Comparative Advantage with Many Goods

Comparative Advantage with Many Goods
In our discussion so far, we have relied on a model in which only two goods are produced and
consumed. This simplified analysis allows us to capture many essential points about comparative
advantage and trade and, as we saw in the last section, gives us a surprising amount of
mileage as a tool for discussing policy issues. To move closer to reality, however, it is necessary
to understand how comparative advantage functions in a model with a larger number of goods.
Setting Up the Model
Again, imagine a world of two countries, Home and Foreign. As before, each country has
only one factor of production, labor. However, let’s assume that each of these countries
consumes and is able to produce a large number of goods—say, N different goods altogether.
We assign each of the goods a number from 1 to N.
The technology of each country can be described by its unit labor requirement for each
good, that is, the number of hours of labor it takes to produce one unit of each good. We
label Home’s unit labor requirement for a particular good as where i is the number we
have assigned to that good. If cheese is assigned the number 7, will mean the unit labor
requirement in cheese production. Following our usual rule, we label the corresponding
Foreign unit labor requirement .
To analyze trade, we next pull one more trick. For any good, we can calculate ,
the ratio of Home’s unit labor requirement to Foreign’s. The trick is to relabel the goods so
that the lower the number, the lower this ratio. That is, we reshuffle the order in which we
number goods in such a way that
(3-6)
Relative Wages and Specialization
We are now prepared to look at the pattern of trade. This pattern depends on only one
thing: the ratio of Home to Foreign wages. Once we know this ratio, we can determine
who produces what.
Let w be the wage rate per hour in Home and be the wage rate in Foreign. The ratio
of wage rates is then . The rule for allocating world production, then, is simply this:
Goods will always be produced where it is cheapest to make them. The cost of making
some good, say good i, is the unit labor requirement times the wage rate. To produce good
i in Home will cost . To produce the same good in Foreign will cost . It will be
cheaper to produce the good in Home if
which can be rearranged to yield
aLi
* /aLi 7 w/w*.
waLi 6 w*aLi
* ,
w*aLi
wa * Li
w/w*
w*
aL1/aL1
* 6 aL2/aL2
* 6 aL3/aL13
* 6 Á 6 aLN/aLN
* .
aLi/aLi
*
aLi
*
aL7
aLi,
CHAPTER 3 Labor Productivity and Comparative Advantage: The Ricardian Model 41
On the other hand, it will be cheaper to produce the good in Foreign if
which can be rearranged to yield
Thus we can restate the allocation rule: Any good for which will be produced
in Home, while any good for which will be produced in Foreign.
We have already lined up the goods in increasing order of (equation (3-6)). This
criterion for specialization tells us that there is a “cut” in the lineup determined by the ratio
of the two countries’ wage rates, . All the goods to the left of that point end up being
produced in Home; all the goods to the right end up being produced in Foreign. (It is possible,
as we will see in a moment, that the ratio of wage rates is exactly equal to the ratio of
unit labor requirements for one good. In that case this borderline good may be produced in
both countries.)
Table 3-2 offers a numerical example in which Home and Foreign both consume and
are able to produce five goods: apples, bananas, caviar, dates, and enchiladas.
The first two columns of this table are self-explanatory. The third column is the ratio of
the Foreign unit labor requirement to the Home unit labor requirement for each good—or,
stated differently, the relative Home productivity advantage in each good. We have labeled
the goods in order of Home productivity advantage, with the Home advantage greatest for
apples and least for enchiladas.
Which country produces which goods depends on the ratio of Home and Foreign wage
rates. Home will have a cost advantage in any good for which its relative productivity is
higher than its relative wage, and Foreign will have the advantage in the others. If, for
example, the Home wage rate is five times that of Foreign (a ratio of Home wage to
Foreign wage of five to one), apples and bananas will be produced in Home and caviar,
dates, and enchiladas in Foreign. If the Home wage rate is only three times that of Foreign,
Home will produce apples, bananas, and caviar, while Foreign will produce only dates and
enchiladas.
Is such a pattern of specialization beneficial to both countries? We can see that it is by
using the same method we used earlier: comparing the labor cost of producing a good
directly in a country with that of indirectly “producing” it by producing another good and
trading for the desired good. If the Home wage rate is three times the Foreign wage (put
another way, Foreign’s wage rate is one-third that of Home), Home will import dates and
enchiladas. A unit of dates requires 12 units of Foreign labor to produce, but its cost in
terms of Home labor, given the three-to-one wage ratio, is only 4 person-hours (12/4 = 3).
w/w*
aLi/aLi
*
aLi
* /aLi 6 w/w*
aLi
* /aLi 7 w/w*
aLi
* /aLi 6 w/w*.
