Model of a Two-Factor Economy

Model of a Two-Factor Economy
In this chapter, we’ll focus on the simplest version of the factor-proportions model, sometimes
referred to as “2 by 2 by 2”: two countries, two goods, two factors of production. In
our example we’ll call the two countries Home and Foreign. We will stick with the same
two goods, cloth (measured in yards) and food (measured in calories), that we used in the
specific factors model of Chapter 4. The key difference is that in this chapter, we assume
that the immobile factors that were specific to each sector (capital in cloth, land in food)
are now mobile in the long run. Thus land used for farming can be used to build a textile
plant, and conversely, the capital used to pay for a power loom can be used to pay for a
tractor. To keep things simple, we model a single additional factor that we call capital,
which is used in conjunction with labor to produce either cloth or food. In the long run,
both capital and labor can move across sectors, thus equalizing their returns (rental rate
and wage) in both sectors.
Prices and Production
Both cloth and food are produced using capital and labor. The amount of each good produced,
given how much capital and labor are employed in each sector, is determined by a
production function for each good:
where and are the output levels of cloth and food, and are the amounts of
capital and labor employed in cloth production, and and are the amounts of capital
and labor employed in food production. Overall, the economy has a fixed supply of capital
K and labor L that is divided between employment in the two sectors.
We define the following expressions that are related to the two production technologies:
These unit input requirements are very similar to the ones defined in the Ricardian model
(for labor only). However, there is one crucial difference: In these definitions, we speak of
the quantity of capital or labor used to produce a given amount of cloth or food, rather than
the quantity required to produce that amount. The reason for this change from the
Ricardian model is that when there are two factors of production, there may be some room
for choice in the use of inputs.
aLF = labor used to produce one calorie of food
aKF = capital used to produce one calorie of food
aLC = labor used to produce one yard of cloth
aKC = capital used to produce one yard of cloth
KF LF
QC QF KC LC
QF = QF (KF, LF),
QC = QC(KC, LC),
82 PART ONE International Trade Theory
In general, those choices will depend on the factor prices for labor and capital.
However, let’s first look at a special case in which there is only one way to produce
each good. Consider the following numerical example: Production of one yard of
cloth requires a combination of two work-hours and two machine-hours. The production
of food is more automated; as a result, production of one calorie of food requires
only one work-hour along with three machine-hours. Thus, all the unit input requirements
are fixed at and there is no possibility of
substituting labor for capital or vice versa. Assume that an economy is endowed with
3,000 units of machine-hours along with 2,000 units of work-hours. In this special
case of no factor substitution in production, the economy’s production possibility
frontier can be derived using those two resource constraints for capital and labor.
Production of yards of cloth requires machine-hours and
work-hours. Similarly, production of calories of food requires
machine-hours and work-hours. The total
machine-hours used for both cloth and food production cannot exceed the total supply
of capital:
(5-1)
This is the resource constraint for capital. Similarly, the resource constraint for labor states
that the total work-hours used in production cannot exceed the total supply of labor:
(5-2)
Figure 5-1 shows the implications of (5-1) and (5-2) for the production possibilities
in our numerical example. Each resource constraint is drawn in the same way that we
drew the production possibility line for the Ricardian case in Figure 3-1. In this case,
however, the economy must produce subject to both constraints. So the production
possibility frontier is the kinked line shown in red. If the economy specializes in food
production (point 1), then it can produce 1,000 calories of food. At that production
point, there is spare labor capacity: Only 1,000 work-hours out of 2,000 are employed.
