The Specific Factors Model
The specific factors model was developed by Paul Samuelson and Ronald Jones.1 Like
the simple Ricardian model, it assumes an economy that produces two goods and that can
allocate its labor supply between the two sectors. Unlike the Ricardian model, however,
the specific factors model allows for the existence of factors of production besides labor.
Whereas labor is a mobile factor that can move between sectors, these other factors are
assumed to be specific. That is, they can be used only in the production of particular
goods.
1Paul Samuelson, “Ohlin Was Right,” Swedish Journal of Economics 73 (1971), pp. 365–384; and Ronald W.
Jones, “A Three-Factor Model in Theory, Trade, and History,” in Jagdish Bhagwati et al., eds., Trade, Balance of
Payments, and Growth (Amsterdam: North-Holland, 1971), pp. 3–21.
*See Bruce Fallick, “The Industrial Mobility of Displaced Workers,” Journal of Labor Economics 11 (April 1993), pp. 302–323.
†See Gueorgui Kambourov and Iourii Manovskii, “Occupational Specificity of Human Capital,” International Economic
Review 50 (February 2009), pp. 63–115.
Worker mobility varies greatly with the characteristics
of the worker (such as age) and the job
occupation (whether it requires general or jobspecific
skills). Nevertheless, one can measure an
average rate of mobility by looking at the duration
of unemployment following a worker’s displacement.
After four years, a displaced worker in the
United States has the same probability of being
employed as a similar worker who was not
displaced.* This four-year time-span compares with
a lifetime of 15 or 20 years for a typical specialized
machine, and 30 to 50 years for structures (a shopping
mall, office building, or production plant).
So labor is certainly a less specific factor than most
kinds of capital. However, even though most workers
can find new employment in other sectors
within a four-year time-span, switching occupations
entails additional costs: A displaced worker who is
re-employed in a different occupation suffers an
18 percent permanent drop in wages (on average).
This compares with a 6 percent drop if the worker
does not switch occupations.† Thus, labor is truly
flexible only before a worker has invested in any
occupation-specific skills.
52 PART ONE International Trade Theory
Assumptions of the Model
Imagine an economy that can produce two goods, cloth and food. Instead of one factor of
production, however, the country has three: labor (L), capital (K), and land (T for terrain).
Cloth is produced using capital and labor (but not land), while food is produced using land
and labor (but not capital). Labor is therefore a mobile factor that can be used in either sector,
while land and capital are both specific factors that can be used only in the production
of one good. Land can also be thought of as a different type of capital, one that is specific
to the food sector (see box below).
How much of each good does the economy produce? The economy’s output of cloth
depends on how much capital and labor are used in that sector. This relationship is summarized
by a production function that tells us the quantity of cloth that can be produced
given any input of capital and labor. The production function for cloth can be summarized
algebraically as
QC = QC (4-1) 1K,LC2,
In the model developed in this chapter, we assume
that there are two factors of production, land and capital,
that are permanently tied to particular sectors of
the economy. In advanced economies, however, agricultural
land receives only a small part of national
income. When economists apply the specific factors
model to economies like those of the United States or
France, they typically think of factor specificity not
as a permanent condition but as a matter of time. For
example, the vats used to brew beer and the stamping
presses used to build auto bodies cannot be substituted
for each other, and so these different kinds of
equipment are industry-specific. Given time, however,
it would be possible to redirect investment from
auto factories to breweries or vice versa. As a result,
in a long-term sense both vats and stamping presses
can be considered to be two manifestations of a single,
mobile factor called capital.
In practice, then, the distinction between specific
and mobile factors is not a sharp line. Rather, it is a
question of the speed of adjustment, with factors
being more specific the longer it takes to redeploy
them between industries. So how specific are the
factors of production in the real economy?
What Is a Specific Factor?
2Diminishing returns to a single factor does not imply diminishing returns to scale when all factors of production
are adjusted. Thus, diminishing returns to labor is entirely consistent with constant returns to scale in both labor
and capital.
where is the economy’s output of cloth, K is the economy’s capital stock, and is the
labor force employed in cloth. Similarly, for food we can write the production function
(4-2)
where is the economy’s output of food, T is the economy’s supply of land, and
is the labor force devoted to food production. For the economy as a whole, the labor
employed must equal the total labor supply L:
(4-3)
Production Possibilities
The specific factors model assumes that each of the specific factors, capital and land, can
be used in only one sector, cloth and food, respectively. Only labor can be used in either
sector. Thus to analyze the economy’s production possibilities, we need only to ask how
the economy’s mix of output changes as labor is shifted from one sector to the other. This
can be done graphically, first by representing the production functions (4-1) and (4-2), and
then by putting them together to derive the production possibility frontier.
