Discussion and outlook

The modelling approach with respect to economics in this paper is admittedly simplistic. Some obvious and necessary improvements concern credit and bankruptcy (i.e. rules for companies to operate with a negative amount of cash). Instead of those, we want to discuss some issues here that are closer to this paper. These issues are concerned with time, space, and communication.

In this paper, in order to reach a clean model with possible analytic solutions, we have described the models in a language which is rather unnatural with respect to economics. For example, instead of ``one company per time step'' which changes prices one would use rates (for example a probability of $p_{\it ch}$ for each company to change prices in a given time step). However, in the limiting case of $p_{\it ch} \to 0$, at most one and usually zero companies change prices in a given time step. If one also assumes that consumers adaptation is fast enough so that it is always completed before the next price change occurs, then this will result in the same dynamics as our model. Thus, our model is not ``different'' from reality, but it is a limiting case for the limit of fast customer adaptation and slow company adaptation. Our approach is to understand these limiting cases first before we move to the more general cases.

Similar comments refer to the utilization of space. We have already seen that moving from a spatial to a non-spatial model is rather straightforward. There is an even more systematic way to make this transition, which is the increase of the dimensions. In two dimensions on a square grid, every agent has four nearest neighbors. In three dimensions, there are six nearest neighbors. In general, if $D$ is the dimension, there are $2D$ nearest neighbors. If we leave the number $N$ of agents constant and keep periodic boundary conditions ($D$-dimensional torus), then at $D=(N-1)/2$ everybody is a nearest neighbor of everybody. Thus, a non-spatial model is the $D \to \infty$ limiting case of a spatial model.

These concepts can be generalized beyond grids and nearest neighbors - the only two ingredients one needs is that (i) the probability to interact with someone else decreases fast enough with distance, and that (ii) if one doubles distance from $r$ to $2r$, then the number of interactions up to $2r$ is $2^D$ times the number of interactions up to distance $r$.

This should also make clear that space can be seen in a generalized way if one replaces distance by generalized cost. For example, how many more people can you call for ``20 cents a minute or less'' than for ``10 cents a minute or less''? If the answer to this is ``two times as many'', then for the purposes of this discussion you live in a one-dimensional world.

Given this, it is important to note that we have used space only for the communication structure, i.e. the way consumers aquire information (by asking neighbors). This is a rather weak influence of space, as opposed to, for example, transportation costs[1]; it however also assumes a not very sophisticated information structure, as for example in contrast to today's internet. The details of this need to be left to future work.

Last, one needs to consider which part of the economy one wants to model. For example, a stockmarket is a centralized institution, and space plays a weak role at best. In contrast, we had the retail market in mind when we developed the models of this paper. In fact, we implicitely assume perishable goods, since agents have no memory of what they bought and consumed the day before. Also, we assume that consumers spend little effort in selecting the ``right'' place to shop, which excludes major personal investments such as cars or furniture. Also note that our companies have no fixed costs, which implies that there are no capital investments, which excludes for example most manufacturing.