Deriving Production Possibilities Frontiers
INTRODUCTION AND ASSUMPTIONS
Two-good x two-input model
Production functions
Let:
X=x(Lx, Kx) X is produced by the amount of labor and capital distributed to its
Y=y(Ly, Ky) production, ditto for Y.
Other assumptions:
Both L and K necessary for production. The two production functions are
constant returns to scale. This produces several results, most important is that the
curvature of isoquants are the same such that the MRTS is the same along the K/L
ray from the origin.
Ktotal = Kx + Ky, full employment assumptions
Ltotal = Lx + Ly
L and K perfectly mobile between production of X or Y. In other words, L and K are
homogeneous.
Production functions for X and Y differ sufficient but not necessarily if for some wage
rental ratio w/r (slope of an isocost), their
capital labor ratios differ.
Capital
In the diagram to the right there are two
kY
representative isoquants for the production
of goods X and Y. Both are tangent to an
isocost with the slope of w/r. Each tangency
Y
has a ray from the origin passing through it,
kX
the slopes of these rays are the capital labor
ratios used in producing X and Y, and are
labeled kX and kY respectively. X and Y are
w/r
different goods using different technologies
X
because for the same w/r they have different
Labor
capital labor ratios. In this case X is
considered to be a labor intensive good and Y is a capital intensive good. The production
of Y uses relatively more capital than labor then does the production of good X and vice
versa.
EDGEWORTH BOX
Capital
Kt
Country’s
endowment
of Labor
and Capital
The next step in the derivation is to shift
our attention away from the production of
individual goods to how a country’s
endowments of labor and capital are
allocated to the production of both goods.
This will be done with the help of an
Edgeworth Box.
Let Lt and Kt stand for the total amount of
labor and capital in a particular country. In
our production graphs that could be
represented as the following graph. Now
Lt
Good X
Labor
imagine
Labor
this graph is specific for the production of good X.
Good Y
Lt
Let a second graph describe the production of good Y.
Y
Now rotate Y’s graph 180° so that the origin is in the
upper right hand corner.
Country’s
endowment
of Labor
and Capital
To create an Edgeworth Box we now align the two
graphs together so that the horizontal and vertical
distances between the two origins are the endowments
Capital
of labor and capital respectively. The Edgeworth Box
now allows us to study the distribution of labor and
capital between the two goods maintaining full employment. Any point in the box such a
point A describes the distribution of labor
Lx to good X and Ly to good Y as well as
Good Y
Ly
the distribution of capital Kx and Ky .
C
Kt
A
Ky a
Kx
p
We will interested in several different
i
distributions of L and K across X and Y. In
a
particular is the proportional distribution of
l
inputs across the two goods. This is
portrayed as the diagonal between the two
Lx
Lt
Good X
origins. Halfway along the diagonal L and
K are equally distributed (point w).
Another example of a proportional
Ly=1/2Lt Good Y
Kt
distribution would occur one fourth
Ky
along the diagonal from the X origin
C
=
(black dot). Here we are allocating one
a
1/2
fourth of L and K to X and three fourths
Kx
w
p
Kt
of L and K to Y.
=
i
1/2
a
●
Recall that one assumption of the
l
Kt
production functions for X and Y is
constant returns to scale. In other
Lx=1/2Lt
Lt
Good X
Kt
words, a proportional increase in all inputs leads to the same proportion increase in
output. For example a doubling of inputs doubles output. With this in mind we can
begin constructing a Production Possibilities Frontier. If we allocate all resources to the
production of good X, the
upper right-hand corner of
the Edgeworth Box, we’d be
Xt
1/2Xt
able to produce some
C
maximal amount of good X,
a
say Xt described by the
p
●
1/2Yt
i
isoquant that runs through
a
that point. Conversely, if
l
we allocate all resources to
the production of good Y,
Yt
Lt
we’d be located in the lower
left-hand corner, producing
some maximal amount of good Y, say Yt. Now if we allocated our resources one half to
production of good X and the other half to good Y, with constant returns to scale we’ll
produce one-half of Xt and one-half of Yt.
Graphing these quantities of X and
Y in a Production Possibilities
Frontier gives a downward sloping
line between Yt and Xt. All points
on this line correspond to the levels
of X and Y along the diagonal in the
Edgeworth Box.
Good Y
Yt
½Yt
The question we now turn our
attention to is whether we can
produce more X and/or Y located at
the halfway point along the
Xt
Edgeworth Box diagonal. Is it
½Xt
possible to increase the production
of one good while holding the amount of the other constant? In this case we will increase
the production of X while
holding the amount of Y
constant along the ½ Yt
●
isoquant. As we move left to
right along the ½ Yt holding
●
the amount of Y constant we
Xp
move to progressively higher
●
X isoquants lying above the ½
½Xt
Xt isoquant line. We continue
to increase production of X
½Yt
until we can no longer move
Yp
to a higher X isoquant without
moving to a lower Y isoquant. The maximum amount of X (Xp) we can produce
holding Y constant at 1/2Yt is given by the tangency of the X isoquant to the 1/2Yt
isoquant. The same logic if we hold X constant and increase the amount of Y. We’d
move along the 1/2Xt isoquant to the tangency of the Yp isoquant.
At the tangencies we have an important
principle, Pareto efficiency. That is we
cannot increase production of one good
without reducing the production of the
other. Note as we reallocate labor and
capital increasing production we are
shifting more labor and less capital to the
labor intensive good, and doing the
reverse for the capital intensive good.
Good Y
Yt
Yp
½Yt
Xp
These two Pareto efficient combinations
of labor and capital correspond to points
Xt
½Xt
on the Production Possibilities Frontier.
In the diagram to the left we have two
such points. Recall that the initial distribution of L and K was half way along the
diagonal in the Edgeworth Box which when plotted as a PPF resulting in output levels of
1/2Yt and 1/2Xt. Reallocating to become Pareto efficient we can increase production of
one good holding the other constant. For example, holding Y at 1/2Yt we can increase
production of X to Xp. The converse is also true holding X constant and increasing Y.
Xt
●
●
C
a
p
i
a
l
Yt
Box we get a curve between the two
origins called an efficiency locus.
Remapping the quantities of X and Y
along the efficiency locus results in a
Production Possibilities Frontier.
Returning to an Edgeworth Box, we
initially started with a distribution of L
and K halfway along the diagonal. That
resulted through redistributing L and K
to two Pareto efficient allocations. If we
choose some other initial distribution
we’ll have two other Pareto allocations.
If we connect all the possible Pareto
efficient allocations in the Edgeworth
Good Y
Yt
Yp
½Yt
The student should contemplate several
issues. What contributes to the
concavity of a PPF? How are resources
allocated? What does Pareto efficiency
mean?
Xp
½Xt
Xt