# Has this differential calculus inequality approach to optimizing the production possibility curve exist?

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: • I want to try to publish it on ssrn or in a journal if it doesn't exist yet. – FX_NINJA Sep 9 '17 at 0:03
• Do you attend the Harvard of community colleges? – 123 Sep 9 '17 at 1:27
• 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. Sep 9 '17 at 2:07
• @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 Sep 9 '17 at 2:48
• Are you assuming that the PPF is an arc of a circle? Why should it be? – Herr K. Sep 9 '17 at 19:04

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.

• 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 Sep 14 '17 at 3:55
• 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 Sep 14 '17 at 3:59