Figure 4-1 illustrates the relationship between labor input and output of cloth. The
larger the input of labor, for a given capital supply, the larger will be output. In Figure 4-1,
the slope of represents the marginal product of labor, that is, the addition to
output generated by adding one more person-hour. However, if labor input is increased
without increasing capital as well, there will normally be diminishing returns: Because
adding a worker means that each worker has less capital to work with, each successive
increment of labor will add less to production than the last. Diminishing returns are
reflected in the shape of the production function: gets flatter as we move to
the right, indicating that the marginal product of labor declines as more labor is used.
Figure 4-2 shows the same information a different way. In this figure we directly plot the
marginal product of labor as a function of the labor employed. (In the appendix to this
chapter, we show that the area under the marginal product curve represents the total output
of cloth.)
A similar pair of diagrams can represent the production function for food. These diagrams
can then be combined to derive the production possibility frontier for the economy,
as illustrated in Figure 4-3. As we saw in Chapter 3, the production possibility frontier
shows what the economy is capable of producing; in this case it shows how much food it
can produce for any given output of cloth and vice versa.
Figure 4-3 is a four-quadrant diagram. In the lower right quadrant we show the production
function for cloth illustrated in Figure 4-1. This time, however, we turn the figure on
its side: A movement downward along the vertical axis represents an increase in the labor
input to the cloth sector, while a movement to the right along the horizontal axis represents
an increase in the output of cloth. In the upper left quadrant we show the corresponding
production function for food; this part of the figure is also flipped around, so that a movement
to the left along the horizontal axis indicates an increase in labor input to the food
sector, while an upward movement along the vertical axis indicates an increase in food
output.
The lower left quadrant represents the economy’s allocation of labor. Both quantities
are measured in the reverse of the usual direction. A downward movement along
the vertical axis indicates an increase in the labor employed in cloth; a leftward movement
along the horizontal axis indicates an increase in labor employed in food. Since
an increase in employment in one sector must mean that less labor is available for the
other, the possible allocations are indicated by a downward-sloping line. This line,
labeled AA, slopes downward at a 45-degree angle, that is, it has a slope of . To see
why this line represents the possible labor allocations, notice that if all labor were
employed in food production, would equal L, while would equal 0. If one were
then to move labor gradually into the cloth sector, each person-hour moved would
increase LC by one unit while reducing LF by one unit, tracing a line with a slope
LF LC
Production of cloth and food is determined by the allocation of labor. In the lower left quadrant, the allocation of
labor between sectors can be illustrated by a point on line AA, which represents all combinations of labor input to
cloth and food that sum up to the total labor supply L. Corresponding to any particular point on AA, such as point 2,
is a labor input to cloth and a labor input to food . The curves in the lower right and upper left quadrants
represent the production functions for cloth and food, respectively; these allow determination of output
given labor input. Then in the upper right quadrant, the curve PP shows how the output of the two goods varies as
the allocation of labor is shifted from food to cloth, with the output points 1¿, 2¿, 3¿ corresponding to the labor
allocations 1, 2, 3. Because of diminishing returns, PP is a bowed-out curve instead of a straight line.
1QC 2 , QF 2 2
1LF 2 2 1LC 2 2
of , until the entire labor supply L is employed in the cloth sector. Any particular
allocation of labor between the two sectors can then be represented by a point on AA,
such as point 2.
We can now see how to determine production given any particular allocation of labor
between the two sectors. Suppose that the allocation of labor were represented by point 2
in the lower left quadrant, that is, with hours in cloth and hours in food. Then we
can use the production function for each sector to determine output: units of cloth,
units of food. Using coordinates , point 2¿ in the upper right quadrant of Figure 4-3
shows the resulting outputs of cloth and food.
To trace the whole production possibility frontier, we simply imagine repeating this
exercise for many alternative allocations of labor. We might start with most of the labor
allocated to food production, as at point 1 in the lower left quadrant, then gradually
increase the amount of labor used in cloth until very few workers are employed in food, as
at point 3; the corresponding points in the upper right quadrant will trace out the curve
running from 1¿ to 3¿. Thus PP in the upper right quadrant shows the economy’s production
possibilities for given supplies of land, labor, and capital.
In the Ricardian model, where labor is the only factor of production, the production
possibility frontier is a straight line because the opportunity cost of cloth in terms of food
is constant. In the specific factors model, however, the addition of other factors of production
changes the shape of the production possibility frontier PP to a curve. The curvature
of PP reflects diminishing returns to labor in each sector; these diminishing returns are the
crucial difference between the specific factors and the Ricardian models.
Notice that when tracing PP we shift labor from the food to the cloth sector. If we
shift one person-hour of labor from food to cloth, however, this extra input will
increase output in that sector by the marginal product of labor in cloth, . To
increase cloth output by one unit, then, we must increase labor input by hours.
