Stolper-Samuelson theorem Totally Explained

Everything about The Stolper-samuelson Theorem totally explained

The Stolper-Samuelson theorem is a basic theorem in trade theory. It describes a relation between the relative prices of output goods and relative factor rewards, specifically, real wages and real returns to capital. The theorem states that — under some economic assumptions (constant returns, perfect competition) — a rise in the relative price of a good will lead to a rise in the return to that factor which is used most intensively in the production of the good, and conversely, to a fall in the return to the other factor.
   It was derived in 1941 from within the framework of the Heckscher-Ohlin model by Paul Samuelson and Wolfgang Stolper, but has subsequently been derived in less restricted models. As a term, it's applied to all cases where the effect is seen. Jones & Scheinkman (1977) show that under very general conditions the factor returns change with output prices as predicted by the theorem. If considering the change in real returns under increased international trade a robust finding of the theorem is that returns to the scarce factor will go down, ceteris paribus. A further robust corollary of the theorem is that a compensation to the scarce-factor exists which will overcome this effect and make increased trade Pareto optimal.
   The original Heckscher-Ohlin model was a two factor model with a labour market specified by a single number. Therefore, the early versions of the theorem could make no predictions about the effect on the unskilled labour force in a high income country under trade liberalization. However, more sophisticated models with multiple classes of worker productivity have been shown to produce the Stolper-Samuelson effect within each class of labour: Unskilled workers producing traded goods in a high-skill country will be worse off as international trade increases, because, relative to the world market in the good they produce, an unskilled first world production-line worker is a less abundant factor of production than capital.
   The Stolper-Samuelson theorem is closely linked to the factor price equalization theorem, which states that, regardless of international factor mobility, factor prices will tend to equalize across countries that don't differ in technology.

Derivation

Considering a two-good economy that produces only wheat and cloth, with labour and land being the only factors of production, wheat a land-intensive industry and cloth a labour-intensive one, and assuming that the price of each product equals its marginal cost, the theorem can be derived.
The price of cloth should be:
(1)

P(C)= ar + bw

,
with

P(C)

standing for the price of cloth,

r

standing for rent paid to landowners, $w$ for wage levels and $a$ and $b$ respectively standing for the amount of land and labour uses.
Similarly, the price of wheat would be:
(2)

P(W) = cr + dw

With $P(W)$ standing for the price of wheat, $r$ and $w$ for rent and wages, and $c$ and $d$ for the respective amount of land and labour used.
   If, then, cloth experiences a rise in its price, at least one of its factors must also become more expensive, for equation 1 to hold true, since the relative amounts of labour and land are not affected by changing prices. It can be assumed that it would be labour, the intensively used factor in the production of cloth, that would rise.
   When wages rise, rent must fall, in order for equation 2 to hold true. But a fall in rent also affects equation 1. For it to still hold true, then, the rise in wages must be more than proportional to the rise in cloth prices.
   A rise in the price of a product, then, will more than proportionally raise the return to the most intensively used factor, and a fall on the return to the less intensively used factor.

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