Mathematical model for the economic dynamics and periodic crisis described by Karl Marx

Karl Marx spent 30 years analyzing the capitalist mode of production and it's dynamics in his book "The Capital". One of those dynamics were the competition between capitalists and its consequences, one of which being periodic crisis, another-the tendency of the rate of profit to fall. In this paper I will attempt to model these dynamics using mathematics. For the purpose of this model, I will define parameters and assume an economy of n private firms.