This is “The Magnification Effect for Prices”, section 5.7 from the book Policy and Theory of International Trade (v. 1.0).
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The magnification effect for prices is a more general version of the Stolper-Samuelson theorem. It allows for simultaneous changes in both output prices and compares the magnitudes of the changes in output and factor prices.
The simplest way to derive the magnification effect is with a numerical example.
Suppose the exogenous variables of the model take the values in Table 5.6 "Numerical Values for Exogenous Variables" for one country.
Table 5.6 Numerical Values for Exogenous Variables
a_{LS} = 3 | a_{KS} = 4 | P_{S} = 120 |
a_{LC} = 2 | a_{KC} = 1 | P_{C} = 40 |
where a_{LC} = unit labor requirement in clothing production a_{LS} = unit labor requirement in steel production a_{KC} = unit capital requirement in clothing production a_{KS} = unit capital requirement in steel production P_{S} = the price of steel P_{C} = the price of clothing |
With these numbers, $\frac{{a}_{KS}}{{a}_{LS}}\left(\frac{4}{3}\right)>\frac{{a}_{KS}}{{a}_{LC}}\left(\frac{1}{2}\right)$, which means that steel production is capital intensive and clothing is labor intensive.
The following are the zero-profit conditions in the two industries:
The equilibrium wage and rental rates can be found by solving the two constraint equations simultaneously.
A simple method to solve these equations follows.
First, multiply the second equation by (−4) to get
3w + 4r = 120and
−8w − 4r = −160.Adding these two equations vertically yields
−5w − 0r = −40,which implies $w=\frac{-40}{-5}=8$. Plugging this into the first equation above (any equation will do) yields 3∗8 + 4r = 120. Simplifying, we get $r=\frac{120-24}{4}=24$. Thus the initial equilibrium wage and rental rates are w = 8 and r = 24.
Next, suppose the price of clothing, P_{C}, rises from $40 to $60 per rack. This changes the zero-profit condition in clothing production but leaves the zero-profit condition in steel unchanged. The zero-profit conditions now are the following:
Follow the same procedure to solve for the equilibrium wage and rental rates.
First, multiply the second equation by (–4) to get
3w + 4r = 120and
−8w − 4r = −240.Adding these two equations vertically yields
−5w − 0r = −120,which implies $w=\frac{-120}{-5}=24$. Plugging this into the first equation above (any equation will do) yields 3∗24 + 4r = 120. Simplifying, we get $r=\frac{120-72}{4}=12$. Thus the new equilibrium wage and rental rates are w = 24 and r = 12.
The Stolper-Samuelson theorem says that if the price of clothing rises, it will cause an increase in the price paid to the factor used intensively in clothing production (in this case, the wage rate to labor) and a decrease in the price of the other factor (the rental rate on capital). In this numerical example, w rises from $8 to $24 per hour and r falls from $24 to $12 per hour.
The magnification effect for prices ranks the percentage changes in output prices and the percentage changes in factor prices. We’ll denote the percentage change by using a ^ above the variable (i.e., $\stackrel{\wedge}{X}$= percentage change in X).
Table 5.7 Calculating Percentage Changes in the Goods and Factor Prices
$\stackrel{\wedge}{{P}_{C}}=\frac{60-40}{40}\ast 100=+50\%$ | The price of clothing rises by 50 percent. |
$\stackrel{\wedge}{w}=\frac{24-8}{8}\ast 100=+200\%$ | The wage rate rises by 200 percent. |
$\stackrel{\wedge}{r}=\frac{12-24}{24}\ast 100=-50\%$ | The rental rate falls by 50 percent. |
$\stackrel{\wedge}{{P}_{S}}=+0\%$ | The price of steel is unchanged. |
where w = the wage rate r = the rental rate |
The rank order of the changes in Table 5.7 "Calculating Percentage Changes in the Goods and Factor Prices" is the magnification effect for pricesA relationship in the H-O model that specifies the magnitude of factor price changes in response to changes in the output prices. It is used to identify the real wage and real rent effects of output price changes.:
$$\stackrel{\wedge}{w}>\stackrel{\wedge}{{P}_{C}}>\stackrel{\wedge}{{P}_{S}}>\stackrel{\wedge}{r}\text{.}$$The effect is initiated by changes in the output prices. These appear in the middle of the inequality. If output prices change by some percentage, ordered as above, then the wage rate paid to labor will rise by a larger percentage than the price of steel changes. The size of the effect is magnified relative to the cause.
The rental rate changes by a smaller percentage than the price of steel changes. Its effect is magnified downward.
Although this effect was derived only for the specific numerical values assumed in the example, it is possible to show, using more advanced methods, that the effect will arise for any output price changes that are made. Thus if the price of steel were to rise with no change in the price of clothing, the magnification effect would be
$$\stackrel{\wedge}{r}>\stackrel{\wedge}{{P}_{S}}>\stackrel{\wedge}{{P}_{C}}>\stackrel{\wedge}{w}\text{.}$$This implies that the rental rate would rise by a greater percentage than the price of steel, while the wage rate would fall.
The magnification effect for prices is a generalization of the Stolper-Samuelson theorem. The effect allows for changes in both output prices simultaneously and provides information about the magnitude of the effects. The Stolper-Samuelson theorem is a special case of the magnification effect in which one of the endowments is held fixed.
Although the magnification effect is shown here under the special assumption of fixed factor proportions and for a particular set of parameter values, the result is much more general. It is possible, using calculus, to show that the effect is valid under any set of parameter values and in a more general variable proportions model.
The magnification effect for prices can be used to determine the changes in real wages and real rents whenever prices change in the economy. These changes would occur as a country moves from autarky to free trade and when trade policies are implemented, removed, or modified.
Consider a country producing milk and cookies using labor and capital as inputs and described by a Heckscher-Ohlin model. The following table provides prices for goods and factors before and after a tariff is eliminated on imports of cookies.
Table 5.8 Goods and Factor Prices
Initial ($) | After Tariff Elimination ($) | |
---|---|---|
Price of Milk (PM) | 5 | 6 |
Price of Cookies (PC) | 10 | 8 |
Wage (w) | 12 | 15 |
Rental rate (r) | 20 | 15 |
Consider the following data in a Heckscher-Ohlin model with two goods (wine and cheese) and two factors (capital and labor).
a_{KC} = 5 hours per pound (unit capital requirement in cheese)
a_{KW} = 10 hours per gallon (unit capital requirement in wine)
a_{LC} = 15 hours per pound (unit labor requirement in cheese)
a_{LW} = 20 hours per gallon (unit labor requirement in wine)
P_{C} = $80 (price of cheese)
P_{W} = $110 (price of wine)