Stolper-Samuelson Theorem

The Relationship between commodity and factor prices

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  =      aL1 w       +    aK1 r.

      P2  =      aL2 w       +    aK2 r.

Iso-unit cost curve

It is well known that the unit cost function g1(w,r) = aL1 w + aK1 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.

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/aK1 - waL1/aK1 .

Recall that the input-output coefficients are not fixed, but are actually functions of factor prices, i.e., aij = aij(w,r). The slope of the isoprice curve p1 is

aL1/aK1 = L1/y1 ÷ K1/y1 = L1/K1 = 1/k1 , k1= K1/L1.

 

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.

The Stolper-Samuelson Theorem

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

Figure 13. The Stolper-Samuelson Theorem


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. Fortunately, in February 1943, UK invented a new kind of radar at higher frequencies that cannot be detected by German submarines, and subsequently, most of German submarines were destroyed.) What was the impact of war on the interest rate? (US interest data are available only for a later period.)


A diary of desperate soldiers on a rafter after their ship was sunk by a U-boat.

Is there evidence to support the Stolper-Samuelson Theorem?

During World War II and the 50s, interest rates were low in Canada (probably in the US, but US interest rate data are not available). After trade volume increased in the 60s and 70s, Canadian interest rates rose considerably. This would be consistent with the Stolper-Samuelson Theorem if Canada had been abundant in capital, because the transition from autarky until 1950s to free trade would have raised the interest rate. There is some evidence that US wage rates of unskilled workers also declined since the 1970s.

Magnification Effect

alarm clock
The maginification effect states that an increase in the price of a capital-intensive good increases the return to capital more than proportionately.

Proof:

 

 

 

 

 

Here, Δ reads "change in" and the percentage change in x is written as x with a hat.

Divide both sides by p2:

 

 (The percentage change in product price is a weighted average of percentage change factor price changes.)

where ^w = Δw/w is the "percentage change in" w, and θL2 = aL2 w/p2 = wL2/p2y2 is the share of labor in industry 2. Moreover,

θL2 + θK2 = 1 (For example, labor share 75% + capital share 25% = 100%). That is, the sum of labor and capital shares is unity in every industry.

 

Which is rising faster, r or p2? Subtracting ^p2 from ^r, we get

 

 

Then the rental rate must rise faster than the output price.

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

triball An increase in the price of a capital intensive good raises (r/w), and hence makes all industries less capital intensive.