Conversely, if the economy specializes in cloth production (point 2), then it can
produce 1,000 yards of cloth. At that production point, there is spare capital capacity:
Only 2,000 machine-hours out of 3,000 are employed. At production point 3, the economy
is employing all of its labor and capital resources (1,500 machine-hours and 1,500
work-hours in cloth production, and 1,500 machine-hours along with 500 work-hours
in food production).1
The important feature of this production possibility frontier is that the opportunity cost
of producing an extra yard of cloth in terms of food is not constant. When the economy is
producing mostly food (to the left of point 3), then there is spare labor capacity. Producing
two fewer units of food releases six machine-hours that can be used to produce three yards
of cloth: The opportunity cost of cloth is 2/3. When the economy is producing mostly cloth
(to the right of point 3), then there is spare capital capacity. Producing two fewer units of
food releases two work-hours that can be used to produce one yard of cloth: The opportunity
cost of cloth is 2. Thus, the opportunity cost of cloth is higher when more units of
cloth are being produced.
aLC * QC + aLF * QF … L, or 2QC + QF … 2,000
aKC * QC + aKF * QF … K, or 2QC + 3QF … 3,000
3QF = aKF * QF 1QF = aLF * QF
2QC = aLC * QC QF
QC 2QC = aKC * QC
aKC = 2; aLC = 2; aKF = 3; aLF = 1;
1The case of no factor substitution is a special one in which there is only a single production point that fully
employs both factors; some factors are left unemployed at all the other production points on the production possibilities
frontier. In the more general case below with factor substitution, this peculiarity disappears, and both
factors are fully employed along the entire production possibility frontier.
CHAPTER 5 Resources and Trade: The Heckscher-Ohlin Model 83
Production possibility frontier:
slope = opportunity cost of cloth
in terms of food
Labor constraint
slope=−2
Capital constraint
slope = −2/3
Quantity of food, QF
2,000
750 1,000 1,500
1,000
500
Quantity of
cloth, QC
1
3
2
Figure 5-1
The Production Possibility Frontier Without Factor Substitution: Numerical Example
If capital cannot be substituted for labor or vice versa, the production possibility frontier in the
factor-proportions model would be defined by two resource constraints: The economy can’t use
more than the available supply of labor (2,000 work-hours) or capital (3,000 machine-hours). So
the production possibility frontier is defined by the red line in this figure. At point 1, the economy
specializes in food production, and not all available work-hours are employed. At point 2, the
economy specializes in cloth, and not all available machine-hours are employed. At production
point 3, the economy employs all of its labor and capital resources. The important feature of the
production possibility frontier is that the opportunity cost of cloth in terms of food isn’t constant:
It rises from 2/3 to 2 when the economy’s mix of production shifts toward cloth.
Now let’s make the model more realistic and allow the possibility of substituting capital
for labor and vice versa in production. This substitution removes the kink in the
production possibility frontier; instead, the frontier PP has the bowed shape shown in
Figure 5-2. The bowed shape tells us that the opportunity cost in terms of food of producing
one more unit of cloth rises as the economy produces more cloth and less food.
That is, our basic insight about how opportunity costs change with the mix of production
remains valid.
Where on the production possibility frontier does the economy produce? It depends on
prices. Specifically, the economy produces at the point that maximizes the value of production.
Figure 5-3 shows what this implies. The value of the economy’s production is
where and are the prices of cloth and food, respectively. An isovalue line—a line
along which the value of output is constant—has a slope of . The economy produces
at the point Q, the point on the production possibility frontier that touches the highest
possible isovalue line. At that point, the slope of the production possibility frontier is
equal to . So the opportunity cost in terms of food of producing another unit of
cloth is equal to the relative price of cloth.
-PC /PF
-PC /PF
PC PF
V = PC * QC + PF * QF,
84 PART ONE International Trade Theory
Choosing the Mix of Inputs
As we have noted, in a two-factor model producers may have room for choice in the use of
inputs. A farmer, for example, can choose between using relatively more mechanized
equipment (capital) and fewer workers, or vice versa. Thus, the farmer can choose how
much labor and capital to use per unit of output produced. In each sector, then, producers
will face not fixed input requirements (as in the Ricardian model) but trade-offs like the
one illustrated by curve II in Figure 5-4, which shows alternative input combinations that
can be used to produce one calorie of food.
