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.


Output prices and input prices

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.


Ricardo vs. HO

(i) Output price includes not only wage but also rent.

(ii) Input-output coefficients (e.g., aL2 ) now depend on input prices (w,r).



  2. Iso-unit cost curve/Factor Price Frontier
  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.
Iso-unit cost curve of p1 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 - w(aL1/aK1).

Intuitively, p1/aK1 is the vertical intercept, where w = 0, and aL1/aK1 is the slope of the curve, except that its slope is a function of factor prices (w,r).

 its slope 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 ,

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.

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 the capital-intensive good increases the return to capital and decreases the return to the other factor (labor).

Likewise, an increase in the price of the labor-intensive good increases wage and reduces rent.

(L and the labor-intensive industry are friends, L and the capital-intensive industry are enemies.)

  When does the price of a good rise? Prices of importable goods rise and those of the exportable goods fall dramatically during the war. Price changes are moderate when tariffs are imposed.


  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.


p2 = aL2w + aK2r.



Δp2 = aL2Δw + aK2Δr = aL2w(Δw/w) + aK2r(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 = θL2^w + θK2^r = θL2^w + (1 - θL2)^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

θL2 = aL2w/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.


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 20% × 3/4 + 10% × 1/4 = 17.5%. 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. An increase in the price of the labor (capital) intensive good increases wages (rent) more than proportionately. (The labor-intensive good and labor are friends.)

 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.