1. Rybczynski
We will use the Edgeworth-Bowley box and results from factor price equalization (FPE) to derive Rybczynski’s theorem
It applies to the effect of an increase in the available level of an input, say
labor,
at constant prices for final goods
From FPE we know that if the prices of final goods do not change, then the
rewards to factors of production do not change
That, in turn, implies that the capital-intensity with which the two goods are produced (the
capital-labor ratio) does not change
Suppose that
good X uses capital relatively intensively
Kx/Lx > Ky/Ly

2. Rybczynski Ox Oy K L0
Ky/Ly
Kx/Lx
The capital-labor intensities for goods X and Y determine the allocation of capital and labor (on the contract curve)

3. Rybczynski
More labor available increases the Edgeworth Box Ox Oy K L0
Oy’
Ky/Ly
Kx/Lx L1
The Y-origin is shifted to the right

4. Rybczynski
The goods prices do not change, so K/L intensities do not change Ox Oy K L0
Oy’
Ky/Ly
Kx/Lx E0 E1 L1
The equilibrium moves from E0 to E1

5. Rybczynski
Less capital and labor is allocated to produce good X Oy
Oy’ K
Kx/Lx E0
Ky/Ly E1 Ox L0 L1
The production of X falls; similarly, the production of Y rises

6. Rybczynski
Rybczynski’s result makes perfect sense. After all the economy-wide K/L ratio must be a weighted average of Kx/Lx and Ky/Ly. If the economy-wide K/L ratio falls as a result of an increase in labor, while the sectoral rates Kx/Lx and Ky/Ly do not change because the prices of final goods do not change, then the weight attached to the relatively capital-intensive sector (good X) must decrease, pulling capital and labor away and causing a fall in its output.
Rybczynski’s result can be nicely illustrated in goods space to show the impact on the production possibility frontier (ppf).
In this respect it is important to note that the demonstration of the Rybczynski result above using the Edgeworth Box is exclusively based on straight lines. Any other change in the available labor supply results in changes in produced output
equiproportional

7. Rybczynski
If the economy produces at point A and labor rises10 units, causing a rise in the output of Y of, say, 4 units and a fall in the output of X by 3 units, then X Y
a further increase in labor of 10 units also leads to 4 more Y and 3 less X
This process continues until X is no longer produced; it is summarized by the blue
Labor Rybczynski line
Labor Rybczynski line A
A similar procedure can be used to derive the dotted red Capital Rybczynski line indicating the effect of increasing the supply of capital
Capital Rybczynski line

8. Rybczynski
We can draw the ppf through the initial production point A; the income line must be tangent there X Y
As the labor stock increases production moves along the Labor Rybczynski line to the eq. at point B
Labor Rybczynski line
The increase in labor also shifts the ppf outward; it must be tangent to a parallel income line at point B B A
The outward shift of the ppf is biased toward good Y, that is the labor-intensive good

9. Rybczynski