Price formation is an important aspect of economic activity. Our interest was in price formation in ``everyday'' situations, such as for retail prices. For this, we assumed that companies are price setters and agents are price takers, in the sense that their only strategy option is to go someplace else. In our abstracted situation, this means that companies with too low prices will exit because they cannot cover costs, while companies with too high prices will exit because they lose their customers.
We have used space in order to simplify and structurize the way in which information about alternative shopping places is found. This prevents the singularity of ``Bertrand-style'' models, where the market share of each company is independent from history, leading to potentially huge and totally unrealistic fluctuations.
By doing this, one notices that the spatial dynamics can be separated from the price formation dynamics itself. This makes intuitively sense since, in generalized terms, we are dealing with evolutionary dynamics, which often does not depend on the details of the particular fitness function. We have therefore started with an investigation of a spatial competition model without prices. For this model, we have looked at cluster size distributions, and compared them with real world company size distributions. In contrast to investigations in the literature, which find log-normal distributions, we find a scaling law a better fit of our data. In the models, we find log-normal distributions or scaling laws, depending on the particular rules.
We then added price formation to our spatial model. We showed that the price, in simple scenarios, converges towards the competitive price (which is here the unit cost of production), and that it is able to track slowly varying production costs, as it should. This predicts that prices should lag behind costs of production. We indeed find this in the data of consumer price index vs. production price index for the United States since 1941.