What input choice will producers actually make? It depends on the relative costs of
capital and labor. If capital rental rates are high and wages low, farmers will choose to produce
using relatively little capital and a lot of labor; on the other hand, if the rental rates
are low and wages high, they will save on labor and use a lot more capital. If w is the wage
Isovalue lines
PP
Q
slope = –PC
/PF
Quantity of food, QF
Quantity of cloth, QC
Figure 5-3
Prices and Production
The economy produces at the
point that maximizes the value
of production given the prices it
faces; this is the point that is on
the highest possible isovalue
line. At that point, the opportunity
cost of cloth in terms of
food is equal to the relative
price of cloth, PC /PF .
Quantity of food, QF
PP
Quantity of cloth, QC
Figure 5-2
The Production Possibility
Frontier with Factor Substitution
If capital can be substituted for
labor and vice versa, the production
possibility frontier no longer
has a kink. But it remains true
that the opportunity cost of cloth
in terms of food rises as the
economy’s production mix shifts
toward cloth and away from food.
CHAPTER 5 Resources and Trade: The Heckscher-Ohlin Model 85
2The optimal choice of the labor-capital ratio is explored at greater length in the appendix to this chapter.
rate and r the rental cost of capital, then the input choice will depend on the ratio of these
two factor prices, .2 The relationship between factor prices and the ratio of labor to
capital use in production of food is shown in Figure 5-5 as the curve FF.
There is a corresponding relationship between and the labor-capital ratio in cloth
production. This relationship is shown in Figure 5-5 as the curve CC. As drawn, CC is
shifted out relative to FF, indicating that at any given factor prices, production of cloth
will always use more labor relative to capital than will production of food. When this is
true, we say that production of cloth is labor-intensive, while production of food is
capital-intensive. Notice that the definition of intensity depends on the ratio of labor to
capital used in production, not the ratio of labor or capital to output. Thus a good cannot
be both capital- and labor-intensive.
w/r
w/r
Capital input
per calorie, aKF
II
Labor input
per calorie, aLF
Input combinations
that produce one
calorie of food
Figure 5-4
Input Possibilities in Food
Production
A farmer can produce a calorie of
food with less capital if he or she
uses more labor, and vice versa.
Wage-rental
ratio, w/r
CC
Labor-capital
ratio, L / K
FF
Figure 5-5
Factor Prices and Input Choices
In each sector, the ratio of labor to
capital used in production depends
on the cost of labor relative to the
cost of capital, . The curve FF
shows the labor-capital ratio
choices in food production, while
the curve CC shows the corresponding
choices in cloth production.
At any given wage-rental ratio,
cloth production uses a higher
labor-capital ratio; when this is the
case, we say that cloth production
is labor-intensive and that food production
is capital-intensive.
w/r
86 PART ONE International Trade Theory
The CC and FF curves in Figure 5-5 are called relative factor demand curves; they are
very similar to the relative demand curve for goods. Their downward slope characterizes
the substitution effect in the producers’ factor demand. As the wage w rises relative to the
rental rate r, producers substitute capital for labor in their production decisions. The previous
case we considered with no factor substitution is a limiting case, where the relative
demand curve is a vertical line: The ratio of labor to capital demanded is fixed and does
not vary with changes in the wage-rental ratio w/r. In the remainder of this chapter, we
consider the more general case with factor substitution, where the relative factor demand
curves are downward sloping.
Factor Prices and Goods Prices
Suppose for a moment that the economy produces both cloth and food. (This need not be
the case if the economy engages in international trade, because it might specialize completely
in producing one good or the other; but let us temporarily ignore this possibility.)
Then competition among producers in each sector will ensure that the price of each good
equals its cost of production. The cost of producing a good depends on factor prices: If
wages rise, then other things equal to the price of any good whose production uses labor
will also rise.
