Understanding Rybczynski's Theorem in Economics

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

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

4. Oy K Ky/Ly Ox L0 Rybczynski The goods prices do not change, so K/L intensities do not change Oy’ 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. Y Labor Rybczynski line Capital Rybczynski line 0 X 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 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 A A similar procedure can be used to derive the dotted red Capital Rybczynski line indicating the effect of increasing the supply of capital

8. Y Labor Rybczynski line 0 X Rybczynski We can draw the ppf through the initial production point A; the income line must be tangent there As the labor stock increases production moves along the Labor Rybczynski line to the eq. at point B 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