Numerical Example: Calculating a Real Wage

Consider the real wage of a worker in terms of cheese. Suppose the worker earns $10 per hour and the price of cheese is $5 per pound. The real wage can be found by dividing the wage by the price to get

$$\frac{w}{P_C}=\frac{\$10\text{/}hr}{\$5\text{/}lb}=2lbs\text{/}hr\text{.}$$

This means the worker can buy two pounds of cheese with every hour of work.

The Real Wage of Cheese Workers in Terms of Cheese

The real wage of cheese workers in terms of cheese is the quantity of cheese that a cheese worker can buy with a unit of work. It is calculated by dividing the worker’s wage by the price of cheese, written as $\frac{w_C}{P_C}$. Since zero profit results in each producing industry, we can simply rewrite the relationship derived above to construct the following formula for the real wage:

$$\frac{w_C}{P_C}=\frac{1}{a_{LC}}\text{.}$$

This means that the real wage of a worker in terms of how much cheese can be purchased is equal to labor productivity in cheese production. In other words, the amount of cheese that a worker can buy per period of work is exactly the same as the amount of cheese the worker can make in that same period.

The Real Wage of Cheese Workers in Terms of Wine

The real wage of cheese workers in terms of wine is the quantity of wine that a cheese worker can buy with a unit of work. It is calculated by dividing the cheese worker’s wage by the price of wine and is written as $\frac{w_C}{P_W}$. Using the relationship between wages and prices when zero profit results in the cheese industry implies that

$$\frac{w_C}{P_W}=\frac{\left(\frac{P_C}{a_{LC}}\right)}{P_W}=\frac{1}{a_{LC}}\frac{P_C}{P_W}\text{.}$$

This means that the real wage of cheese workers in terms of wine is the product of labor productivity in the cheese industry and the price ratio. Labor productivity gives the quantity of cheese a cheese worker makes in an hour of work. The price ratio gives the quantity of wine that exchanges for each unit of cheese. The product gives the quantity of wine that a cheese worker can buy with a unit of work. To calculate the autarky real wage, simply plug in the autarky price ratio. To calculate the free trade real wage, plug in the free trade price ratio.

The Real Wage of Wine Workers in Terms of Wine

The real wage of wine workers in terms of wine is the quantity of wine that a wine worker can buy with a unit of work. It is calculated by dividing the worker’s wage by the price of wine, written as w_W/P_W. Since zero profit results in each producing industry, we can rewrite the relationship to get

$$\frac{w_W}{P_W}=\frac{1}{a_{LW}}\text{.}$$

As with cheese, the real wage of a worker in terms of how much wine can be purchased is equal to labor productivity in wine production. In other words, the amount of wine that a worker can buy per period of work is exactly the same as the amount of wine the worker can make in that same period.

The Real Wage of Wine Workers in Terms of Cheese

The real wage of wine workers in terms of cheese is the quantity of cheese that a wine worker can buy with a unit of work. It is calculated by dividing the wine worker’s wage by the price of cheese, written as (w_W/P_C). Using the relationship between prices and wages when zero profit results in the wine industry implies that

$$\frac{w_W}{P_C}=\frac{\left(\frac{P_W}{a_{LW}}\right)}{P_C}=\frac{1}{a_{LW}}\frac{P_W}{P_C}\text{.}$$

This means that the real wage of wine workers in terms of cheese is the product of labor productivity in the wine industry and the price ratio. Labor productivity gives the quantity of wine a wine worker makes in an hour of work. The price ratio gives the quantity of cheese that exchanges for each unit of wine. The product gives the quantity of cheese that a wine worker can buy with a unit of work. To solve for the autarky real wage, simply plug in the autarky price ratio. To find the free trade real wage, plug in the free trade price ratio.

Real Wages in Autarky

To calculate autarky real wages, we simply plug the autarky price ratio into the real wage formulae.

Recall that the autarky price ratio is ${\left(\frac{P_C}{P_W}\right)}^{Aut}=\frac{a_{LC}}{a_{LW}}$. Plugging this in and simplifying yields the results in Table 2.13 "Autarky Real Wages".

Notice that in autarky, the real wage of cheese workers is exactly the same as the real wage of wine workers with respect to purchases of both goods. This occurs because labor is assumed to be homogeneous—that is, all labor is the same—and because there is free mobility between industries. (If workers were paid different wages, the lower-wage workers would move to the higher-wage industry.)