The importance of a particular factor’s price to the cost of producing a good depends,
however, on how much of that factor the good’s production involves. If food production
makes use of very little labor, for example, then a rise in the wage will not have much
effect on the price of food, whereas if cloth production uses a great deal of labor, a rise in
the wage will have a large effect on the price. We can therefore conclude that there is a
one-to-one relationship between the ratio of the wage rate to the rental rate, , and the
ratio of the price of cloth to that of food, . This relationship is illustrated by the
upward-sloping curve SS in Figure 5-6.3
PC /PF
w/r
3This relationship holds only when the economy produces both cloth and food, which is associated with a given
range for the relative price of cloth. If the relative price rises beyond a given upper-bound level, then the economy
specializes in cloth production; conversely, if the relative price drops below a lower-bound level, then the
economy specializes in food production.
Relative price of
cloth, PC
/PF
SS
Wage-rental
ratio, w/r
Figure 5-6
Factor Prices and Goods Prices
Because cloth production is laborintensive
while food production is
capital-intensive, there is a
one-to-one relationship between
the factor price ratio and the
relative price of cloth ; the
higher the relative cost of labor,
the higher must be the relative
price of the labor-intensive good.
The relationship is illustrated by
the curve SS.
PC /PF
w/r
CHAPTER 5 Resources and Trade: The Heckscher-Ohlin Model 87
Let’s look at Figures 5-5 and 5-6 together. In Figure 5-7, the left panel is Figure 5-6
(of the SS curve) turned counterclockwise 90 degrees, while the right panel reproduces
Figure 5-5. By putting these two diagrams together, we see what may seem at first to be
a surprising linkage of the prices of goods to the ratio of labor to capital used in the
production of each good. Suppose that the relative price of cloth is (left panel
of Figure 5-7); if the economy produces both goods, the ratio of the wage rate to the
capital rental rate must equal . This ratio then implies that the ratios of labor to
capital employed in the production of cloth and food must be and ,
respectively (right panel of Figure 5-7). If the relative price of cloth were to rise to the
level indicated by , the ratio of the wage rate to the capital rental rate would
rise to . Because labor is now relatively more expensive, the ratios of labor to
capital employed in the production of cloth and food would therefore drop to
and .
We can learn one more important lesson from this diagram. The left panel already tells
us that an increase in the price of cloth relative to that of food will raise the income of
workers relative to that of capital owners. But it is possible to make a stronger statement:
Such a change in relative prices will unambiguously raise the purchasing power of workers
and lower the purchasing power of capital owners by raising real wages and lowering
real rents in terms of both goods.
(LF/KF)2
(LC/KC)2
(w/r)2
(PC/PF)2
(LC/KC)1 (LF/KF)1
(w/r)1
(PC/PF)1
Relative price
of cloth, PC /PF
CC
Wage-rental, w/r
Laborcapital
ratio,
Increasing Increasing L/K
(PC /PF
2 (PC /PF
1 (LF /KF )1 (LC /KC )2 (LC /KC ) 1
SS FF
(w/r)1
(w/r)2
) ) (LF /KF
)2
Figure 5-7
From Goods Prices to Input Choices
Given the relative price of cloth , the ratio of the wage rate to the capital rental rate must equal .
This wage-rental ratio then implies that the ratios of labor to capital employed in the production of cloth and
food must be and . If the relative price of cloth rises to , the wage-rental ratio must rise
to (w/r)2. This will cause the labor-capital ratio used in the production of both goods to drop.
(LC/KC)1 (LF /KF)1 (PC/PF)2
(PC/PF)1 (w/r)1
88 PART ONE International Trade Theory
How do we know this? When increases, the ratio of labor to capital falls in both
cloth and food production. But in a competitive economy, factors of production are paid
their marginal product—the real wage of workers in terms of cloth is equal to the marginal
productivity of labor in cloth production, and so on. When the ratio of labor to capital falls
in producing either good, the marginal product of labor in terms of that good increases—
so workers find their real wage higher in terms of both goods. On the other hand, the marginal
product of capital falls in both industries, so capital owners find their real incomes
lower in terms of both goods.
