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ments of output. The section of the production surface shows a straight
line so that the contour lines corresponding to output levels A, B, C, appear
as the points R, Q, P, with RQ=QP. On an isoquant map this is trans-
lated as generating equal distances (ML = LK) between isoquants corre-
sponding to output levels separated by equal output increments (AB=BC).
If a productive process were to be characterized by increasing returns
to scale, the section of the production surface would be convex from below
[as in Figure 8-8(b)] so that equal output increments would correspond to
unequal distances between contour lines; the higher the output level, the
shorter will be the distance between isoquants corresponding to a given
output increment. Thus L1K1 (corresponding to output increment BC)
is shorter than M1LÌ (corresponding to output increment AB, which is
equal to BC). If there were decreasing returns to scale, the situation would
be reversed, as in Figure 8-8(c), with L2K2 (which corresponds to output in-
crement BC) longer than M2L2 (which corresponds to the equal increment
AB at a lower level of output).
While intuitively it might seem almost obvious that constant returns
to scale must prevail universally, with a doubling of all factors in a given
combination yielding a doubled output, and so on, it is impossible to
make any a priori generalizations to this effect. In the real world, moreover,
it is extremely difficult to discover cases where an increase has occurred in
all factors. Usually it is discovered that some important ingredient in a
productive process (for example, managerial skill) has stayed unchanged
during an increase in all other inputs. Where this has been the case,
the changes in output cannot be attributed to a pure change in scale. Along
with the change in scale, in such cases there has occurred also a shift in the
proportions in which the factors, whose input was increased, are combined
with the factor whose input was not increased.

THE LAWS OF VARIABLE PROPORTIONS: THE PROBLEM
We have already noticed some of the consequences of an alteration in
factor proportions. We saw that as the proportion used of one factor
increased (relative to a second factor), substitution of the first for the second
becomes more and more difficult, if a given output level was to be main-
tained. Our focus of attention, in that situation, was a change in factor
proportions under the condition that the level of output be unchanged.
But the problem of changed factor proportions is of importance in several
other aspects. One such problem, for example, is the effect upon output
of changes in factor proportions, under the condition that total cost of
production be unchanged. This will be taken up in a later section of
this chapter.
At this point we are interested in yet another aspect of the problem
166 MARKET THEORY AND THE PRICE SYSTEM


of effects of variations in factor proportions. We are concerned with the
effect exerted by an increase in the ratio of the quantity in which one
factor is employed, relative to the quantity in which a second factor is
employed upon (1) the output per unit (a) of the factor being used rela-
tively more freely, (b) of the factor being used relatively more sparingly;
and (2) the incremental eßect upon output brought about by additional
input (a) of the factor being used relatively more freely, and (b) of the
factor being used relatively more sparingly. Our examination of these mat-
ters will be confined to the simplest case, that of a process of production yield-
ing constant returns to scale. It is clear that as the ratio of employment of
one factor to that of a second is increased from very low values to very high
ones, there is an initial stage where the first is spread very sparsely, so to
speak, over the second factor, and a final stage where the second factor is
spread very sparsely over the first. This symmetry between the initial and
the final stages will be reflected in the above measurements of the efficiency
of the two factors. The behavior, during the initial stage, of the output per
unit of the factor that is being used sparingly in this stage will be mirrored,
during the final stage, in the behavior of the output per unit of the other
factor. And the same will be the case with the incremental effects on output
of additional inputs of the two factors in these two stages.
Inquiries have been made by economists throughout the history of the
science into the effects upon both the per-unit efficiency and the marginal
effectiveness of factors between which the input proportions are under-
going variations. These investigations have tended to focus attention on
one particular way that an alteration in input proportions can be achieved,
the attention paid to this case arising in part from its supposed relevance to
real-world situations. The case most frequently considered involved suc-
cessive increments in the input of one factor to a fixed quantity of another
factor. In the history of economic thought this case has been dealt with
under the name "the law of diminishing returns;" in the real world the case
was exemplified whenever an alteration occurs in the quantity of labor and
capital applied to the cultivation of a given acreage of land.
While we too will investigate the effects on production efficiency of vari-
ations in input proportions, by references to this case, it must be stressed that
the importance of the case lies purely in the change in input proportions
that it exemplifies. The fixed quantity of the one factor is not to be
thought of as one of those "other things" that are so often held unchanged
in economics. It is, on the contrary, the means through which factor pro-
portions can be altered under particular circumstances. For this reason
it is probably better to use the newer term laws of variable proportions in
place of "law of diminishing returns." What these laws describe, once
again, can be visualized with the aid of an isoquant map drawn to express
the results determined by these laws. In the diagram (Figure 8-9), the iso-
167
PRODUCTION THEORY

quants (on a production surface characterized by constant returns to scale)
are drawn convex to the origin (that is, with what we found to be their typi-
cal shape, due to the imperfect substitutability of the factors). However, the
isoquant lines have now been extended to the point where they slope up-
wards in their outer regions.




