<p>In optimal control theory there is the <strong>Hamilton–Jacobi–Bellman (HJB)</strong> equation which per se is a PDE.
(As usual when it comes to advanced Math in conjecutre with Econ) It's more like a cooking recipy to solve PDE's. The HJB equation is both necessary and sufficient for an optimum.</p>

<p>Optimal control is virtually used in all fields of micro (and thus HJB as well). In macro it's usually applied when it comes to micro-foundations. The ramsey(-cass-koopmans) model for instance (I've seen that adressed with HJB equations as well).</p>

<p>Another sub-field where PDEs are typically solved is <strong>Differential Game Theory</strong> (which goes somewhat in a pure math direction but for instance has applications in Industrial Organization). See <a href="https://www.math.psu.edu/cim/lecturenotes/game-lnew.pdf" rel="nofollow">here</a> for instance for an intro. That's absolutely not my field but one famous guy of whom I know that uses that in IO is Luca Lambertini.</p>

CC BY-SA 3.0
2016-02-10T14:23:12.180Z
Very interesting. I presume you mean continuous time models, as it probably wouldn't be a PDE with discrete time. Is there a particular continuous time optimal control theory paper that comes to mind for you?
Shane
CC BY-SA 3.0
2016-02-10T18:29:42.643Z
http://epubs.siam.org/doi/abs/10.1137/09076742X this could be one, through crosscitations you could get a pretty good overview I think
Fitzroy Hogsflesh
CC BY-SA 3.0
2016-02-10T18:34:33.713Z
Another example just came to my mind. I updated my answer accordingly
Fitzroy Hogsflesh
CC BY-SA 3.0
2016-02-11T14:28:56.463Z
I started a new bounty to give to your answer, which was very useful. It'll let me award it tomorrow.
Shane
CC BY-SA 3.0
2016-02-10T08:52:00.967Z
2016-02-10T18:34:14.227Z
2016-02-10T18:34:14.227Z
Fitzroy Hogsflesh
<p>Here are a couple of suggestions. There are some PDEs in some recent continuous-time models, for instance in:</p>

<ul>
<li>A Continuous-Time Version of the Principal-Agent Problem, by Sannikov, Review of Economic Studies (2008)</li>
<li>Persistent private information, by Williams, Econometrica (2011)</li>
</ul>

<p>I would not be surprised if Yuliy Sannikov used PDEs in some of his other papers (although I have nothing precise in mind).</p>

<p>You can refer to the review by Achdou et al. (Partial differential equation models in macroeconomics, Philosophical Transactions of the Royal Society, 2014) for other references.</p>

CC BY-SA 3.0
2016-02-08T22:01:01.203Z
@Shane my pleasure! But don't hesitate to wait a couple of days in case someone comes up with more ideas.
Oliv
CC BY-SA 3.0
2016-02-08T18:45:34.807Z
2016-02-08T18:45:34.807Z
Oliv
<p>Where, in economics, are <strong>partial differential equations</strong> used?</p>

<p>I'm particularly interested in micro theory applications, but would also be keen to know of other applications.</p>

CC BY-SA 3.0
2016-02-05T21:21:38.283Z
Here is one http://economics.stackexchange.com/questions/5571/compute-evolution-of-a-distribution-over-time/5700#5700 ;)
Martin Van der Linden
<microeconomics><reference-request><mathematical-economics>
Partial differential equations in economics
Shane
# Partial differential equations in economics

(Asked by: username_0 on 2016-02-05)

Where, in economics, are **partial differential equations** used?

I'm particularly interested in micro theory applications, but would also be keen to know of other applications.

**Tags:** microeconomics, reference-request, mathematical-economics
> **Comments:**
> **username_1**: Here is one http://economics.stackexchange.com/questions/5571/compute-evolution-of-a-distribution-over-time/5700#5700 ;)  

---

## 2 answers

### Answer by username_3 on 2016-02-08. Score: 3 | Accepted answer

Here are a couple of suggestions. There are some PDEs in some recent continuous-time models, for instance in:

* A Continuous-Time Version of the Principal-Agent Problem, by Sannikov, Review of Economic Studies (2008)
* Persistent private information, by Williams, Econometrica (2011)

I would not be surprised if Yuliy Sannikov used PDEs in some of his other papers (although I have nothing precise in mind).

You can refer to the review by Achdou et al. (Partial differential equation models in macroeconomics, Philosophical Transactions of the Royal Society, 2014) for other references.

> **Comments:**
> **username_3**: @Shane my pleasure! But don't hesitate to wait a couple of days in case someone comes up with more ideas.  

---

### Answer by username_2 on 2016-02-10. Score: 2

In optimal control theory there is the **Hamilton–Jacobi–Bellman (HJB)** equation which per se is a PDE.
(As usual when it comes to advanced Math in conjecutre with Econ) It's more like a cooking recipy to solve PDE's. The HJB equation is both necessary and sufficient for an optimum.

Optimal control is virtually used in all fields of micro (and thus HJB as well). In macro it's usually applied when it comes to micro-foundations. The ramsey(-cass-koopmans) model for instance (I've seen that adressed with HJB equations as well).

Another sub-field where PDEs are typically solved is **Differential Game Theory** (which goes somewhat in a pure math direction but for instance has applications in Industrial Organization). See [here](https://www.math.psu.edu/cim/lecturenotes/game-lnew.pdf) for instance for an intro. That's absolutely not my field but one famous guy of whom I know that uses that in IO is Luca Lambertini.

> **Comments:**
> **username_0**: Very interesting. I presume you mean continuous time models, as it probably wouldn't be a PDE with discrete time. Is there a particular continuous time optimal control theory paper that comes to mind for you?  
> **username_2**: http://epubs.siam.org/doi/abs/10.1137/09076742X this could be one, through crosscitations you could get a pretty good overview I think  
> **username_2**: Another example just came to my mind. I updated my answer accordingly  
> **username_0**: I started a new bounty to give to your answer, which was very useful. It'll let me award it tomorrow.  

---

