An example for a closed-form solution for a system of linear partial dierential equations

where s; ;mw and mb are known real constants. Further we assume that mw 6= mb. Those equations are motivated in economic papers by Bayer and Walde (2010 a,b,c). The authors analyse the distribution of wealth a in a world of uncertain labour income, i.e. labour income z that moves stochastically between a low state b and a high state w. This stochastic movement is modeled using two mutually independent Poisson processes. Arrival rates are s for the transition from state w to b and for the opposite direction. The state w can be called “employment”and state b “unemployment”. Individuals can save in this setup implying a distribution of wealth that is a function of labour market history. The above linear system results from Fokker-Planck equations describing the joint density of a and z under the assumption of a utility function characterised by so-called “constant absolute risk aversion”. As far as we can tell, general closed form solutions for a linear PDE system (1) do not seem to exist.2 A posting by Serre (2011) supports this opinion by stating that a general closed-form solution for a linear PDE system does not exist. Special solutions, i.e. solutions for special initial functions, are well-known. An example are exponential-type solutions of the form ee (Israel, 2011). After having consulted, inter alia, Polyanin (2001, 2011), Polyanin et al. (2001), Evans (2010), Farlow (1993) and Zachmanoglou and Thoe (1986), we concluded that solutions of the “gamma-type”suggested here are not as widely known as one would expect.3