H




Figure 8-9
This pattern of isoquant map corresponds to a particular set of tech-
nical conditions that, according to the laws of variable proportions, are
typical of production processes. A portion of an isoquant that slopes up-
wards is to be interpreted as the situation where a withdrawal of one of the
factors from the productive process, keeping the input of the other factor
constant (for example, a movement from the point Z in the diagram to the
point E), actually increases the level of output (shown in the diagram by the
fact £ is on a higher isoquant than Z). A positively sloping isoquant thus
corresponds to the case where the marginal increment of product associated
with an increase in the input of one of the factors is negative. The lines
OP, OQ, separate the upward sloping portions of the isoquants from the
other regions. Thus, between the lines OP and OQ, all isoquants are nega-
tively sloped. The lines OP, OQ, are called ridge lines and pass, by their
definition, through all those points where isoquants are vertical (such as
points M, D, F), and through all the points where isoquants are horizontal
(such as points, N, C, E).

THE LAWS OF VARIABLE PROPORTIONS
The behavior of output according to the laws of variable proportions
can be examined by considering in the diagram (Figure 8-9) the line GH
drawn parallel to the Ax axis. Points on this line correspond to combina-
tions of the input of a fixed quantity (OG) of factor A2, together with the
input of different quantities of factor Ax. As we move to the right along
1 68 MARKET THEORY AND THE PRICE SYSTEM


the line GH, we are considering the effects of combining greater and greater
quantities of At with the fixed quantity of A2.8 In so doing, of course, we
are decreasing the ratio of the quantity of A2 employed relative to the
quantity of A1 employed. (Thus if straight lines were drawn joining the
origin to points M and E, we would find that the slope of a straight line OE
would be less than that of a straight line OM.)
Now as we move from G toward M (where the line GH is intersected by
the ridge line OP), we cross higher and higher isoquant lines; total output
is steadily increasing. But so long as the point M has not been reached, the
isoquants slope upward since we are outside the ridge line. This means
that at any point between G and M, output could be greater if there were
less of the fixed factor A2. In this range there is too much of the fixed
factor in relation to the variable factor. While the marginal increment of
output corresponding to increases in the input of the variable input is posi-
tive (for all points in this range), that corresponding to increases in the fixed
input is negative. (That is, for any point between G and M the output is
higher than it would have been if the quantity of the fixed input had been
greater.)
As we move further to the right, from the point M to the point E, we
are in the region between the two ridge lines. Within this range, move-
ment to the right still brings us to higher isoquant lines; successive incre-
ments of the variable factor bring about progressively higher levels of
output. Also, in this region, the isoquants slope downwards. Marginal
increments of output corresponding to increases in either factor would be
positive. At any point between M and E, output is lower than it would
have been if the quantity of either input would have been greater.
As we move still further to the right, we reach the region outside the
second ridge line OQ. In this range, every increase in the input of the
variable factor decreases output (shown by the intersection of GH with
lower isoquants). Output is higher than it would be if the input of the
variable factors (Ax) were greater, but lower than it would be if the quantity
of the fixed input (A2) had been greater. There is too much here of the
variable factor A1 in relation to the quantity of fixed factor A2 available.
The fixed factor is being overworked.
From these considerations it is possible to develop a rather complete
description of the effect that different input proportions will have upon
both the per-unit and the incremental effectiveness of the factors. We must

8 When we talk of "a movement to the right" along a line, we do not, of course, mean
a temporal succession of cases (each one of which is more to the right than the ones
actually earlier in time). Different points on an insoquant map refer to alternative situa-
tions possible at one moment in time. A "movement to the right" means, then, that
we proceed to consider successively the situations more to the right as alternatives to
those more to the left, which we consider first.
169
PRODUCTION THEORY



remember that for the case of constant returns to scale, which we are con-
sidering, points on an isoquant map that lie on the same straight line
through the origin correspond to situations where the per-unit output of
any one of the factors is the same for both points, and where the marginal
effectiveness of any one of the factors is the same for both points. 9 This
means that these measures of factor effectiveness depend only on the ratio




