Archive for September, 2019

Article on mathematical economics

September 28, 2019

I now find myself in the for me a priori surprising situation of being one of the authors of an article in the Journal of Mathematical Economics (85, 17-37). The title is ‘The invariant distribution of wealth and employment status in a small open economy with precautionary savings’. Without having a global overview of the subject I want to explain roughly what is being described by the mathematical model in this article. The idea is that we have a population of individuals who may lose their jobs and find new ones in a way which involves a lot of randomness. It is supposed that when they have employment these individuals save some money in order to be able to support themselves better during periods where they are unemployed. They question is whether these assumptions lead to a long-term stationary distribution of wealth and employment in the population. The starting model is a stochastic one but the consideration of stationary distributions lead to an ODE problem.

So how did I get involved in this project? Klaus Wälde, a professor of economics at the University of Mainz, had a project on this subject with Christian Bayer, a mathematician from the Weierstrass Institute in Berlin. The two of them submitted a paper which contained some partial results on stationary distributions. However they did not have a complete existence proof and the referees insisted that they try to improve on that. Wälde looked for some assistance on this point in the mathematics department in Mainz and ended up talking to me about the matter. The concrete mathematical problem to be solved concerned a certain boundary value problem for a system of ODE with rather singular boundary conditions. At this point the universality of mathematics comes in. It turned out that this problem coming from economics could be solved with the help of a theorem which I proved with Bernd Schmidt in 1991 to treat a problem coming from astrophysics. Along the way I was also able to show that an analyticity assumption which had been made in the formulation of the economics model was unnecessary. I have no intention of going further into the area of mathematical economics – I want to remain focussed on mathematical biology. It is nevertheless a nice feeling to think that with the mathematical training I have I could in principle contribute to almost any field of science.