# 5.7: The Magnification Effect for Prices

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Learning Objectives

1. Learn how the magnification effect for prices represents a generalization of the Stolper-Samuelson theorem by incorporating the relative magnitudes of the changes.

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 $$\PageIndex{1}$$ for one country.

 $$a_{LS} = 3$$ $$a_{KS} = 4$$ $$P_S = 120$$ $$a_{LC} = 2$$ $$a_{KC} = 1$$ $$P_C = 40$$

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, $$\frac{a_{KS}}{a_{LS}} \left( \frac{4}{3} \right) > \frac{a_{KC}}{a_{LC}} \left( \frac{1}{2} \right)$$, which means that steel production is capital intensive and clothing is labor intensive.

The following are the zero-profit conditions in the two industries:

• Zero-profit steel: $$3w + 4r = 120$$
• Zero-profit clothing: $$2w + r = 40$$

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

$3w + 4r = 120 \nonumber$

and

$−8w − 4r = −160 \nonumber .$

Adding these two equations vertically yields

$−5w − 0r = −40 \nonumber ,$

which implies $$w=\frac{−40}{−5} = 8$$. Plugging this into the first equation above (any equation will do) yields $$3 \times 8 + 4r = 120$$. 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:

• Zero-profit steel: $$3w + 4r = 120$$
• Zero-profit clothing: $$2w + r = 60$$

Follow the same procedure to solve for the equilibrium wage and rental rates.

First, multiply the second equation by (–4) to get

$3w + 4r = 120 \nonumber$

and

$−8w − 4r = −240 \nonumber .$

Adding these two equations vertically yields

$−5w − 0r = −120 \nonumber ,$

which implies $$w = \frac{−120}{−5} = 24$$. Plugging this into the first equation above (any equation will do) yields $$3 \times 24 + 4r = 120$$. 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.

## Percentage Changes in the Goods and Factor Prices

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., $$\hat X$$= percentage change in $$X$$).

 $$\hat P_C = \frac{60−40}{40} \times 100= +50\%$$ The price of clothing rises by 50 percent. $$\hat w = \frac{24−8}{8} \times 100= +200\%$$ The wage rate rises by 200 percent. $$\hat r = \frac{12−24}{24} \times 100= −50\%$$ The rental rate falls by 50 percent. $$\hat P_S = +0 \%$$ The price of steel is unchanged.

where

• $$w$$ = the wage rate
• $$r$$ = the rental rate

The rank order of the changes in Table $$\PageIndex{2}$$ is the magnification effect for prices:

$\hat w > \hat P_C > \hat P_S > \hat r \nonumber .$

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

$\hat r > \hat P_S > \hat P_C > \hat w \nonumber .$

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.

## Key Takeaways

• The magnification effect for prices shows that if the product prices change by particular percentages with one greater than the other, then the factor prices will change by percentages that are larger than the larger product price change and smaller than the smaller. It is in this sense that the factor price changes are magnified relative to the product price changes.
• If the percentage change in the price of the capital-intensive good exceeds the percentage change in the price of the labor-intensive good, for example, then the rental rate on capital will change by a greater percentage than the price of the capital-intensive good changed, while the wage will change by less than the price of the labor-intensive good.

Exercise $$\PageIndex{1}$$

1. 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 $$\PageIndex{3}$$: 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
1. Calculate and display the magnification effect for prices in response to the tariff elimination.
2. Which product is capital intensive?
3. Which product is labor intensive?
2. 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)

1. Solve for the equilibrium wage and rental rate.
2. Suppose the price of cheese falls from $80 to$75. Solve for the new equilibrium wage and rental rates.
3. Calculate the percentage changes in the goods prices and factor prices and write the magnification effect for prices.
4. Identify which good is labor intensive and which is capital intensive.

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