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,
Π_{1} = p_{1}y_{1}  wL_{1}  rK_{1} = 0. Π_{2} = p_{2}y_{2}  wL_{2}  rK_{2} = 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. P_{1} = a_{L1} w + a_{K1} r. P_{2} = a_{L2} w + a_{K2} r.

Ricardo vs. HO  (i) Output price includes not only wage but also rent. (ii) Inputoutput coefficients (e.g., a_{L2} ) now depend on input prices (w,r). 
4. Magnification Effect  
What is it?  The magnification effect states that an increase in the price of a capitalintensive good increases the return to capital more than proportionately. 
Proof  p_{2} = a_{L2}w + a_{K2}r.

Δp_{2} = a_{L2}Δw + a_{K2}Δr = a_{L2}w(Δw/w) + a_{K2}r(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 p_{2}: 

^p_{2} = 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_{L2}w/p_{2} = wL_{2}/p_{2}y_{2} 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 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 capitalintensive 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 laborintensive 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 laborintensive good and labor are friends.) 
Price and Factor Intensities  An increase in the price of a capitalintensive good raises (r/w), and hence all industries become less capital intensive to minimize costs. 