Stolper-Samuelson Theorem

1. The Relationship between commodity and factor prices

Commodity prices and factor prices	The SS theorem links commodity prices and factor prices.
Competitive markets	If a country suddenly opens up to trade, how will factor prices change? FPE theorem presupposes that all markets are perfectly competitive. Perfect competition implies that in the long run profits are zero in every industry. Specifically, Π₁ = p₁y₁ - wL₁ - rK₁ = 0. Π₂ = p₂y₂ - wL₂ - rK₂ = 0.
	Per unit profits are also zero. Dividing the two profit functions by the respective outputs, we get the relationship between prices and factor prices. price = unit labor cost + unit capital cost. P₁ = a_L1 w + a_K1 r. P₂ = a_L2 w + a_K2 r.

2. Iso-unit cost curve/Factor price frontier

	It is well known that the unit cost function g₁(w,r) = a_L1 w + a_K1 r is concave in factor prices, w and r. That is, as either factor price rises, unit cost rises at a decreasing rate. Moreover, the iso-unit cost contour of (w,r) is convex to the origin, as shown by two curves in Figure 12.
Iso-unit cost curve	An iso-unit cost curve, also known as factor price frontier, is a locus of factor price combinations along which the unit cost of a good remains constant. Paul Samuelson first considered this notion and called it factor price frontier. These are derived from the above two unit cost equations. r = p₁/a_K1 - wa_L1/a_K1 .
its slope	Recall that the input-output coefficients are not fixed, but are actually functions of factor prices, i.e., a_ij = a_ij(w,r). The slope of the isoprice curve p₁ is a_L1/a_K1 = L₁/y₁ ÷ K₁/y₁ = L₁/K₁ = 1/k₁, where k₁= K₁/L₁. Thus, a more capital-intensive industry has a flatter curve.
	Recall that the slope is changing as w or r changes. That is why the iso-unit cost curves are not linear. A pair of output prices (p₁,p₂) results in a unique combination of factor prices, (w,r)^e, an equilibrium set of wage and rental. Figure 12. Equilibrium wage and rental. Note that k₂ > k₁ implies that p₂ is flatter than p₁ everywhere.

3. The Stolper-Samuelson Theorem

The SS Theorem	An increase in the price of a capital-intensive good increases the return to capital and decreases the return to the other factor (labor).

	When does the price of a good rise? Prices of importable goods rise and those of the exportable goods fall dramatically during war. Price changes moderately when tariffs are imposed.
	Corollary: An increase in the price of a capital-intensive good decreases the wage-rental ratio, w/r. Remark: Free trade increases the domestic price of the exportable and increases the return to the abundant factor. After France was defeated in May 1940, German Unterseebooten (U-boats, submarines) began to attack many merchant ships in order to cut off enemy or neutral shipping. Trans-Atlantic trade was more or less haulted until the US was dragged into WWII in December 1941. (About 2900 Allied ships were sunk by German U-boats. Five days after Japan's attack on Pearl Harbor, Germany declared war against the United States and Germany began to sink US merchant ships. What was the impact of war on the interest rate? (US interest data are available only for a later period.)

4. Magnification Effect

What is it?	The magnification effect states that an increase in the price of a capital-intensive good increases the return to capital more than proportionately.
Proof	p₂ = a_L2w + a_K2r.
	Δp₂ = a_L2Δw + a_K2Δr = a_L2w(Δw/w) + a_K2r(Δr/r).
	Here, Δ reads "change in" and the percentage change in x is written as x with a hat. That is, ^x = Δx/x. Divide both sides by p₂:
	^p₂ = θ_L2^w + θ_K2^r = θ_L2^w + (1 - θ_L2)^r for example = 75% × ^w + 25% × ^r
labor share, capital share	The percentage change in product price is a weighted average of percentage change factor price changes.) Note ^w = Δw/w is the "percentage change in" w, and θ_L2 = a_L2w/p₂ = wL₂/p₂y₂ is the share of labor in industry 2. Moreover, θ_L2 + θ_K2 = 1 (For example, labor share 75% + capital share 25% = 100%). That is, the sum of labor and capital shares is unity in every industry.
Intuition	Intuitive Reason: If both the rental rate and wage were to double, the product price must also double to breakeven. Recall that if the labor share is 75%, then the capital share is 25%. If the wage rate rises by 20% and the interest rate by 10%, then the product price rises by 10% × 3/4 + 20% × 1/4 = 12.5%. If the rental rate were to remain constant, then a 10% increase in output price must be accompanied by 10%/.75 = 13.33%. However, by the SS Theorem, we know that a 10% rise in the output price (of a capital-intensive good) results in a reduction in the wage rate, and hence the interest rate must rise even faster than 13.33% (a magnification effect on the interest rate) to more than offset the negative effect of the falling wage rate. Similarly, a rise in the price of a labor-intensive good reduces the interest rate and hence increases the wage rate more than proportionately. This is the magnification effect on the wage rate. The return to the friend factor increases more than proportionately.
Price and Factor Intensities	An increase in the price of a capital intensive good raises (r/w), and hence all industries become less capital intensive to minimize costs.