Meanwhile, each unit of labor input shifted out of food production will lower output in
that sector by the marginal product of labor in food, . To increase output of cloth
by one unit, then, the economy must reduce output of food by units. The
slope of PP, which measures the opportunity cost of cloth in terms of food—that is, the
number of units of food output that must be sacrificed to increase cloth output by
one unit—is therefore
We can now see why PP has the bowed shape it does. As we move from l¿ to 3¿, rises
and falls. We saw in Figure 4-2, however, that as rises, the marginal product of labor
in cloth falls; correspondingly, as falls, the marginal product of labor in food rises. As
more and more labor is moved to the cloth sector, each additional unit of labor becomes
less valuable in the cloth sector and more valuable in the food sector: The opportunity cost
(foregone food production) of each additional cloth unit rises, and PP thus gets steeper as
we move down it to the right.
We have now shown how output is determined, given the allocation of labor. The next
step is to ask how a market economy determines what the allocation of labor should be.
Prices, Wages, and Labor Allocation
How much labor will be employed in each sector? To answer this we need to look at supply
and demand in the labor market. The demand for labor in each sector depends on the
price of output and the wage rate. In turn, the wage rate depends on the combined demand
for labor by food and cloth producers. Given the prices of cloth and food together with the
wage rate, we can determine each sector’s employment and output.
First, let us focus on the demand for labor. In each sector, profit-maximizing employers
will demand labor up to the point where the value produced by an additional person-hour
equals the cost of employing that hour. In the cloth sector, for example, the value of an
additional person-hour is the marginal product of labor in cloth multiplied by the price of
one unit of cloth: If w is the wage rate of labor, employers will therefore hire
workers up to the point where
But the marginal product of labor in cloth, already illustrated in Figure 4-2, slopes
downward because of diminishing returns. So for any given price of cloth , the value
of that marginal product, will also slope down. We can therefore think of
equation (4-4) as defining the demand curve for labor in the cloth sector: If the wage
rate falls, other things equal, employers in the cloth sector will want to hire more
workers.
Similarly, the value of an additional person-hour in food is . The demand
curve for labor in the food sector may therefore be written
(4-5)
The wage rate w must be the same in both sectors, because of the assumption that labor
is freely mobile between sectors. That is, because labor is a mobile factor, it will move
from the low-wage sector to the high-wage sector until wages are equalized. The wage
rate, in turn, is determined by the requirement that total labor demand (total employment)
equals total labor supply. This equilibrium condition for labor is represented in
equation (4-3).
By representing these two labor demand curves in a diagram (Figure 4-4), we can see
how the wage rate and employment in each sector are determined given the prices of food
and cloth. Along the horizontal axis of Figure 4-4 we show the total labor supply L.
Measuring from the left of the diagram, we show the value of the marginal product of
labor in cloth, which is simply the curve from Figure 4-2 multiplied by . This is
the demand curve for labor in the cloth sector. Measuring from the right, we show the
value of the marginal product of labor in food, which is the demand for labor in food. The
equilibrium wage rate and allocation of labor between the two sectors is represented by
point 1. At the wage rate , the sum of labor demanded in the cloth and food
sectors just equals the total labor supply L.
There is a useful relationship between relative prices and output that emerges clearly
from this analysis of labor allocation; this relationship applies to more general situations
than that described by the specific factors model. Equations (4-4) and (4-5) imply that
or, rearranging, that
(4-6)
The left side of equation (4-6) is the slope of the production possibility frontier at the
actual production point; the right side is minus the relative price of cloth. This result tells us
that at the production point, the production possibility frontier must be tangent to a line
whose slope is minus the price of cloth divided by that of food. As we will see in the following
chapters, this is a very general result that characterizes production responses to changes
in relative prices along a production possibility frontier. It is illustrated in Figure 4-5: If the
relative price of cloth is , the economy produces at point 1.
What happens to the allocation of labor and the distribution of income when the prices of
food and cloth change? Notice that any price change can be broken into two parts: an equalproportional
change in both and , and a change in only one price. For example, suppose
that the price of cloth rises 17 percent and the price of food rises 10 percent. We can analyze the
effects of this by first asking what happens if cloth and food prices both rise by 10 percent, and
then by finding out what happens if only cloth prices rise by 7 percent. This allows us to separate
the effect of changes in the overall price level from the effect of changes in relative prices.
An Equal-Proportional Change in Prices Figure 4-6 shows the effect of an equalproportional
increase in and . rises from to ; rises from to . If the
prices of both goods increase by 10 percent, the labor demand curves will both shift up by
10 percent as well. As you can see from the diagram, these shifts lead to a 10 percent
increase in the wage rate from (point 1) to (point 2). However, the allocation of
labor between the sectors and the outputs of the two goods does not change.
