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John Creighto

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Trade: Production Possibility Boundaries

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) [tex]{x^2 \over C_{x,1}^2}+{y^2 \over C_{y,1}^2}=1[/tex]

2) [tex]{x^2 \over C_{x,2}^2}+{y^2 \over C_{y,2}^2}=1[/tex]

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) [tex]y=J/x <=> xy=J[/tex]

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) [tex]{x^2 \over C_{x,1}^2}+{{J_1/x}^2 \over C_{y,1}^2}=1[/tex]

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

Multiplying both sides by [tex]x^2[/tex] and rearranging:

6) [tex]{x^4 \over C_{x,1}^2}-x^2+{J^2 \over C_{y,1}^2}=0[/tex]

7) [tex]{x^4 \over C_{x,2}^2}-x^2+{J^2 \over C_{y,2}^2}=0[/tex]

These equations have roots:

8) [tex]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}} [/tex]9) [tex]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}} [/tex]

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)[tex]{1-{4 J_1^2 \over C_{x,1}^2 C_{y,1}^2}=0[/tex]11) [tex]{1-{4 J_2^2 \over C_{x,2}^2 C_{y,2}^2}=0[/tex]

Which gives:

12)[tex]J_1={C_{x,1} C_{y,1} \over 2}[/tex]13) [tex]J_2={C_{x,2} C_{y,2} \over 2}[/tex]

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

That is:

[tex]x_1={C_{x,1} \over \sqrt{2}}[/tex]

[tex]y_1={C_{y,1} \over \sqrt{2}}[/tex]

[tex]x_1={C_{x,2} \over \sqrt{2}}[/tex]

[tex]y_1={C_{y,2} \over \sqrt{2}}[/tex]

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

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) [tex]{x^2 \over C_{x,1}^2}+{y^2 \over C_{y,1}^2}=1[/tex]

2) [tex]{x^2 \over C_{x,2}^2}+{y^2 \over C_{y,2}^2}=1[/tex]

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) [tex]y=J/x <=> xy=J[/tex]

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) [tex]{x^2 \over C_{x,1}^2}+{{J_1/x}^2 \over C_{y,1}^2}=1[/tex]

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

Multiplying both sides by [tex]x^2[/tex] and rearranging:

6) [tex]{x^4 \over C_{x,1}^2}-x^2+{J^2 \over C_{y,1}^2}=0[/tex]

7) [tex]{x^4 \over C_{x,2}^2}-x^2+{J^2 \over C_{y,2}^2}=0[/tex]

These equations have roots:

8) [tex]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}} [/tex]9) [tex]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}} [/tex]

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)[tex]{1-{4 J_1^2 \over C_{x,1}^2 C_{y,1}^2}=0[/tex]11) [tex]{1-{4 J_2^2 \over C_{x,2}^2 C_{y,2}^2}=0[/tex]

Which gives:

12)[tex]J_1={C_{x,1} C_{y,1} \over 2}[/tex]13) [tex]J_2={C_{x,2} C_{y,2} \over 2}[/tex]

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

That is:

[tex]x_1={C_{x,1} \over \sqrt{2}}[/tex]

[tex]y_1={C_{y,1} \over \sqrt{2}}[/tex]

[tex]x_1={C_{x,2} \over \sqrt{2}}[/tex]

[tex]y_1={C_{y,2} \over \sqrt{2}}[/tex]

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

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