The trouble some issues encountered while modelling the economic growh together with a new exponential modelling

 

The following text is AI (ChatGPT plus) generated with reference to my current issue in mathematical modelling of the macroeconomic growth theory. 

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This week I had an interesting little incident at the boundary between economics and mathematics.

I was checking an exponential model for an economic growth / Total Factor Productivity (TFP) problem. After going back to the mathematical formulation and implementing it carefully in MATLAB, the objective function $Q(\phi)$ showed a clear minimum.

At first sight, that sounds encouraging: the optimisation problem has a well-defined solution.

However, when I reconstructed the corresponding TFP trajectory using the parameters associated with that minimum, the result did not reproduce the observed trajectory satisfactorily.

So I found myself in an interesting situation:

the mathematics said, “Here is the optimum,” while the economic data replied, “Not so fast.” 😅

This does not necessarily mean that the exponential approach is mathematically wrong. Rather, it raises a more interesting question: is this particular model structure appropriate for representing the empirical behaviour of TFP?

At the moment, I am therefore also comparing the result with a penalised smoothing spline, which appears to represent the trajectory more naturally.

For me, this has been a useful reminder that finding a mathematically neat optimum is not the same thing as finding a good empirical model. Optimisation, model specification, and interpretation all have to work together.

The investigation is still ongoing, so this is not a final conclusion — just one of those small research episodes where economics and mathematics refuse to cooperate quite as politely as expected.