Our model mimiques the behavior of a population of consumers of a good or a service. The demand of each consumer depends on his/her intrinsic preference, on the price and on the aggregate demand of the whole population or of the consumer's peers. The intrinsic preferences and the susceptibility to the aggregate demand are distributed over the population in accordance to two probability laws. These laws turn our model into a stochastic process. The time component of the model allows us for modeling the change in the demand of each consumer. The model may be studied in two regimes: when the price is fixed, and when the price changes in time depending on the current value of the aggregate demand. In the first case, we investigate the long time limit of the aggregate demand. We find that it is a continuous or a discontinuous function of price depending on the values of the variance of the probability laws that represent the intrinsic preferences and the social susceptibility. We shall argue that the formation of price in real life may be explained through the lenses of this phase transition. In the second case, we investigate the joint behavior of price and demand. We show that depending on the variance of the probability laws, these quantities either oscillate or converge. This phase transition allows one to understand the formation of market booms and crashes, as we shall argue.
We shall discuss different mathematical treatments of the proposed model. One of them required understanding of a specific dynamical system. The reported results have been obtained in collaboration with Fernando Pigeard de A. Prado, Antonio Luiz Pereira and Paolo Tommasini.