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Equilibrium distributions and relaxation times in gas-like economic models: an analytical derivation

A step by step procedure to derive analytically the exact dynamical evolution equations of the probability density functions (PDF) of well known kinetic wealth exchange economic models is shown. This technique gives a dynamical insight into the evolution of the PDF, e.g., allowing the calculation of its relaxation times. Their equilibrium PDFs can also be calculated by finding its stationary solutions. This gives as a result an integro-differential equation, which can be solved analytically in some cases and numerically in others. This should provide some guidance into the type of probability density functions that can be derived from particular economic agent exchange rules, or for that matter, any other kinetic model of gases with particular collision physics.