From Frank Ramsey to René Thom: A classical problem in the calculus of variations leading to an implicit differential equation

In 1928, motivated by conversations with Keynes, Ramsey formulated an infinite-horizon problem in the calculus of variations. This problem is now classical in economic theory, and its solution lies at the heart of our understanding of economic growth. On the other hand, from the mathematical point of view, it was never solved in a satisfactory manner: In this paper, we give what we believe is the first complete mathematical treatment of the problem, and we show that its solution relies on solving an implicit differential equation. Such equations were first studied by Thom, and we use the geometric method he advocated. We then extend the Ramsey problem to non-constant discount rates, along the lines of Ekeland and Lazrak. In that case, there is time-inconsistency, meaning that optimal growth no longer is a relevant concept for economics, and has to be replaced with equlibrium growth. We briefly define what we mean by equilibrium growth, and proceed to prove that such a path actually exists, The problem, once again, reduces to solving an implicit differential equation, but this time the dimension is higher, and the analysis is more complicated: geometry is not enough, and we have to appeal to the central manifold theorem.