Production Possibilities Frontiers (PPF)
A production possibilities frontier (PPF) represents the set of all combinations of the
quantities of goods a country can produce simultaneously. In Chapter 3, all PPF diagrams are
depicted as concave with respect to the origin. But what is the rationale for this? Is the PPF
always shaped this way?
While the concavity assumption is natural and suitable under a wide range of circumstances,
the PPF can take other forms as well. The shape of the PPF depends crucially on the number
of production factors, substitutability between these factors, and the relationship between
factor inputs and output. This note discusses why this is the case by considering several
prominent examples. Throughout this note we will assume that two countries, Japan and
Vietnam, produce two goods, rice and motorcycles, as in the textbook.
Case 1: A Single Factor of Production, Constant Returns to Scale
In both Japan and Vietnam, each good is produced with a single factor of production, labor.
The amount of a production factor required to produce one unit of output is called the input
coefficient. When the production factor is labor, this value may also be called the labor
coefficient.
When the input coefficient of a particular firm or industry is invariant to the scale of
production, this firm or industry is said to be subject to constant returns to scale (CRS). In a
CRS firm or industry, doubling factor inputs always double the amount of output. Table 1
provides the general notations and numerical examples of labor coefficients, assuming that
both industries are subject to CRS.
When the coefficient of one industry differs between Vietnam and Japan, the two countries
must be producing the same good with different technologies. According to our numerical
examples, the labor coefficient of the motorcycle industry is larger in Vietnam than in Japan.
This implies that the production technique of the Japanese motorcycle industry is superior to
that of the Vietnamese motorcycle industry.
Using the numerical examples in Table 1, draw the PPF for each country in Figure 1. What
does the slope of the PPF represent? What kind of trade is beneficial to both countries?
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Table 1
Input Coefficients: Case 1
General specification
Vietnam
Japan
Example
Rice
Motorcycles
Rice
Motorcycles
a VR
a VM
2
4
J
J
2
2
a
a
R
M
Labor endowment
Example
LV
100
J
100
Vietnam
Japan
L
Figure 1
Production Possibilities Frontiers: Case 1
Vietnam
Japan
QM
QM
QR
QR
Case 2: Two Factors of Production, Constant Returns to Scale
Let us next consider a situation in which both rice and motorcycles are produced with two
factors of production, labor and capital. For simplicity, we assume that the input coefficients
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differ between the two industries but are the same in both countries. This assumption implies
that the two countries use the same production techniques.
Table 2 shows the input coefficients and factor endowments in the two industries, together
with their numerical examples. Since the input coefficients are all constants, doubling both
labor and capital simultaneously will double output. This case is another example of CRS.
In Figure 2, draw the PPFs that correspond to the numerical examples in Table 2. You will
notice that not all production factors are used even when production is conducted on the PPF.
This is because the fixed input coefficients imply no substitutability between labor and capital.
Is this assumption realistic? When labor and capital are imperfect substitutes―that is, when
one factor can stand in for the other to a certain limited extent―how will the shape of the PPF
change?
Table 2
Input-Output Coefficients: Case 2
General specification
Rice
Motorcycles
Example
Labor
Capital
Labor
Capital
aR
aM
bR
bM
4
2
2
4
Factor endowment
Vietnam
Japan
Example
Labor
Capital
Labor
Capital
LV
LJ
KV
KJ
200
100
100
200
3
Figure 2
Production Possibilities Frontiers: Case 2
Vietnam
Japan
QM
QM
QR
QR
Case 3: Three Factors of Production, Constant Returns to Scale1
There are now three factors of production: labor, capital, and land. Rice is produced with
labor and land while motorcycles are produced with labor and capital. Since land and capital
are used only in one industry, these production factors are called specific factors, in the sense
that they are specific to a particular industry. Both industries are subject to CRS, with the two
production factors in each industry imperfectly substitutable. There is no difference in
technology between the two countries.
Let us write the production functions of the rice and motorcycles industries as follows:
QR  QR T , LR 
QM  QM  K , LM 
(1)
(2)
where T denotes the amount of land inputs. CRS implies that doubling T and LR
simultaneously doubles QR and that doubling K and LM simultaneously doubles QM. However,
since all land is allocated to the rice industry and all capital is allocated to the motorcycle
industry, T in (1) and K in (2) are constants. Increasing labor input alone in (1) or (2) is
subject to diminishing returns to scale (DRS), in the sense that increasing labor inputs raises
output by increasingly smaller amounts.
