# Actions in economics

Mirowski 1989 argues economics and physics have frequently informed each other's theoretical development, and neoclassical economics has the same Lagrangian and Hamiltonian formalism frequently seen in physics. I don't know how mainstream this approach has ever been, but his account doesn't make explicit what would be an action in economics. For example, I can imagine a firm's profit maximization, or a portfolio's ROI variance minimization, being formalized as an extremal action principle, but only if such quantities are expressible in the form $$\int_a^bL(t,\,q,\,\dot{q})dt$$ or a suitable generalization. Are there any specific examples like that?

• Can you define"action" for a non-physicist? Mar 19 at 7:51
– J.G.
Mar 19 at 9:05
• The cited Wikipedia article Action (physics) states: Action has dimensions of energy⋅time or momentum⋅length, and its SI unit is joule-second. Action is only of interest when the total energy of the system is conserved. This suggest that an economic analogue for conservation of energy must be assumed to apply the principle of Action (physics) as a principle of Action (economics). Mar 19 at 15:15
• @SystemTheory I advise against taking that too literally. No economic action needs the units of angular momentum; not even all actions in physics do (see e.g. general relativity's geodesics). There's also no energy conservation if the Lagrangian $L$ satisfies $\partial L/\partial t\ne0$. In any case, one can discover the conservation laws an action implies rather than having to know them in advance, so we don't have to decide up front what would be considered energy in economics.
– J.G.
Mar 19 at 15:32
• @J.G. Good advice. However I did not state or imply that economic Action would have physical units or any units whatsoever. In the domain of classical physics we use the principle of conservation of energy to write a differential equation for the total energy of the system under study. Then we can solve for unknown variables in terms of known variables and initial conditions. The units applied are arbitrary, the numbers applied are arbitrary (we select the zero or origin), the math is based on many physical definitions, and conservation of energy holds until violated by experimental evidence. Mar 19 at 16:16

Select examples from a book suggestion due to @markleeds, Optimal Control Theory with applications in Economics:

• Polluting growth has action $$\int(bz-\dot{z}+\ddot{z})dt$$.
• Optimal extraction of a natural resource has action $$\int(C(-\dot{x},\,t)+q\dot{x})e^{-st}dt$$.
• Consumption vs rapid attainment of production capacity has action $$\int[\dot{x}-x+q-a]dt$$.
• Resource extraction in an open economy has action $$\int-U(\bar{c}-e^{rt}\dot{x}+ay-\dot{y})dt$$.
• Consumption versus investment can give rise to many actions, including:
• $$\int(\dot{x}-1)U(x)dt$$
• $$\int e^{-\rho t}(\dot{K}+\delta K-f)U(K)dt$$
• $$\int\frac{dt}{K-\dot{K}}$$
• $$\int e^{\dot{K}-aK}dt$$
• $$\int(\dot{K}-cK)^ae^{-bt}dt$$
• I'm glad that it looks like it helped ? Mar 20 at 18:34
• @markleeds In a "raises as many questions as it answers" way, yes. Some of my quoted examples may be missing Lagrange multipliers. For example, the first Lagrangian not only includes a seemingly unnecessary total derivative, but simply has $b=0$ as its Euler-Lagrange equation.
– J.G.
Mar 20 at 18:37
• I don't know either of the subjects much at all but langrange multipliers I think come up more in calculus of variations than in optimal control. There are so many books, I couldn't begin to suggest on which to look at. Mar 21 at 16:06