In fact, when and change in the same proportion, no real changes occur. The
wage rate rises in the same proportion as the prices, so real wage rates, the ratios of the
wage rate to the prices of goods, are unaffected. With the same amount of labor employed
in each sector, receiving the same real wage rate, the real incomes of capital owners and
landowners also remain the same. So everyone is in exactly the same position as before.
This illustrates a general principle: Changes in the overall price level have no real effects,
that is, do not change any physical quantities in the economy. Only changes in relative
prices—which in this case means the price of cloth relative to the price of food, —
affect welfare or the allocation of resources.
A Change in Relative Prices Consider the effect of a price change that does affect
relative prices. Figure 4-7 shows the effect of a change in the price of only one good, in
this case a 7 percent rise in from to . The increase in shifts the cloth labor
demand curve in the same proportion as the price increase and shifts the equilibrium
from point 1 to point 2. Notice two important facts about the results of this shift. First,
although the wage rate rises, it rises by less than the increase in the price of cloth. If
wages had risen in the same proportion as the price of cloth (7 percent increase), then
wages would have risen from to . Instead, wages rise by a smaller proportion,
from to .
Second, when only rises, in contrast to a simultaneous rise in and , labor shifts
from the food sector to the cloth sector and the output of cloth rises while that of food
falls. (This is why w does not rise as much as : Because cloth employment rises, the
marginal product of labor in that sector falls.)
of cloth can also be seen directly by looking at
the production possibility curve. In Figure 4-8, we show the effects of the same rise in the
price of cloth, which raises the relative price of cloth from to . The production
point, which is always located where the slope of PP equals minus the relative
price, shifts from 1 to 2. Food output falls and cloth output rises as a result of the rise in the
relative price of cloth.
Since higher relative prices of cloth lead to a higher output of cloth relative to that of
food, we can draw a relative supply curve showing as a function of . This relative
supply curve is shown as RS in Figure 4-9. As we showed in Chapter 3, we can also
draw a relative demand curve, which is illustrated by the downward-sloping line RD. In
the absence of international trade, the equilibrium relative price and output
are determined by the intersection of relative supply and demand.
Relative Prices and the Distribution of Income
So far we have examined the following aspects of the specific factors model: (1) the determination
of production possibilities given an economy’s resources and technology and
(2) the determination of resource allocation, production, and relative prices in a market
economy. Before turning to the effects of international trade, we must consider the effect
of changes in relative prices on the distribution of income.
Look again at Figure 4-7, which shows the effect of a rise in the price of cloth. We have
already noted that the demand curve for labor in the cloth sector will shift upward in proportion
to the rise in , so that if rises by 7 percent, the curve defined by
also rises by 7 percent. We have also seen that unless the price of food also rises by at least
A Rise in the Price of Cloth
The cloth labor demand curve rises in proportion to the 7 percent increase in , but the wage rate
rises less than proportionately. Labor moves from the food sector to the cloth sector. Output of cloth
rises; output of food falls.
7 percent, w will rise by less than . Thus, if only cloth prices rise by 7 percent, we would
expect the wage rate to rise by only, say, 3 percent.
Let’s look at what this outcome implies for the incomes of three groups: workers, owners
of capital, and owners of land. Workers find that their wage rate has risen, but less than
in proportion to the rise in . Thus their real wage in terms of cloth (the amount of cloth
they can buy with their wage income), , falls, while their real wage in terms of food,
, rises. Given this information, we cannot say whether workers are better or worse off;
this depends on the relative importance of cloth and food in workers’ consumption (determined
by the workers’ preferences), a question that we will not pursue further.
Owners of capital, however, are definitely better off. The real wage rate in terms of cloth
has fallen, so the profits of capital owners in terms of what they produce (cloth) rises. That
is, the income of capital owners will rise more than proportionately with the rise in .
Since PC in turn rises relative to PF, the income of capitalists clearly goes up in terms of
both goods. Conversely, landowners are definitely worse off. They lose for two reasons:
The real wage in terms of food (the good they produce) rises, squeezing their income, and
the rise in cloth price reduces the purchasing power of any given income. The chapter
appendix describes the welfare changes of capitalists and landowners in further detail.
If the relative price had moved in the opposite direction and the relative price of cloth
had decreased, then the predictions would be reversed: Capital owners would be worse
off, and landowners would be better off. The change in the welfare of workers would again
be ambiguous because their real wage in terms of cloth would rise, but their real wage in
terms of food would fall. The effect of a relative price change on the distribution of
income can be summarized as follows:
• The factor specific to the sector whose relative price increases is definitely better off.
• The factor specific to the sector whose relative price decreases is definitely worse off.
• The change in welfare for the mobile factor is ambiguous.