1
This case is discussed in detail in Krugman et al. (2012), pp.81−93.
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Figure 3
Production Functions with Diminishing Returns to Scale
Rice
Motorcycles
QM
QR
LR
LM
Figure 3 illustrates this relationship graphically. In this figure, DRS sets in faster in the rice
industry than in the motorcycle industry, which means that labor is less substitutable for land
than for capital.
Figure 4 illustrates how to find the PPF corresponding to the previous production functions.
In this four-quadrant diagram, the production function for rice is depicted in the lower-right
quadrant whereas the production function for motorcycles is drawn in the upper-left quadrant.
The straight line in the lower-left quadrant represents the allocation of labor between the two
industries and has a slope of −1.
Choosing a particular value of LR determines where the economy is located in the lower-right
quadrant (A). It also determines the amount of LM as L  LR and hence the location of the
economy in the upper-left quadrant (C). In the upper-right quadrant, the intersection of the
vertical line that goes through A and the horizontal line that passes through C (point D)
represents the amounts of rice and motorcycles corresponding to this particular combination
of LR and LM. By repeating the same exercise for a number of values of LR, you will be able to
find the PPF for this country.
The PPF in Figure 4 is concave with respect to the origin. This PPF would not be concave if
the two industries use only labor and are not subject to DRS. It will become more concave
when the production functions of one or both industries are subject to more stringent DRS.
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Figure 4
Production Possibilities Frontiers: Case 3
QM
D
C
Labor input
for motorcycles
L
LM
B
QR
LR
A
L
Labor input
for rice
Now consider the following situation. Vietnam and Japan have the same size of labor force
(i.e., the same number of workers). However, there is more land in Vietnam than in Japan
while there is more capital in Japan than in Vietnam. Since land and capital are specific
factors, the endowment of these factors affects the shape of the production function in each
industry and hence also that of the PPF.
By referring to Figures 3 and 4, draw an example of the two countries’ PPFs in Figure 5 and
compare their shapes. Are the derived PPFs consistent with what are depicted in the textbook?
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Figure 5
Production Possibilities Frontier: Case 4
Vietnam
Japan
QM
QM
QR
QR
Case 4: A Single Factor of Production, Increasing Returns to Scale
Let us return to the case of labor being the only factor of production. This time, however, we
assume that there are two types of labor. One type engages in “blue-collar” work (physical
production) while the other engages in “white-collar” work (marketing, accounting, logistics,
etc.). While all workers are homogeneous in the sense that they can switch freely between
blue-collar and white-collar work, they cannot engage in both types of work simultaneously.
For simplicity, let us assume that the two countries have the same production techniques and
hence the same production functions. We assume that each industry has the following
(reverse) production function:
LR  bQR
LM  a  bQM .
(3)
(4)
In these equations, a represents white-collar workers while bQ represents blue-collar workers.
The number of blue-collar workers increases proportionately with output whereas the number
of white-collar workers is invariant to the size of production. Since white-collar work is
(presumably) unimportant in rice production, a is included only in (4).
Dividing both sides of (3) and (4) by the amount of output, we find the following equations:
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In (6), a rise in QM
LR
b
QR
(5)
LM
a

b
QM QM
(6)
reduces a / QM and hence also LM / QM . This implies that the
motorcycle industry is now subject to increasing returns to scale (IRS), that is, the labor
requirement per output falls at higher levels of production.
For simplicity, let us assume that the total number of workers is the same in the two countries.
Since the two countries share the same production functions (3) and (4), their PPFs are
identical. Draw this PPF in the two panels of Figure 5 and consider whether there is any
possibility that the two countries benefit from international trade.
Figure 6
Production Possibilities Frontier: Case 4
Vietnam
Japan
QM
QM
QR
QR
Note that what we have discussed here is only a particular example of IRS. It is conceivable,
for example, that the blue-collar labor in (3) and (4) is also subject to IRS. In later chapters we
will discuss in more detail how IRS can affect the structure of international trade.
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