<p>Select examples from a book suggestion due to @markleeds, <em>Optimal Control Theory with applications in Economics</em>:</p>
<ul>
<li>Polluting growth has action <span class="math-container">$\int(bz-\dot{z}+\ddot{z})dt$</span>.</li>
<li>Optimal extraction of a natural resource has action <span class="math-container">$\int(C(-\dot{x},\,t)+q\dot{x})e^{-st}dt$</span>.</li>
<li>Consumption vs rapid attainment of production capacity has action <span class="math-container">$\int[\dot{x}-x+q-a]dt$</span>.</li>
<li>Resource extraction in an open economy has action <span class="math-container">$\int-U(\bar{c}-e^{rt}\dot{x}+ay-\dot{y})dt$</span>.</li>
<li>Consumption versus investment can give rise to many actions, including:
<ul>
<li><span class="math-container">$\int(\dot{x}-1)U(x)dt$</span></li>
<li><span class="math-container">$\int e^{-\rho t}(\dot{K}+\delta K-f)U(K)dt$</span></li>
<li><span class="math-container">$\int\frac{dt}{K-\dot{K}}$</span></li>
<li><span class="math-container">$\int e^{\dot{K}-aK}dt$</span></li>
<li><span class="math-container">$\int(\dot{K}-cK)^ae^{-bt}dt$</span></li>
</ul>
</li>
</ul>

CC BY-SA 4.0
2021-03-20T18:34:19.527Z
I'm glad that it looks like it helped ?
mark leeds
CC BY-SA 4.0
2021-03-20T18:37:29.750Z
@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.
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2021-03-21T16:06:29.770Z
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.
mark leeds
CC BY-SA 4.0
2021-03-20T08:28:05.767Z
2021-03-20T08:28:05.767Z
J.G.
<p><a href="https://en.wikipedia.org/wiki/Philip_Mirowski#Career" rel="nofollow noreferrer">Mirowski 1989</a> 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 <span class="math-container">$\int_a^bL(t,\,q,\,\dot{q})dt$</span> or a suitable generalization. Are there any specific examples like that?</p>

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2021-03-19T07:51:06.280Z
Can you define"action" for a non-physicist?
Michael Greinecker
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2021-03-19T09:05:58.320Z
@MichaelGreinecker My integral already did that. See also [this link](https://en.wikipedia.org/wiki/Action_(physics)#Mathematical_definition), which I'll add to the question if you think it would help.
J.G.
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2021-03-19T15:15:34.517Z
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).
SystemTheory
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2021-03-19T15:32:36.120Z
@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.
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2021-03-19T16:16:07.907Z
@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.
SystemTheory
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2021-03-19T16:37:23.367Z
@SystemTheory Let's put physics aside and focus on my question of whether an action, be it of conversed Hamiltonian or otherwise (and note my integral includes the latter case), appears in economics. It doesn't even need $t$ to be time.
J.G.
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2021-03-19T18:36:31.230Z
That seems to amount to the question of whether the calculus of variations has uses in economics.
Michael Greinecker
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2021-03-19T18:43:33.350Z
@MichaelGreinecker Well, judging by Mirowski they do, but I'm not clear how the action would look.
J.G.
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2021-03-20T01:10:33.267Z
As Michael alluded to with this point about calculus of variations, google for or get a book on "optimal control in economics" . the one by weber is okay. I'm reading it now but there are many others with lots of examples.
mark leeds
CC BY-SA 4.0
2021-03-20T07:51:33.873Z
@markleeds Is _that_ what he was telling me to do? Thanks, I didn't know. I'd like to think if someone asked the Physics SE for an action someone would say "Einstein-Hilbert" rather than "read an unnamed book with this author's surname; it's in their somewhere", but thanks, I'll give this information a go.
J.G.
CC BY-SA 4.0
2021-03-20T12:03:52.247Z
@J.G. I think what Michael was trying to say is that in this case the question is tantamount to ask what is the use of calculus of variations (based of your description of what you mean by action), calculus of variations is term which most economists would immediately recognize and be familiar with, but the term action as used here would be less familiar to average economists. So I think Michael was rather suggesting an edit than suggesting you should look at some unnamed literature.
1muflon1
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2021-03-20T12:08:34.033Z
@1mutton1 Well, I actually wanted to see some actions, so I gave my question the right name.
J.G.
CC BY-SA 4.0
2021-03-20T18:29:16.670Z
HI J.G.I can't speak for Michael but calculus of variations is sort of a pre-req for optimal control. I recently asked an expert if I should understand calc of variations first and then go to optimal control or go straight to optimal control. He said to go to optimal control straight away but he's a giant and brilliant so that may only work for some people. These are both giant fields and take time but I was just trying to get across to you that I think that's the name of the field that you should be looking at. Atleast, I'm pretty sure of that. You'll find out quickly enough if I'm wrong.
mark leeds
<mathematical-economics><theory>
Actions in economics
J.G.
# Actions in economics