The specific factors model was developed by Paul Samuelson and Ronald Jones.1 Like
the simple Ricardian model, it assumes an economy that produces two goods and that can
allocate its labor supply between the two sectors. Unlike the Ricardian model, however,
the specific factors model allows for the existence of factors of production besides labor.
Whereas labor is a mobile factor that can move between sectors, these other factors are
assumed to be specific. That is, they can be used only in the production of particular
goods.
1Paul Samuelson, “Ohlin Was Right,” Swedish Journal of Economics 73 (1971), pp. 365–384; and Ronald W.
Jones, “A Three-Factor Model in Theory, Trade, and History,” in Jagdish Bhagwati et al., eds., Trade, Balance of
Payments, and Growth (Amsterdam: North-Holland, 1971), pp. 3–21.
*See Bruce Fallick, “The Industrial Mobility of Displaced Workers,” Journal of Labor Economics 11 (April 1993), pp. 302–323.
†See Gueorgui Kambourov and Iourii Manovskii, “Occupational Specificity of Human Capital,” International Economic
Review 50 (February 2009), pp. 63–115.
Worker mobility varies greatly with the characteristics
of the worker (such as age) and the job
occupation (whether it requires general or jobspecific
skills). Nevertheless, one can measure an
average rate of mobility by looking at the duration
of unemployment following a worker’s displacement.
After four years, a displaced worker in the
United States has the same probability of being
employed as a similar worker who was not
displaced.* This four-year time-span compares with
a lifetime of 15 or 20 years for a typical specialized
machine, and 30 to 50 years for structures (a shopping
mall, office building, or production plant).
So labor is certainly a less specific factor than most
kinds of capital. However, even though most workers
can find new employment in other sectors
within a four-year time-span, switching occupations
entails additional costs: A displaced worker who is
re-employed in a different occupation suffers an
18 percent permanent drop in wages (on average).
This compares with a 6 percent drop if the worker
does not switch occupations.† Thus, labor is truly
flexible only before a worker has invested in any
occupation-specific skills.
52 PART ONE International Trade Theory
Assumptions of the Model
Imagine an economy that can produce two goods, cloth and food. Instead of one factor of
production, however, the country has three: labor (L), capital (K), and land (T for terrain).
Cloth is produced using capital and labor (but not land), while food is produced using land
and labor (but not capital). Labor is therefore a mobile factor that can be used in either sector,
while land and capital are both specific factors that can be used only in the production
of one good. Land can also be thought of as a different type of capital, one that is specific
to the food sector (see box below).
How much of each good does the economy produce? The economy’s output of cloth
depends on how much capital and labor are used in that sector. This relationship is summarized
by a production function that tells us the quantity of cloth that can be produced
given any input of capital and labor. The production function for cloth can be summarized
algebraically as
QC = QC (4-1) 1K,LC2,
In the model developed in this chapter, we assume
that there are two factors of production, land and capital,
that are permanently tied to particular sectors of
the economy. In advanced economies, however, agricultural
land receives only a small part of national
income. When economists apply the specific factors
model to economies like those of the United States or
France, they typically think of factor specificity not
as a permanent condition but as a matter of time. For
example, the vats used to brew beer and the stamping
presses used to build auto bodies cannot be substituted
for each other, and so these different kinds of
equipment are industry-specific. Given time, however,
it would be possible to redirect investment from
auto factories to breweries or vice versa. As a result,
in a long-term sense both vats and stamping presses
can be considered to be two manifestations of a single,
mobile factor called capital.
In practice, then, the distinction between specific
and mobile factors is not a sharp line. Rather, it is a
question of the speed of adjustment, with factors
being more specific the longer it takes to redeploy
them between industries. So how specific are the
factors of production in the real economy?
What Is a Specific Factor?
2Diminishing returns to a single factor does not imply diminishing returns to scale when all factors of production
are adjusted. Thus, diminishing returns to labor is entirely consistent with constant returns to scale in both labor
and capital.
where is the economy’s output of cloth, K is the economy’s capital stock, and is the
labor force employed in cloth. Similarly, for food we can write the production function
(4-2)
where is the economy’s output of food, T is the economy’s supply of land, and
is the labor force devoted to food production. For the economy as a whole, the labor
employed must equal the total labor supply L:
(4-3)
Production Possibilities
The specific factors model assumes that each of the specific factors, capital and land, can
be used in only one sector, cloth and food, respectively. Only labor can be used in either
sector. Thus to analyze the economy’s production possibilities, we need only to ask how
the economy’s mix of output changes as labor is shifted from one sector to the other. This
can be done graphically, first by representing the production functions (4-1) and (4-2), and
then by putting them together to derive the production possibility frontier.
