🧵1/7) On the difference between dynamical systems in #physics vs #economics. Example 1. The case of Ramsey-Cass-Koopmans (RCK). Instability and learning the initial condition for c.

2/7) Consider the RCK problem. As described before, equation (4) is the difficulty. It is not a typical boundary value problem, for two reasons: 1) the boundary condition falls at ♾, and 2) no value is prescribed at ♾. To solve this, we resort to the concept of steady state.

3/7) This system has two steady states. The one we are interested in, denoted by superscript *, is a saddle. The other is a sink.

4/7) The path converging to the saddle satisfies eq. (4). Any path converging to the sink violates it: along those paths consumption goes to zero, c -> 0, driving the limit to infinity.

5/7) We can approximate ♾ with a large T (here T = 25) and turn the original problem into a Dirichlet boundary value problem, solvable with BVP solvers. The solution yields an initial condition for consumption c, which depends on T.

6/7) Now we have an initial value problem we can solve and see how it generalizes for T' > T. Here T' = 75. Will the path hover around the saddle steady state, which satisfies the transversality condition, eq. (4)?

7/7) The answer is no. The optimal path lies on a saddle, so even small approximation errors grow and throw the system off the saddle path.

Dynare doesn't have this problem 😏

I think I should have used a simple OLS :D

Hello? BASED department?

Very interesting! If you miss just an epsilon>0 the critical point is this an example of a turnpike solution?

Yes, it is very related to turnpike theorems of Samuelson and McKenzie

Nice to see some physicists stepping in and debunking economics.

This is not about debunking economics. It is about the mathematical differences and the computational challenges arising from them. See this post:

Yeah, I’ve seen some physicists calling out the limits of economics. Your post caught my eye—I’ll definitely check out what you shared.

@MahdiKahou Economist here, even though not a macroeconomist. Just be aware that economists are, of course, aware of these limitations. There's just often a reason why models are still set up the way they are. Might be they are just otherwise not feasible, or still yield useful conclusions.

Ah, of course. The problem is that economists actually know the limits of their models, but they tend to brush them aside and keep building their theories anyway. Take the Sonnenschein–Mantel–Debreu theorem, for instance, especially when dealing with aggregate market demand—it ends up having pretty strong implications at the macro level, even though those foundations are far from solid.

Thanks for the post, humble question: the error you are refering to is c* - c_0(T)?

This seems very much related to the results of the Boldrin-Montrucchio theorem

But the point is that we are trying to find the optimal path for the agent (unlike objects in physics, which do not choose). When you draw a path violating the transversality condition you are not showing alternative dynamics for the economy; you are just making a mistake

I have to say, I truste the hjb solution even less. Solving a cauchy problem on a bounded domain. No terminal condition, no spatial boundary treatment, artificial boundary, transparent. I don't buy it