<p>I don't really understand your optimization problem as it stands. Just as a few examples, when you write out</p>

<p>$$\min \sum_m \sum_v \left( \frac{\partial \vec f_v (\vec Q_v)}{\partial \vec f_m (\vec Q_m)} - \frac{\partial \vec f_m (\vec Q_m)}{\partial \vec f_m (\vec Q_m)} \right)$$</p>

<p>I have questions about why you are minimizing the difference between optimal conditions for output that has all equal opportunity costs. The justification is unclear to me. The function $\vec f_v(\vec Q_v)$ also leaves me confused looking at your work. Why is $f$ taking a vector of $n$ aggregate outputs as its argument? $f(q)$ is defined as the adjusted cost of purchasing $q \cdot n$ resources, so how does $f$ transform a vector as opposed to a scalar and what interpretation are you trying to give it?</p>

<p>Or in your optimization problem when you write out</p>

<p>$$\frac{dy}{d\sqrt{r^2 - x^2}} \rightarrow - \frac{x}{\sqrt{r^2 - x^2}} = y$$</p>

<p>I have no idea what you are differentiating with respect to and how you arrived at the equality on the right. In step c.) later on you substitute $x = 0$ which doesn't have a justification either. (If it does, I think you should try to be more explicit about what is happening.)</p>

<hr>

<p>I think as is, the method of finding an optimal point on the production possibilities frontier is cumbersome, and I don't think the optimization problem is well defined.</p>

<p><strong>If you wanted to refine your method</strong>, you'd have to give</p>

<ul>
<li>conditions under where an optimum exists (the conditions you state are not sufficient)</li>
<li>identification strategies for an optimum</li>
<li>a situation where this method is more practical than the industry standard</li>
</ul>

<p>So for that last point, why can't someone just have a production function (where they can derive the PPF) and a budget constraint from prices, regularity conditions and then solve? Since you seem to have some familiarity with higher level math, you may find Chapter 5 and 15 of Mas-Colell, Whinston, and Greene a good resources for seeing the foundation of how production problems are approached. Chapter 15 in particular more explicitly talks about production possibility sets.</p>

CC BY-SA 3.0
2017-09-14T03:55:18.950Z
I asked my economics Professor and apparently Lagrange Multipliers are the concept I was touching at. I didn't do the math for solving them, I just drew a picture in my head.
FX_NINJA
CC BY-SA 3.0
2017-09-14T03:59:36.340Z
Lagrangeans are a very robust tool in optimization. It is sound advice to learn them. Many a model needs a solid grasp of them :)
Kitsune Cavalry
CC BY-SA 3.0
2017-09-09T05:20:52.387Z
2017-09-09T05:20:52.387Z
Kitsune Cavalry
<p>I just started micro-economics at my community college and my teacher mentioned the derivative of the PPF for two output resources. I thought about it a while and came up with this approach. Some of the notation may be incorrect as the highest math course I completed was college algebra (I tested out of it but still have todo trigonometry).</p>

<p>Does this approach I came up with already exist?</p>

<p>Here is the equations:
<a href="https://i.sstatic.net/UqcAq.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/UqcAq.png" alt="enter image description here"></a></p>

CC BY-SA 3.0
2017-09-09T00:03:08.470Z
I want to try to publish it on ssrn or in a journal if it doesn't exist yet.
FX_NINJA
CC BY-SA 3.0
2017-09-09T01:27:05.330Z
Do you attend the Harvard of community colleges?
123
CC BY-SA 3.0
2017-09-09T02:07:43.323Z
If you want to publish this, you'd need to explain your notation (e.g. what is $r$?) and be more explicit about your assumptions.
Herr K.
CC BY-SA 3.0
2017-09-09T02:48:50.140Z
@Herr k. I should have addent that i used unit circle notation to represent the curve as an inequality. R would be radius
FX_NINJA
CC BY-SA 3.0
2017-09-09T19:04:12.327Z
Are you assuming that the PPF is an arc of a circle? Why should it be?
Herr K.
CC BY-SA 3.0
2017-09-09T19:14:44.317Z
@herr K i dont assume it to be. In the solving example i set it to a circle for the function f's output
FX_NINJA
<microeconomics><linear-algebra><linear-programming>
Has this differential calculus inequality approach to optimizing the production possibility curve exist?
FX_NINJA
# Has this differential calculus inequality approach to optimizing the production possibility curve exist?