Figure 4-1 illustrates the relationship between labor input and output of cloth. The
larger the input of labor, for a given capital supply, the larger will be output. In Figure 4-1,
the slope of represents the marginal product of labor, that is, the addition to
output generated by adding one more person-hour. However, if labor input is increased
without increasing capital as well, there will normally be diminishing returns: Because
adding a worker means that each worker has less capital to work with, each successive
increment of labor will add less to production than the last. Diminishing returns are
reflected in the shape of the production function: gets flatter as we move to
the right, indicating that the marginal product of labor declines as more labor is used.
Figure 4-2 shows the same information a different way. In this figure we directly plot the
marginal product of labor as a function of the labor employed. (In the appendix to this
chapter, we show that the area under the marginal product curve represents the total output
of cloth.)
A similar pair of diagrams can represent the production function for food. These diagrams
can then be combined to derive the production possibility frontier for the economy,
as illustrated in Figure 4-3. As we saw in Chapter 3, the production possibility frontier
shows what the economy is capable of producing; in this case it shows how much food it
can produce for any given output of cloth and vice versa.
Figure 4-3 is a four-quadrant diagram. In the lower right quadrant we show the production
function for cloth illustrated in Figure 4-1. This time, however, we turn the figure on
its side: A movement downward along the vertical axis represents an increase in the labor
input to the cloth sector, while a movement to the right along the horizontal axis represents
an increase in the output of cloth. In the upper left quadrant we show the corresponding
production function for food; this part of the figure is also flipped around, so that a movement
to the left along the horizontal axis indicates an increase in labor input to the food
sector, while an upward movement along the vertical axis indicates an increase in food
output.
The lower left quadrant represents the economy’s allocation of labor. Both quantities
are measured in the reverse of the usual direction. A downward movement along
the vertical axis indicates an increase in the labor employed in cloth; a leftward movement
along the horizontal axis indicates an increase in labor employed in food. Since
an increase in employment in one sector must mean that less labor is available for the
other, the possible allocations are indicated by a downward-sloping line. This line,
labeled AA, slopes downward at a 45-degree angle, that is, it has a slope of . To see
why this line represents the possible labor allocations, notice that if all labor were
employed in food production, would equal L, while would equal 0. If one were
then to move labor gradually into the cloth sector, each person-hour moved would
increase LC by one unit while reducing LF by one unit, tracing a line with a slope
LF LC
Production of cloth and food is determined by the allocation of labor. In the lower left quadrant, the allocation of
labor between sectors can be illustrated by a point on line AA, which represents all combinations of labor input to
cloth and food that sum up to the total labor supply L. Corresponding to any particular point on AA, such as point 2,
is a labor input to cloth and a labor input to food . The curves in the lower right and upper left quadrants
represent the production functions for cloth and food, respectively; these allow determination of output
given labor input. Then in the upper right quadrant, the curve PP shows how the output of the two goods varies as
the allocation of labor is shifted from food to cloth, with the output points 1¿, 2¿, 3¿ corresponding to the labor
allocations 1, 2, 3. Because of diminishing returns, PP is a bowed-out curve instead of a straight line.
1QC 2 , QF 2 2
1LF 2 2 1LC 2 2
of , until the entire labor supply L is employed in the cloth sector. Any particular
allocation of labor between the two sectors can then be represented by a point on AA,
such as point 2.
We can now see how to determine production given any particular allocation of labor
between the two sectors. Suppose that the allocation of labor were represented by point 2
in the lower left quadrant, that is, with hours in cloth and hours in food. Then we
can use the production function for each sector to determine output: units of cloth,
units of food. Using coordinates , point 2¿ in the upper right quadrant of Figure 4-3
shows the resulting outputs of cloth and food.
To trace the whole production possibility frontier, we simply imagine repeating this
exercise for many alternative allocations of labor. We might start with most of the labor
allocated to food production, as at point 1 in the lower left quadrant, then gradually
increase the amount of labor used in cloth until very few workers are employed in food, as
at point 3; the corresponding points in the upper right quadrant will trace out the curve
running from 1¿ to 3¿. Thus PP in the upper right quadrant shows the economy’s production
possibilities for given supplies of land, labor, and capital.
In the Ricardian model, where labor is the only factor of production, the production
possibility frontier is a straight line because the opportunity cost of cloth in terms of food
is constant. In the specific factors model, however, the addition of other factors of production
changes the shape of the production possibility frontier PP to a curve. The curvature
of PP reflects diminishing returns to labor in each sector; these diminishing returns are the
crucial difference between the specific factors and the Ricardian models.
Notice that when tracing PP we shift labor from the food to the cloth sector. If we
shift one person-hour of labor from food to cloth, however, this extra input will
increase output in that sector by the marginal product of labor in cloth, . To
increase cloth output by one unit, then, we must increase labor input by hours.
