Dynamical Analysis On An Evolutionary Model Of Boundedly Rational Consumers

Under an assumption that consumer is boundedly rational, and an adaptive evolutionaryapproach is the iteration of consumer choices, this paper build a dynamical model describingconsumers’ choices.Firstly, we introduce an approach to solving consumer behavior by traditional utilitymaximization; based on this, the consumers update their consumption according the adaptiveprocess at each time period, and a simply discrete time model of boundedly rational consumeris proposed. By qualitative analysis, we found the consumption bundles convergence to theoptimal consumption when the reactivity parameter is lower than a certain threshold value.when, the consumption bundles does not converge to the optimal consumption, butto the unpredictable chaos eventually by the flip bifurcation. Thus, it is assumed thatconsumers learn to estimate the optimal consumption bundle, and continue to adjust thereaction parameters according to the distance between the actual consumption bundles and theestimated consumption bundles. Then the least square method is used to estimate the optimalconsumption bundle. We demonstrate that the consumer’s optimal choice convergence to thefully rational consumer choice under the endogenous learning mechanism, then we proposethat the endogenous reaction parameter should meet four conditions, and a specific reactionfunction is given according to the conditions. Numerical simulations are practiced by Matlabbefore and after introducing endogenous learning mechanisms, the experimental results showthat endogenous learning mechanisms can help consumers make the choice of a fully rationalconsumer ultimately.Secondly, on the basis of the study of one-dimensional model of the discrete system,under the assumption of boundedly rational consumers, we considered the impact ofpreferences, which is represented by the S-shaped curve. A dynamic model on consumerchoices and preferences is established, we prove the existence of nonnegative fixed points ofthis model, we analyze the local stability of the fixed point according Jury conditions; westudy the stability of the fixed point when eigenvalue is-1by using center manifold theorem,using numerical methods to study feasible domain of attraction, which reflect whether initialconsumers choice and preference is reasonable, The increase in commodity prices is notconducive to the consumers to make rational choices.