In this paper we discuss a concept of dynamic memory and an application of

Derivatives of non-integer orders are applied to generalize notion of elasticity in framework of economic dynamics with memory. Elasticity of Y with respect to X is defined for the case of a finite-interval fading memory of changes of X and Y. We define generalizations of point price elasticity of demand to the case of processes with memory. In these generalizations we take into account dependence of demand not only from current price (price at current time), but also all changes of prices for some time interval. For simplification, we will assume that there is one parameter, which characterizes a degree of damping memory over time. The properties of the suggested fractional elasticities and examples of calculations of these elasticities of demand are suggested. Mathematics subject classification (2010): 26A33.

Abstract.A generalization of the economic model of logistic growth, which takes into account the effects of memory and crises, is suggested. Memory effect means that the economic factors and parameters at any given time depend not only on their values at that time, but also on their values at previous times. For the mathematical description of the memory effects, we use the theory of derivatives of non-integer order. Crises are considered as sharp splashes (bursts) of the price, which are mathematically described by the delta-functions. Using the equivalence of fractional differential equations and the Volterra integral equations, we obtain discrete maps with memory that are exact discrete analogs of fractional differential equations of economic processes. We derive logistic map with memory, its generalizations, and "economic" discrete maps with memory from the fractional differential equations, which describe the economic natural growth with competition, power-law memory and crises.Keywords: model of logistic growth, logistic map, chaos, discrete map with memory, hereditarity, memory effects, power-law memory, derivatives of non-integer order MSC: 26A33; 34A08

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