(Asked by: username_0 on 2017-09-08)

I just started micro-economics at my community college and my teacher mentioned the derivative of the PPF for two output resources. I thought about it a while and came up with this approach. Some of the notation may be incorrect as the highest math course I completed was college algebra (I tested out of it but still have todo trigonometry).

Does this approach I came up with already exist?

Here is the equations:
[![enter image description here](https://i.sstatic.net/UqcAq.png)](https://i.sstatic.net/UqcAq.png)

**Tags:** microeconomics, linear-algebra, linear-programming

> **Comments:**
> **username_0**: I want to try to publish it on ssrn or in a journal if it doesn't exist yet.  
> **username_1**: Do you attend the Harvard of community colleges?  
> **username_2**: If you want to publish this, you'd need to explain your notation (e.g. what is $r$?) and be more explicit about your assumptions.  
> **username_0**: @Herr k. I should have addent that i used unit circle notation to represent the curve as an inequality. R would be radius  
> **username_2**: Are you assuming that the PPF is an arc of a circle? Why should it be?  
> **username_0**: @herr K i dont assume it to be. In the solving example i set it to a circle for the function f's output  

---

## 1 answers

### Answer by username_3 on 2017-09-09. Score: 1

I don't really understand your optimization problem as it stands. Just as a few examples, when you write out

$$\min \sum\_m \sum\_v \left( \frac{\partial \vec f\_v (\vec Q\_v)}{\partial \vec f\_m (\vec Q\_m)} - \frac{\partial \vec f\_m (\vec Q\_m)}{\partial \vec f\_m (\vec Q\_m)} \right)$$

I have questions about why you are minimizing the difference between optimal conditions for output that has all equal opportunity costs. The justification is unclear to me. The function $\vec f\_v(\vec Q\_v)$ also leaves me confused looking at your work. Why is $f$ taking a vector of $n$ aggregate outputs as its argument? $f(q)$ is defined as the adjusted cost of purchasing $q \cdot n$ resources, so how does $f$ transform a vector as opposed to a scalar and what interpretation are you trying to give it?

Or in your optimization problem when you write out

$$\frac{dy}{d\sqrt{r^2 - x^2}} \rightarrow - \frac{x}{\sqrt{r^2 - x^2}} = y$$

I have no idea what you are differentiating with respect to and how you arrived at the equality on the right. In step c.) later on you substitute $x = 0$ which doesn't have a justification either. (If it does, I think you should try to be more explicit about what is happening.)

---

I think as is, the method of finding an optimal point on the production possibilities frontier is cumbersome, and I don't think the optimization problem is well defined.

**If you wanted to refine your method**, you'd have to give

* conditions under where an optimum exists (the conditions you state are not sufficient)
* identification strategies for an optimum
* a situation where this method is more practical than the industry standard

So for that last point, why can't someone just have a production function (where they can derive the PPF) and a budget constraint from prices, regularity conditions and then solve? Since you seem to have some familiarity with higher level math, you may find Chapter 5 and 15 of Mas-Colell, Whinston, and Greene a good resources for seeing the foundation of how production problems are approached. Chapter 15 in particular more explicitly talks about production possibility sets.

> **Comments:**
> **username_0**: I asked my economics Professor and apparently Lagrange Multipliers are the concept I was touching at. I didn't do the math for solving them, I just drew a picture in my head.  
> **username_3**: Lagrangeans are a very robust tool in optimization. It is sound advice to learn them. Many a model needs a solid grasp of them :)  

---