In this model, then, as in the specific factors model, changes in relative prices have
strong effects on income distribution. Not only does a change in the prices of goods
change the distribution of income; it always changes it so much that owners of one factor
of production gain while owners of the other are made worse off.4
Resources and Output
We can now complete the description of a two-factor economy by describing the relationship
between goods prices, factor supplies, and output. In particular, we investigate how
changes in resources (the total supply of a factor) affect the allocation of factors across
sectors and the associated changes in output produced.
Suppose that we take the relative price of cloth as given. We know from Figure 5-7 that a
given relative price of cloth, say , is associated with a fixed wage-rental ratio (so
long as both cloth and food are produced). That ratio, in turn, determines the ratios of labor to
capital employed in both the cloth and the food sectors: and , respectively.
Now we assume that the economy’s labor force grows, which implies that the economy’s
aggregate labor to capital ratio, , increases. At the given relative price of cloth , we
just saw that the ratios of labor to capital employed in both sectors remain constant. How can
the economy accommodate the increase in the aggregate relative supply of labor if the
relative labor demanded in each sector remains constant at and ? In other
words, how does the economy employ the additional labor hours? The answer lies in the
allocation of labor and capital across sectors: The labor-capital ratio in the cloth sector is higher
than that in the food sector, so the economy can increase the employment of labor to capital
(holding the labor-capital ratio fixed in each sector) by allocating more labor and capital to the
production of cloth (which is labor-intensive).5 As labor and capital move from the food sector
to the cloth sector, the economy produces more cloth and less food.
The best way to think about this result is in terms of how resources affect the economy’s
production possibilities. In Figure 5-8 the curve represents the economy’s
production possibilities before the increase in labor supply. Output is at point 1, where
the slope of the production possibility frontier equals minus the relative price of cloth,
, and the economy produces and of cloth and food. The curve shows
the production possibility frontier after an increase in the labor supply. The production
possibility frontier shifts out to After this increase, the economy can produce more
of both cloth and food than before. The outward shift of the frontier is, however, much
larger in the direction of cloth than of food—that is, there is a biased expansion of production
possibilities, which occurs when the production possibility frontier shifts out
much more in one direction than in the other. In this case, the expansion is so strongly
biased toward cloth production that at unchanged relative prices, production moves from
TT2
TT2 QF 1 QC 1 -PC/PF
TT1
(LC/KC)1 (LF/KF)1
L /K
L /K (PC/PF)1
(LC/KC)1 (LF/KF)1
(PC/PF)1 (w/r)1
PC/PF
5See the appendix for a more formal derivation of this result and additional details.
4This relationship between goods prices and factor prices (and the associated welfare effects) was clarified in a
classic paper by Wolfgang Stolper and Paul Samuelson, “Protection and Real Wages,” Review of Economic
Studies 9 (November 1941), pp. 58–73, and is therefore known as the Stolper-Samuelson effect.
CHAPTER 5 Resources and Trade: The Heckscher-Ohlin Model 89
point 1 to point 2, which involves an actual fall in food output from to and a large
increase in cloth output from to .
The biased effect of increases in resources on production possibilities is the key to understanding
how differences in resources give rise to international trade.6 An increase in the
supply of labor expands production possibilities disproportionately in the direction of cloth
production, while an increase in the supply of capital expands them disproportionately in the
direction of food production. Thus an economy with a high relative supply of labor to capital
will be relatively better at producing cloth than an economy with a low relative supply of
labor to capital. Generally, an economy will tend to be relatively effective at producing goods
that are intensive in the factors with which the country is relatively well endowed.
We will further see below that there is some strong empirical evidence confirming that
changes in a country’s resources lead to growth that is strongly biased toward the sectors
that intensively use the factor whose supply has increased. We document this for the
economies of Japan, South Korea, Taiwan, Hong Kong, and Singapore, which all experienced
very rapid growth in their supply of skilled labor over the last half-century.