(Asked by: username_0 on 2021-03-18)

[Mirowski 1989](https://en.wikipedia.org/wiki/Philip_Mirowski#Career) 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?

**Tags:** mathematical-economics, theory

> **Comments:**
> **username_1**: Can you define"action" for a non-physicist?  
> **username_0**: @username_1 My integral already did that. See also [this link](https://en.wikipedia.org/wiki/Action_(physics)#Mathematical_definition), which I'll add to the question if you think it would help.  
> **username_2**: 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).  
> **username_0**: @username_2 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.  
> **username_2**: @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.  
> **username_0**: @username_2 Let's put physics aside and focus on my question of whether an action, be it of conversed Hamiltonian or otherwise (and note my integral includes the latter case), appears in economics. It doesn't even need $t$ to be time.  
> **username_1**: That seems to amount to the question of whether the calculus of variations has uses in economics.  
> **username_0**: @username_1 Well, judging by Mirowski they do, but I'm not clear how the action would look.  
> **username_3**: As username_1 alluded to with this point about calculus of variations, google for or get a book on "optimal control in economics" . the one by weber is okay. I'm reading it now but there are many others with lots of examples.  
> **username_0**: @username_3 Is _that_ what he was telling me to do? Thanks, I didn't know. I'd like to think if someone asked the Physics SE for an action someone would say "Einstein-Hilbert" rather than "read an unnamed book with this author's surname; it's in their somewhere", but thanks, I'll give this information a go.  
> **username_4**: @J.G. I think what username_1 was trying to say is that in this case the question is tantamount to ask what is the use of calculus of variations (based of your description of what you mean by action), calculus of variations is term which most economists would immediately recognize and be familiar with, but the term action as used here would be less familiar to average economists. So I think username_1 was rather suggesting an edit than suggesting you should look at some unnamed literature.  
> **username_0**: @1mutton1 Well, I actually wanted to see some actions, so I gave my question the right name.  
> **username_3**: HI username_0I can't speak for username_1 but calculus of variations is sort of a pre-req for optimal control. I recently asked an expert if I should understand calc of variations first and then go to optimal control or go straight to optimal control. He said to go to optimal control straight away but he's a giant and brilliant so that may only work for some people. These are both giant fields and take time but I was just trying to get across to you that I think that's the name of the field that you should be looking at. Atleast, I'm pretty sure of that. You'll find out quickly enough if I'm wrong.  

---

## 1 answers

### Answer by username_0 on 2021-03-20. Score: 1

Select examples from a book suggestion due to @username_3, *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:
  + $\int(\dot{x}-1)U(x)dt$
  + $\int e^{-\rho t}(\dot{K}+\delta K-f)U(K)dt$
  + $\int\frac{dt}{K-\dot{K}}$
  + $\int e^{\dot{K}-aK}dt$
  + $\int(\dot{K}-cK)^ae^{-bt}dt$

> **Comments:**
> **username_3**: I'm glad that it looks like it helped ?  
> **username_0**: @username_3 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.  
> **username_3**: 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.  

---

