ñòð. 19 |

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 M1LÃŒ (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 eÃŸect 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 Â£ 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˜ fÂ¯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 factorsâ€”the one being used sparingly

in this stage, A2', and the one being used relatively freely in this stage, A1â€”

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 AÂ± 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 AÂ± 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 marginâ€”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)â€”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)â€”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)â€”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/Â°, and average output is

therefore IIÂ°/GI, which measures the slope of the angle IÂ°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 /Â°, 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 EÂ°, 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

MÂ° (where the slope of the line GMÂ° is also the slope of the output curve it-

self, GMÂ° being tangent to the output curve at this point); after MÂ° the lines

ñòð. 19 |