Rybczynski Theorem

Factor Growth

Production: Factor growth may increase the output of both the exportable and the importable by the same rate. This kind of growth is called neutral growth.

For example, if K and L grow by 5% (^K = ^L = 5%), then neutral growth occurs because both the outputs of the exportable and the importable grow by the same rate.

If the production of the exportable grows faster than that of the importable (^y₁ > ^y₂ > 0) it is called export-biased growth.

If the output of the exportable increases and that of the importable decreases (^y₁ > 0 > ^y₂) the growth is ultra-export biased.

Ultra-import biased growth is similarly defined. The following figure shows neutral, import biased, export biased, ultra export biased and ultra import biased growth.

Instead of shifting outward, sometimes it is possible for the PPF to contract inwards. For instance, during World War II, much resources was diverted away from civilian production to produce defense goods.

Figure 16.

The growth of a factor does not necessarily increase the output of every good.
The growth of a factor does not necessarily increase exports. Figure 2 shows that export may decline, if a consumption bias exists.
Figure 17
For a small country, factor growth always increases national income and welfare (because growth does not affect prices)
If the terms of trade deteriorate sufficiently, growth can be immiserizing, lowering income and welfare.
Figure 18. Immiserizing Growth

The Rybczynski Theorem

a_L1 y₁ + a_L2 y₂ = L, (i.e., L₁ + L₂ = L)

a_K1 y₁ + a_K2 y₂ = K. (i.e., K₁ + K₂ = K.)

_ij

The Rybczynski Theorem

An increase in the endowment of labor increases the output of the labor intensive good and decreases that of the other good (capital intensive good).

The above equations show that the sum of the inputs used in the two industries must add up to the nation's input supplies.

(L₁,K₁) + (L₂,K₂₎ = (L,K).

This relationship between inputs and outputs are shown in Figure 19. The cone of diversification can be used to illustrate Rybczynski Theorem in the output spac e. An increase in the endowment of one factor results in either an ultra-export or import biased growth.

Figure 19. The Rybczynski Theorem

The next diagram also illustrates Rybczynski Theorem.

Figure 20.

The Magnification Effect

You may skip this proof.

a_L1 y₁ + a_L2y₂ =L,

a_L1Δy₁ + a_L2 Δy₂

ΔL = a_L1 y₁ (Δy₁/y₁) + a_L2 y₂ (Δy₂/y₂) = ΔL

divide both sides by L

^L = λ_L1 ^y₁ + λ_L2 ^y₂.

where λL1 = a_L1 y₁/L = L₁/L = the percentage of labor forced employed in industry 1. This shows that the percentage change in labor is a weighted average of the growth rates of the two outputs. That is, labor growth lies somewhere between the two output growth rates, ^y₁ and ^y₂.

If both outputs grew at the same rate, say 10%, then labor demand would also grow by the same rate. However, by the Rybczynski Theorem, the output of one industry actually declines. This implies that the other industry must grow more than proportionately. That is, if labor grows by 10%, one industry declines but the other industry must grow more than 10%. This is the intuition behind the Magnification Effect. But if you insist on proof, it is given below.

λ_L1 + λ_L2 = 1.

^y₁ - ^L = (1 - λ_L1) ^y₁ - λ_L2 ^y₂ = λ_L2 (^y₁ - ^y₂) > 0.

Remark: If both outputs, y₁ and y₂, were to increase 10%, labor demand would also increase 10%. However, by the Rybczinski theorem, an increase in labor decreases the output of the capital-intensive good, y₂. Since y₂ is declining, in order to effect a 10% growth in labor force, output of y₁ must increase by more than 10%, hence a magnification effect.

Exercises

1. Using the isoprice curves, evaluate the impacts of an increase in the price of a labor intensive good to returns to capital and labor. Restate the Stolper-Samuelson theorem and the magnification effect.
2. Evaluate the effects of an increase in the endowment of capital on the production of labor intensive and capital intensive goods. Restate the Rybczynski Theorem and its magnification effect.

What is this cardinal showing off in this portrait? About the time (1517) when Martin Luther pegged the ninety five theses on the church door of Wittenburg, the church was rich.

Washington National Gallery