˜n˜ f¯f




w
Figure 8-10

of input proportions, not on scale. Thus in Figure 8-10, the diagram (which
selects certain features of Figure 8-9 for emphasis) shows (besides the line
GH) the line WZ drawn parallel to the A2 axis, so that the situation at
V on GH is the same (with respect to the per-unit and marginal effective-
ness of the factors) as at the point V on WZ; the situation at B on GH is the
same as at B' on WZ; and so on.
Now as we moved to the right along GH, total output rose steadily until
point E (on the ridge line OQ) and then declined. Since the quantity of
Ao did not change during this movement, it follows that the output attrib-
utable to one unit of A2 rose steadily until E and then declined. This is
an important result. We have seen that movement to the right along GH
is equivalent (insofar as the effectiveness of units of the factors is concerned)
to a movement downward along WZ. We are thus able to state that a move-
ment downward along WZ increases the output per unit of A2 until point Ef,
after which the output per unit of A2 falls. (With constant returns to scale
9
For proofs of these mathematical propositions, see Allen, R. G. D., Mathematical
Analysis for Economists, The Macmillan Co., London, 1938, pp. 317-322.
170 MARKET THEORY AND THE PRICE SYSTEM


the ridge lines are straight lines through the origin; thus, E, E' are both points
on the ridge line.) Said another way, a movement upward along WZ first
increases the per-unit output attributable to A2 and then decreases it. This
is an even more important result. It tells us that with one factor constant
(here Alt held fixed at an input of OW), successive increments of a second
factor bring about first a steady increase and then a steady decrease in the
per-unit output attributable to this second (variable) factor. Similarly, the
output per unit of A1 steadily increases with movement upward along WZ
until M' (on the ridge line OP), after which it declines (since Axh constant
along WZ, and total output rises till M' and then falls). Hence for move-
ment to the right along GH the output per unit of A1 rises until M and then
declines steadily thereafter.
We can restate the results of the previous paragraph in the following
terms. As the ratio of the employment of one factor to that of a second
is steadily increased from very low values to very high values, the follow-
ing changes appear in the output per unit of each of the factors.
1. At first, for each of the factors being used, the per-unit output in-
creases. This is seen for the factor used relatively freely in this stage, from
the behavior of the output per unit of A2, during the movement to the right
along GH from G to M. The same is seen for the factor used relatively
sparingly in this stage, from the behavior of the output per unit of Alt dur-
ing the movement to the right along GH from G to M.
2. A range follows during which the output per unit of the factor whose
relative employment is being decreased rises steadily (this is seen from the
behavior of the output of A2 during the movement to the right along GH,
from M to E); while the output per unit of the factor whose relative employ-
ment is being increased falls steadily (this is seen from the behavior of the
output of A1 during the movement to the right along GH, from M to E).
3. Then there is a final stage where, for each of the factors, the per-unit
output decreases (this is seen for both factors”the one being used sparingly
in this stage, A2', and the one being used relatively freely in this stage, A1”
by the behavior of the per-unit output of each in a movement to the right
along GH, to the right of E).
We are also in a position to set forth the consequences of altered input
proportions upon the effectiveness at the margin of units of the factors.
We have seen that a movement to the right along GH (that is, the addition
of successive increments of input A1 to a fixed input of A2) brought about
a rise in the output per unit of Ax until the ridge line at M, after which it
fell. In other words, so long as the input of Ax (for the given quantity of
A2) is less than indicated by the point M, each additional unit of Ax brought
about such an addition to total output that the output per unit of A± was
raised. This means that in this range the marginal effectiveness of Ax was
PRODUCTION THEORY 1 71


greater than the average effectiveness of Ax. Moreover, in the range along
GH moving from M to E, the effect of adding a unit of Ax brought about so
small an addition to output that the output per unit of Ax was lowered.
This means that in this range the marginal effectiveness of A1 was lower
than the average effectiveness of Av Finally, moving along GH to the
right of E, we found that each additional unit of A1 actually reduced total
output; the marginal effectiveness of A± in this range is therefore negative.
Similarly, for a movement upward along WZ it can be seen that until
the ridge line at Ef, the marginal effectiveness of A2 (added to a fixed input
of Ar) is greater than the average effectiveness of A2; that above E' the mar-
ginal effectiveness is lower than the average effectiveness, and that above M'
the marginal effectiveness is actually negative. Translating the movement
up WZ into the equivalent but reversed movement to the right along GH,
we see that until the point M, the marginal effectiveness of A2 is negative;
that between M and E the marginal effectiveness of A2 is positive but below
the average effectiveness of A2, while to the right of E the marginal effective-
ness is greater than the average effectiveness of A2.
We can restate the results of the preceding paragraphs as follows. As
the ratio of the employment of one factor to that of a second is steadily
increased from very low values to very high values, the following changes
occur in the effectiveness at the margin of additional units of input of each
of the factors.
1. At first the factor that is being used relatively freely in this stage
is negatively effective at the margin”this is seen in the negative marginal
effectiveness of A2 in the movement along GH to the right until M; while
the factor being used relatively sparingly in this stage is positively effective at
the margin (and has a marginal effectiveness greater than its average effec-
tiveness in this stage)”this is seen in the marginal effectiveness of A1 in the
movement to the right along GH to M.
2. A range follows where the factor xohose relative employment is being
decreased is positively and increasingly effective at the margin (although not
as effective as the factor as a whole is, per unit, in this range)”this is seen in
the effectiveness at the margin of A2 along GH from M to E; while the factor
whose relative employment is being increased has an effectiveness at the
margin that is positive but steadily declining (so that it is below the over-all
per-unit effectiveness of the factor in this range)”this is seen in the effective-
ness at the margin of A1 along GH from M to E.
3. There is a final stage where the factor whose relative employment has
been decreased has an effectiveness at the margin that has risen higher than
the over-all per-unit effectiveness of the factor in this range, while the factor
whose relative employment has been increased is negatively effective at the
margin.
172 MARKET THEORY AND THE PRICE SYSTEM