Comparison of Autarky Real Wages between Countries

Suppose the United States has an absolute advantage in the production of both goods. In this case, $\frac{1}{a_{LC}}>\frac{1}{a_{LC}^{\ast }}$ and $\frac{1}{a_{LW}}>\frac{1}{a_{LW}^{\ast }}$. This implies that the real wages of workers in both industries in the United States are higher than the real wages in France. Put another way, workers in France earn lower wages in both industries.

Sometimes cross-country wage comparisons are made and it is suggested that firms in a high-wage country cannot compete with firms in low-wage countries. However, wage comparisons of this kind are not sufficient in this model to determine who will produce what or whether trade can be advantageous. Instead, what matters is relative wage comparisons. In this model, a country will tend to specialize in the good in which it has the greatest real wage advantage. Thus if

$$\frac{\frac{1}{a_{LC}}}{\frac{1}{a_{LC}^{\ast }}}>\frac{\frac{1}{a_{LW}}}{\frac{1}{a_{LW}^{\ast }}}\text{,}$$

then the United States has relatively higher real wages with respect to cheese purchases than it does in wine purchases. When trade opens, the United States will specialize in its comparative advantage good, which, by rearranging the above inequality, can easily be shown to be cheese.

Effects of Free Trade on Real Wages

Suppose two countries, the United States and France, move from autarky to free trade. If the United States has the comparative advantage in cheese production, then $\frac{a_{LC}}{a_{LW}}<\frac{a_{LC}^{\ast }}{a_{LW}^{\ast }}$, which implies ${\left(\frac{P_C}{P_W}\right)}^{Aut}<{\left(\frac{P_C^{\ast }}{P_W^{\ast }}\right)}^{Aut}$. When the two countries move to free trade, the free trade price ratio will lie somewhere between the autarky price ratios. This means that (P_C/P_W) rises in the United States when moving from autarky to free trade, while P_C∗/P_W∗ falls when moving to free trade.

The other major change that occurs is that the United States specializes in cheese production, while France specializes in wine production. This means that real wages in free trade for wine workers in the United States need not be calculated since the United States will no longer have any wine workers. Similarly, real wages for cheese workers in France need not be calculated.

Thus we can calculate the changes in real wages shown in Table 2.14 "Changes in Real Wages (Autarky to Free Trade)".

First, consider the fate of U.S. cheese workers. Since the unit labor requirement for cheese does not change in moving to free trade, there is also no change in the real wage in terms of cheese. However, since the price of cheese in terms of wine rises, U.S. cheese workers can get more wine for each unit of cheese in exchange. Thus the real wage of cheese workers in terms of wine rises. This means cheese workers are at least as well off in free trade as they were in autarky.

The worst outcome occurs if a cheese worker has no demand for wine. Perhaps an individual abstains from alcohol consumption. In this case, the worker would be able to buy just as much cheese in free trade as in autarky, but no more. Such a person would receive no benefit from free trade. However, every worker who demands both wine and cheese will be able to buy more of both goods.

As for the workers who worked in the wine industry in the United States in autarky, they are now cheesemakers earning cheesemaker wages. Since real wages for wine workers were the same as wages for cheese workers in autarky, and since cheese workers are no worse off with free trade, then wine workers must also be no worse off in free trade. Of course, the model assumes that the movement of workers from one industry to another is costless. In the immobile factor model, we address the implications of adjustment costs across industries.

In France, the real wage of winemakers in terms of how much wine they can buy remains constant, while the real wage in terms of cheese must go up. French cheesemakers have all become winemakers because of specialization, which means all French workers are no worse off and most likely better off as a result of free trade.

The likely welfare effect of free trade, then, is that everyone in both trading countries benefits. At the very worst, some individuals will be just as well off as in autarky. This result occurs for any free trade price ratio that lies between the autarky price ratios.

In David Ricardo’s original numerical example, he demonstrated that when both countries specialize in their comparative advantage goods and engage in free trade, both countries can experience gains from trade. However, his demonstration was only true for particular numerical values. By calculating real wage changes, it is shown that it doesn’t matter which price ratio emerges in free trade as long as it is between the autarky prices. Also, because all workers receive the same wage in each country, the real wage calculations tell us that everyone benefits equally in each country.