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We introduce a system of kinetic equations describing an exchange marketconsisting of two populations of agents (dealers and speculators) expressingthe same preferences for two goods, but applying different strategies in theirexchanges. We describe the trading of the goods by means of some fundamentalrules in price theory, in particular by using Cobb-Douglas utility functionsfor the exchange. The strategy of the speculators is to recover maximal utilityfrom the trade by suitably acting on the percentage of goods which areexchanged. This microscopic description leads to a system of linearBoltzmann-type equations for the probability distributions of the goods on thetwo populations, in which the post-interaction variables depend from thepre-interaction ones in terms of the mean quantities of the goods present inthe market. In this case, it is shown analytically that the strategy of thespeculators can drive the price of the two goods towards a zone in which thereis a marked utility for their group. Also, the general system of nonlinearkinetic equations of Boltzmann type for the probability distributions of thegoods on the two populations is described in details. Numerical experimentsthen show how the policy of speculators can modify the final price of goods inthis nonlinear setting.

Kinetic models for goods exchange in a multi-agent market