribution were random.

Fig. 9

Production patterns at large integration times, 1000, for different periodic initial conditions

The fraction of active sites is respectively 6.16, 11.18 and 19.32 % for n=2, 5, 10. Not surprisingly in view of the previously observed metastability, initial conditions do play a role in the final aspect of the patterns. The meso-scale initial periodicity is maintained while the micro structure of the localised peaks of activity reflects the character of the dynamics, independently from the initial conditions.

One might be tempted to investigate further the transition from present socio-economic patterns to future conditions with different energy sources. Let us remind that we are not able to describe the dynamics of the slow technological change that would drive the transition. The above set of remarks does not allow precise predictions: what we can still conclude is that some memory of the present large scale spatial structures could be maintained.