Meanwhile, each unit of labor input shifted out of food production will lower output in
that sector by the marginal product of labor in food, . To increase output of cloth
by one unit, then, the economy must reduce output of food by units. The
slope of PP, which measures the opportunity cost of cloth in terms of food—that is, the
number of units of food output that must be sacrificed to increase cloth output by
one unit—is therefore
We can now see why PP has the bowed shape it does. As we move from l¿ to 3¿, rises
and falls. We saw in Figure 4-2, however, that as rises, the marginal product of labor
in cloth falls; correspondingly, as falls, the marginal product of labor in food rises. As
more and more labor is moved to the cloth sector, each additional unit of labor becomes
less valuable in the cloth sector and more valuable in the food sector: The opportunity cost
(foregone food production) of each additional cloth unit rises, and PP thus gets steeper as
we move down it to the right.
We have now shown how output is determined, given the allocation of labor. The next
step is to ask how a market economy determines what the allocation of labor should be.
Prices, Wages, and Labor Allocation
How much labor will be employed in each sector? To answer this we need to look at supply
and demand in the labor market. The demand for labor in each sector depends on the
price of output and the wage rate. In turn, the wage rate depends on the combined demand
for labor by food and cloth producers. Given the prices of cloth and food together with the
wage rate, we can determine each sector’s employment and output.
First, let us focus on the demand for labor. In each sector, profit-maximizing employers
will demand labor up to the point where the value produced by an additional person-hour
equals the cost of employing that hour. In the cloth sector, for example, the value of an
additional person-hour is the marginal product of labor in cloth multiplied by the price of
one unit of cloth: If w is the wage rate of labor, employers will therefore hire
workers up to the point where
But the marginal product of labor in cloth, already illustrated in Figure 4-2, slopes
downward because of diminishing returns. So for any given price of cloth , the value
of that marginal product, will also slope down. We can therefore think of
equation (4-4) as defining the demand curve for labor in the cloth sector: If the wage
rate falls, other things equal, employers in the cloth sector will want to hire more
workers.
Similarly, the value of an additional person-hour in food is . The demand
curve for labor in the food sector may therefore be written
(4-5)
The wage rate w must be the same in both sectors, because of the assumption that labor
is freely mobile between sectors. That is, because labor is a mobile factor, it will move
from the low-wage sector to the high-wage sector until wages are equalized. The wage
rate, in turn, is determined by the requirement that total labor demand (total employment)
equals total labor supply. This equilibrium condition for labor is represented in
equation (4-3).
By representing these two labor demand curves in a diagram (Figure 4-4), we can see
how the wage rate and employment in each sector are determined given the prices of food
and cloth. Along the horizontal axis of Figure 4-4 we show the total labor supply L.
Measuring from the left of the diagram, we show the value of the marginal product of
labor in cloth, which is simply the curve from Figure 4-2 multiplied by . This is
the demand curve for labor in the cloth sector. Measuring from the right, we show the
value of the marginal product of labor in food, which is the demand for labor in food. The
equilibrium wage rate and allocation of labor between the two sectors is represented by
point 1. At the wage rate , the sum of labor demanded in the cloth and food
sectors just equals the total labor supply L.
There is a useful relationship between relative prices and output that emerges clearly
from this analysis of labor allocation; this relationship applies to more general situations
than that described by the specific factors model. Equations (4-4) and (4-5) imply that
or, rearranging, that
(4-6)
The left side of equation (4-6) is the slope of the production possibility frontier at the
actual production point; the right side is minus the relative price of cloth. This result tells us
that at the production point, the production possibility frontier must be tangent to a line
whose slope is minus the price of cloth divided by that of food. As we will see in the following
chapters, this is a very general result that characterizes production responses to changes
in relative prices along a production possibility frontier. It is illustrated in Figure 4-5: If the
relative price of cloth is , the economy produces at point 1.
What happens to the allocation of labor and the distribution of income when the prices of
food and cloth change? Notice that any price change can be broken into two parts: an equalproportional
change in both and , and a change in only one price. For example, suppose
that the price of cloth rises 17 percent and the price of food rises 10 percent. We can analyze the
effects of this by first asking what happens if cloth and food prices both rise by 10 percent, and
then by finding out what happens if only cloth prices rise by 7 percent. This allows us to separate
the effect of changes in the overall price level from the effect of changes in relative prices.
An Equal-Proportional Change in Prices Figure 4-6 shows the effect of an equalproportional
increase in and . rises from to ; rises from to . If the
prices of both goods increase by 10 percent, the labor demand curves will both shift up by
10 percent as well. As you can see from the diagram, these shifts lead to a 10 percent
increase in the wage rate from (point 1) to (point 2). However, the allocation of
labor between the sectors and the outputs of the two goods does not change.