The laws of variable proportions can now be expressed compactly in the
form of a table.

Effectiveness of Factor (AJ Effectiveness of Factor (A2)
Being Used in Smaller and
Being Used in Greater and
Smaller Proportion.
Greater Proportion.
Average Average Effectiveness
Effectiveness
Ratio of Ai/Ag
at the
effectiveness effectiveness
at the
margin
Margin
Stage 1 Greater than
Increasing
Increasing Negative
Very low At/A2 average
ratio
Falling (but Positive, increas-
Stage 2
positive) and ing, but less than
Increasing
Intermediate Falling
less than the the average
Ax/A2 ratio
average effectiveness
Greater than
Stage 3
Falling the
Falling Negative
Very high At/A2
ratio average

The interest these laws have held for economists over the past century
and a half, we have noticed, has been largely confined to the special case
where successive increments of a variable factor (such as labor) are added
to a given quantity of a "fixed" factor (such as land). The traditional "law
of diminishing returns" was formulated for this case, either (a) in terms of
the average product of the variable input (that is, its product per unit) or
(b) in terms of the marginal increment of product brought about by unit
additions to the variable input.10 The central point in either formulation
was that eventually the average product and the marginal increment of
product would both diminish. One or two points may be noticed concern-
ing these formulations. First of all, they do not assert that these variables
will always be decreasing. In fact, it will be seen from our analysis that if
there is any point (on a production surface characterized by constant returns
to scale) where the addition of a unit of one factor by itself will diminish
total output, then there is a range where the average product of that factor
is increasing. Marginal increment of product also may be increasing ini-
tially, but the point where it begins to decline will be before the point where
average product begins to decline. (This has sometimes caused unnecessary
confusion as to the point where "diminishing returns set in," due to con-
fusion between the two formulations of the law.)
10 For the proof that these two formulations are not mathematically equivalent (as
economists have sometimes believed), see Menger, K., "The Laws of Returns, A Study in
Meta-Economics," Economic Activity Analysis (edited by Morgenstem, O.), John Wiley
and Sons, Inc., New York, 1954.
173
PRODUCTION THEORY

Most of these considerations can be seen in Figure 8-11, which is a
vertical section of the production surface along the line GH. The curve
shown is thus the curve of total output corresponding to increasing input
of A1 (with a fixed input of A2). The curve that has been drawn is contin-
uous; thus, we can observe the way average output changes for very small
changes in input, and also the way marginal output changes continuously.11

Output
of
X




H Input of 4 ,
IM E
Figure 8-11


The average output of any quantity of input Ax is shown by the slope of the
straight line joining the origin to the total output curve at the relevant
point. Thus for input GI of Alf that output is 7/°, and average output is
therefore II°/GI, which measures the slope of the angle I°GI. Marginal
output at any level of input of A1 is shown by the slope of the total output
curve itself at the relevant point, since this is the limit of the rate per unit
of input at which the curve rises for very small increments of input.
It will be seen that until /°, the output curve rises more and more
steeply (corresponding to rising marginal product of Ax) and thereafter
rises less steeply (corresponding to falling, but positive, marginal product).12
At the point E°, when total output is at a maximum, the slope is zero
(horizontal), corresponding to zero marginal product for Ax\ thereafter the
slope is downward, corresponding to negative marginal product. It will
be seen further that straight lines drawn joining the origin to successive
points on the output curve have steeper and steeper slopes until the point
M° (where the slope of the line GM° is also the slope of the output curve it-
self, GM° being tangent to the output curve at this point); after M° the lines

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