1. Feb 6, 2009

### John Creighto

Many years ago in an economics class I remember learning about production possibility boundaries as a model of international trade.
http://tutor2u.net/economics/revision-notes/as-markets-production-possibility-frontier.html

I was thinking a bit about how these models may very with time.

A simple production possibility boundary would be an ellipse.
Consider a two nation model:

1) $${x^2 \over C_{x,1}^2}+{y^2 \over C_{y,1}^2}=1$$

2) $${x^2 \over C_{x,2}^2}+{y^2 \over C_{y,2}^2}=1$$

Each country would try to find the optimal consumption based on some objective function. A simple objective function is a hyperbola function which is equivalent to the function:

3) $$y=J/x <=> xy=J$$
Where J is the objective function (AKA utility function)

If there is no trade then each country tries to maximize J.

Subsituting 3) into equations 1) and 2) gives:

4) $${x^2 \over C_{x,1}^2}+{{J_1/x}^2 \over C_{y,1}^2}=1$$

5) $${x^2 \over C_{x,2}^2}+{{J_2/x}^2 \over C_{y,2}^2}=1$$

Multiplying both sides by $$x^2$$ and rearranging:

6) $${x^4 \over C_{x,1}^2}-x^2+{J^2 \over C_{y,1}^2}=0$$

7) $${x^4 \over C_{x,2}^2}-x^2+{J^2 \over C_{y,2}^2}=0$$

These equations have roots:

8) $$x_1^2={1 \pm \sqrt{1-{4 J_1^2 \over C_{x,1}^2 C_{y,1}^2}} \over {2 \over C_{x,1}^2}}$$

9) $$x_2^2={1 \pm \sqrt{1-{4 J_2^2 \over C_{x,2}^2 C_{y,2}^2}} \over {2 \over C_{x,2}^2}}$$

Since x must be in the top right quadrant x is taken as positive and the positive square root is taken.

J can be found by setting setting the describable equal to zero in equations 8) and 9) to give:

10)$${1-{4 J_1^2 \over C_{x,1}^2 C_{y,1}^2}=0$$

11) $${1-{4 J_2^2 \over C_{x,2}^2 C_{y,2}^2}=0$$

Which gives:

12)$$J_1={C_{x,1} C_{y,1} \over 2}$$

13) $$J_2={C_{x,2} C_{y,2} \over 2}$$

Now with regards to trade each nation starts at their equilibrium given by equations: 1), 2), 8), 9), 12), 13)

That is:

$$x_1={C_{x,1} \over \sqrt{2}}$$

$$y_1={C_{y,1} \over \sqrt{2}}$$

$$x_1={C_{x,2} \over \sqrt{2}}$$

$$y_1={C_{y,2} \over \sqrt{2}}$$

And moves along their production possibility in exchange for receiving goods from the other nation. (More to come .........)

Last edited: Feb 7, 2009
2. Feb 6, 2009

### John Creighto

The above example dealt with no trade. To Keep equations general the quantities of each good should be transformed so the utility function is hyperbolic (y=1/x) as in the above equations.

Consider country two keeping its consumption at the equilibrium defined in the previous post where there is not trade and trading with country one. This would give country one the following set of consumption possibilities.

1) $${(x-{C_{x,1}-C_{x,2} \over \sqrt{2}})^2 \over C_{x,2}^2}+{(y-{C_{y,1}-C_{y,2} \over \sqrt{2}})^2 \over C_{y,2}^2}=1$$

In this example country two is willing to adjust it's production for trade provided it is compensated enough to keep it's consumption as is.

(still more to come.......)

Last edited: Feb 7, 2009
3. Feb 7, 2009

### JorgeLobo

Sophomoric and pedantic. Failing communist style central planning - countries don't produce.

Models are easy - now validate it.

4. Feb 7, 2009

### John Creighto

Clearly it is an idealization. And yes I expect it to be flawed. I'm not even convinced that "capitalist countries maximize their use of resources. Clearly communist countries are less likely to properly allocate resources. However, do communist countries produce further from their production possibility boundary then capitalist countries or do they just tend to produce at a point which is near a less optimal point on the boundary then capitalist countries?

In a centrally planed economy as badly as the resources are mis-allocated in theory they all should be used. In capitalist countries there is massive amounts of "creative destruction" during an economic downturn combined with large unemployment. One would be think that during these downturns more could be produced without an opportunity cost.

Anyway, often in economics simplified arguments are used to try to argue a point. In an economics class I took many years ago a simmilar production possibility argument was used as a justification for trade (although it was graphical and not formula based. I also forget how the terms of trade were agreed upon).

While I believe this argument is flawed because we never have full employment and often let capital deteriorate without use, I am curious how the conclusions might change if we make small changes to the model.

I'm more interested in the mathematical properties then any predictive power such a model might have. The equations were deliberataly simple so that it should be easy to code algorthims from the above equations.

Anyway here is were I want to go with this, the act of trading should shift the production possibility boundaries over time as countries start to reinvest capital in order to produce more in areas which they have a comparative advantage.

I'm curious about the production possibility boundary as a short term choice of consumption/production possibilities. I'm curious as to what extent a short term optimization might be negative in the long term. I'm curious if in such a coupled system what inherent properties it might have such as limit cycles and chaos.

Actually perhaps the model holds well in the short term for both communist and capitalist countries. However, because of market/planing failures we do not achieve our long term optimal allocation of resources in an optimal fashion.

I'm not even sure that such models will always lead to a justification of trade because by investing in areas were countries have a comparative advantage they could hinder their overall efficiency.