In fact, when and change in the same proportion, no real changes occur. The
wage rate rises in the same proportion as the prices, so real wage rates, the ratios of the
wage rate to the prices of goods, are unaffected. With the same amount of labor employed
in each sector, receiving the same real wage rate, the real incomes of capital owners and
landowners also remain the same. So everyone is in exactly the same position as before.
This illustrates a general principle: Changes in the overall price level have no real effects,
that is, do not change any physical quantities in the economy. Only changes in relative
prices—which in this case means the price of cloth relative to the price of food, —
affect welfare or the allocation of resources.
A Change in Relative Prices Consider the effect of a price change that does affect
relative prices. Figure 4-7 shows the effect of a change in the price of only one good, in
this case a 7 percent rise in from to . The increase in shifts the cloth labor
demand curve in the same proportion as the price increase and shifts the equilibrium
from point 1 to point 2. Notice two important facts about the results of this shift. First,
although the wage rate rises, it rises by less than the increase in the price of cloth. If
wages had risen in the same proportion as the price of cloth (7 percent increase), then
wages would have risen from to . Instead, wages rise by a smaller proportion,
from to .
Second, when only rises, in contrast to a simultaneous rise in and , labor shifts
from the food sector to the cloth sector and the output of cloth rises while that of food
falls. (This is why w does not rise as much as : Because cloth employment rises, the
marginal product of labor in that sector falls.)
of cloth can also be seen directly by looking at
the production possibility curve. In Figure 4-8, we show the effects of the same rise in the
price of cloth, which raises the relative price of cloth from to . The production
point, which is always located where the slope of PP equals minus the relative
price, shifts from 1 to 2. Food output falls and cloth output rises as a result of the rise in the
relative price of cloth.
Since higher relative prices of cloth lead to a higher output of cloth relative to that of
food, we can draw a relative supply curve showing as a function of . This relative
supply curve is shown as RS in Figure 4-9. As we showed in Chapter 3, we can also
draw a relative demand curve, which is illustrated by the downward-sloping line RD. In
the absence of international trade, the equilibrium relative price and output
are determined by the intersection of relative supply and demand.
Relative Prices and the Distribution of Income
So far we have examined the following aspects of the specific factors model: (1) the determination
of production possibilities given an economy’s resources and technology and
(2) the determination of resource allocation, production, and relative prices in a market
economy. Before turning to the effects of international trade, we must consider the effect
of changes in relative prices on the distribution of income.
Look again at Figure 4-7, which shows the effect of a rise in the price of cloth. We have
already noted that the demand curve for labor in the cloth sector will shift upward in proportion
to the rise in , so that if rises by 7 percent, the curve defined by
also rises by 7 percent. We have also seen that unless the price of food also rises by at least
A Rise in the Price of Cloth
The cloth labor demand curve rises in proportion to the 7 percent increase in , but the wage rate
rises less than proportionately. Labor moves from the food sector to the cloth sector. Output of cloth
rises; output of food falls.
7 percent, w will rise by less than . Thus, if only cloth prices rise by 7 percent, we would
expect the wage rate to rise by only, say, 3 percent.
Let’s look at what this outcome implies for the incomes of three groups: workers, owners
of capital, and owners of land. Workers find that their wage rate has risen, but less than
in proportion to the rise in . Thus their real wage in terms of cloth (the amount of cloth
they can buy with their wage income), , falls, while their real wage in terms of food,
, rises. Given this information, we cannot say whether workers are better or worse off;
this depends on the relative importance of cloth and food in workers’ consumption (determined
by the workers’ preferences), a question that we will not pursue further.
Owners of capital, however, are definitely better off. The real wage rate in terms of cloth
has fallen, so the profits of capital owners in terms of what they produce (cloth) rises. That
is, the income of capital owners will rise more than proportionately with the rise in .
Since PC in turn rises relative to PF, the income of capitalists clearly goes up in terms of
both goods. Conversely, landowners are definitely worse off. They lose for two reasons:
The real wage in terms of food (the good they produce) rises, squeezing their income, and
the rise in cloth price reduces the purchasing power of any given income. The chapter
appendix describes the welfare changes of capitalists and landowners in further detail.
If the relative price had moved in the opposite direction and the relative price of cloth
had decreased, then the predictions would be reversed: Capital owners would be worse
off, and landowners would be better off. The change in the welfare of workers would again
be ambiguous because their real wage in terms of cloth would rise, but their real wage in
terms of food would fall. The effect of a relative price change on the distribution of
income can be summarized as follows:
• The factor specific to the sector whose relative price increases is definitely better off.
• The factor specific to the sector whose relative price decreases is definitely worse off.
• The change in welfare for the mobile factor is ambiguous.
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