waLi 7 w*aLi
* ,
TABLE 3-2 Home and Foreign Unit Labor Requirements
Good
Home Unit Labor
Requirement aLi
Foreign Unit Labor
Requirement (aLi)
*
Relative Home
Productivity
Advantage (aLi )
* /aLi
Apples 1 10 10
Bananas 5 40 8
Caviar 3 12 4
Dates 6 12 2
Enchiladas 12 9 0.75
42 PART ONE International Trade Theory
This cost of 4 person-hours is less than the 6 person-hours it would take to produce the
unit of dates in Home. For enchiladas, Foreign actually has higher productivity along with
lower wages; it will cost Home only 3 person-hours to acquire a unit of enchiladas through
trade, compared with the 12 person-hours it would take to produce it domestically. A similar
calculation will show that Foreign also gains; for each of the goods Foreign imports, it
turns out to be cheaper in terms of domestic labor to trade for the good rather than produce
the good domestically. For example, it would take 10 hours of Foreign labor to produce a
unit of apples; even with a wage rate only one-third that of Home workers, it will require
only 3 hours of labor to earn enough to buy that unit of apples from Home.
In making these calculations, however, we have simply assumed that the relative wage
rate is 3. How does this relative wage rate actually get determined?
Determining the Relative Wage in the Multigood Model
In the two-good model, we determined relative wages by first calculating Home wages in
terms of cheese and Foreign wages in terms of wine. We then used the price of cheese relative
to that of wine to deduce the ratio of the two countries’ wage rates. We could do this
because we knew that Home would produce cheese and Foreign wine. In the many-good
case, who produces what can be determined only after we know the relative wage rate, so
we need a new procedure. To determine relative wages in a multigood economy, we must
look behind the relative demand for goods to the implied relative demand for labor. This is
not a direct demand on the part of consumers; rather, it is a derived demand that results
from the demand for goods produced with each country’s labor.
The relative derived demand for Home labor will fall when the ratio of Home to
Foreign wages rises, for two reasons. First, as Home labor becomes more expensive relative
to Foreign labor, goods produced in Home also become relatively more expensive,
and world demand for these goods falls. Second, as Home wages rise, fewer goods will be
produced in Home and more in Foreign, further reducing the demand for Home labor.
We can illustrate these two effects using our numerical example as illustrated in Table 3-2.
Suppose we start with the following situation: The Home wage is initially 3.5 times the
Foreign wage. At that level, Home would produce apples, bananas, and caviar while Foreign
would produce dates and enchiladas. If the relative Home wage were to increase from 3.5 to
3.99, the pattern of specialization would not change. However, as the goods produced in
Home became relatively more expensive, the relative demand for these goods would decline
and the relative demand for Home labor would decline with it.
Suppose now that the relative wage were to increase slightly from 3.99 to 4.01. This
small further increase in the relative Home wage would bring about a shift in the pattern
of specialization. Because it is now cheaper to produce caviar in Foreign than in Home,
the production of caviar shifts from Home to Foreign. What does this imply for the relative
demand for Home labor? Clearly it implies that as the relative wage rises from a little
less than 4 to a little more than 4, there is an abrupt drop-off in the relative demand, as
Home production of caviar falls to zero and Foreign acquires a new industry. If the relative
wage continues to rise, relative demand for Home labor will gradually decline, then
drop off abruptly at a relative wage of 8, at which point production of bananas shifts to
Foreign.
We can illustrate the determination of relative wages with a diagram like Figure 3-5.
Unlike Figure 3-3, this diagram does not have relative quantities of goods or relative prices
of goods on its axes. Instead it shows the relative quantity of labor and the relative wage
rate. The world demand for Home labor relative to its demand for Foreign labor is shown
by the curve RD. The world supply of Home labor relative to Foreign labor is shown by
the line RS.
CHAPTER 3 Labor Productivity and Comparative Advantage: The Ricardian Model 43
The relative supply of labor is determined by the relative sizes of Home’s and Foreign’s
labor forces. Assuming that the number of person-hours available does not vary with the
wage, the relative wage has no effect on relative labor supply and RS is a vertical line.
Our discussion of the relative demand for labor explains the “stepped” shape of RD.
Whenever we increase the wage rate of Home workers relative to that of Foreign workers,
the relative demand for goods produced in Home will decline and the demand for Home
labor will decline with it. In addition, the relative demand for Home labor will drop off
abruptly whenever an increase in the relative Home wage makes a good cheaper to produce
in Foreign. So the curve alternates between smoothly downward-sloping sections
where the pattern of specialization does not change and “flats” where the relative demand
shifts abruptly because of shifts in the pattern of specialization. As shown in the figure,
these “flats” correspond to relative wages that equal the ratio of Home to Foreign productivity
for each of the five goods.
The equilibrium relative wage is determined by the intersection of RD and RS. As
drawn, the equilibrium relative wage is 3. At this wage, Home produces apples, bananas,
and caviar while Foreign produces dates and enchiladas. The outcome depends on the relative
size of the countries (which determines the position of RS) and the relative demand
for the goods (which determines the shape and position of RD).
If the intersection of RD and RS happens to lie on one of the flats, both countries produce
the good to which the flat applies.