Diversity of preferences in an unpredictable environment

The origin of preferences was viewed as related to the dominant eigenvalue of a Leslie matrix modelling reproductive strategies. In a variable environment, however, the coexistence of varying preferences no longer requires optimality, but is identified with the mathematical property of viability: a state of the population is viable if there exists at least one solution starting from it and remaining in the set of constraints until a given time horizon (or forever). The coexistence kernel of two competitors with varying preferences is computed for the case of scalar and 2 x 2 Leslie matrices, with either measurable or differentiable preferences. The homologue of indifference curves is the regulation map, the correspondence associating the set of viable preferences to a given state of the population. Among these viable trajectories, some are also optimal in the sense of dominance discounted in time. These viable optimal solutions are obtained as specific trajectories in an auxiliary dynamic system, and the associated maximal values constitute one boundary of the viability kernel of this auxiliary system (theorem). Hence, the perpetuation of varying preferences allows the diversity of economic preferences, as shown here using the example of the comparative history of fertility from mid-nineteenth century to nowadays in France and England.