  A generalization of the economic model of natural growth, which takes into
account the power-law memory effect, is suggested. The memory effect means the
dependence of the process not only on the current state of the process, but
also on the history of changes of this process in the past. For the
mathematical description of the economic process with power-law memory we used
the theory of derivatives of non-integer order and fractional-order
differential equation. We propose equations take into account the effects of
memory with one-parameter power-law damping. Solutions of these fractional
differential equations are suggested. We proved that the growth and downturn of
output depend on the memory effects. We demonstrate that the memory effect can
lead to decrease of output instead of its growth, which is described by model
without memory effect. Memory effect can lead to increase of output, rather
than decrease, which is described by model without memory effect.
