# Intuition behind Euler Lagrange equation in economics

When being exposed to the concept of the Euler Lagrange equation as a mathematical concept, many ideas from the physical sciences are used to explain its relevance in terms of choice of shortest path, brachistochrone problem which are great for educational purposes however in the context of economics it is difficult.

The paper that comes to mind which is explict about the Euler Lagrange condition in terms of solving is Grossman 1972 (the famous grossman model of demand for health). The continuous time variant of the model (on page 28-29 of the PDF) has the following condition associated with it (notation changed for simplicity from time denoted by $$i$$ to being denoted by $$t$$) we get:

$$\frac{\partial Q}{\partial H(t)}-\frac{\partial}{\partial t}\frac{\partial Q}{\partial \dot{H}(t)}=0$$ or $$\frac{\partial Q}{\partial H(t)}=\frac{\partial}{\partial t}\frac{\partial Q}{\partial \dot{H}(t)}$$

What is the economic interpretation of this result?

• The Euler-Lagrange equation is usually derived using the idea that local deviations should not improve the objective function. This also makes sense in economics: local devations (in terms of changing ones behaviour) do not lead to a higher lifetime utility. The only difference between econ and physics is that in economics we generally maximise instead of minimise things.
– tdm
May 3, 2021 at 13:46