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an additional output of BC (other things, especially the quantity of factor

Alf remaining unchanged). The quantity BC is termed the marginal in-

crement of product corresponding to the input increment HN.ГЎ This

quantity, as we shall see, has considerable significance for entrepreneurial

decision making. An entrepreneur is always faced with the alternative

of purchasing an additional quantity of a particular factor. To assess

the attractiveness of any such alternative, it is first necessary for the en-

trepreneur to judge what difference this increment of factor will make

to output. This difference is the marginal increment of product generated

by the additional quantity of factor.

Output

of

X

0 J N Input of As

Figure 8-3

In Figure 8-3 an analogous diagram is drawn to show the alternative

outputs that can be produced with different quantities of input of the

factor Alt the quantity of factor A2, this time, being held unchanged at

OE. The curve OC is thus the projection, onto the XAX plane, of the

vertical section through the production surface at C parallel to this plane.

The quantity BC is the marginal increment of output associated with

the input increment JN of factor Ax.

4 For continuous total product curves (such as in Figure 8-2), the slope of the curve

at any point (such as at C) measures the rale output increases at with increasing input

(of A2) when the input level is shown by the abscissa of the point (such as the quantity

ON). In the literature this rate of output increase is known as the marginal product of

A2 (when it is employed in volume ON). The notion of a marginal increment of prod-

uct corresponding to specific increments of input, used in the text, does not require the

postulation of perfectly divisible inputs or outputs. The marginal increment of product

has the dimensions of products; marginal product has the dimensions of product per

unit of input. For small input increments, marginal increment of product is thus ap-

proximately equal to marginal product times the increment in input.

PRODUCTION THEORY 1 57

At any point on the production surface, the relationships between the

marginal increments of output corresponding to the various variable

factors spell out the alternatives open to the entrepreneur. As we shall

see the first question asked by an entrepreneur concerning a given process

of production is whether it is the cheapest method of producing the given

quantity of output. This is the question of whether the process, corre-

sponding to a point on the production surface, is cheaper than any other

point on the same isoquant. This question resolves itself into two com-

ponents. The one component asks which other physical combinations of

factors are able to yield the same output; the second component concerns

the money costs of these different input combinations. Leaving aside the

latter problem at this stage, it is clear that the first part of the question

asks about the various additional quantities of, say, factor Ar required to

keep the level of output unchanged when various quantities of the other

factor, A2, are withdrawn from the productive process.

The relationships can be visualized with the aid of Figure 8-4. Here

MKL is an (enlarged) portion of the production surface bounded by (a)

the solid line KL, a small portion of an isoquant line; (b) KM, the line

of intersection of the production surface through Г…7 by a vertical plane

perpendicular to one of the factor axes, say, At (so that the line MQ is

horizontal, and is indicating increasing input of A2, toward Q); and (c)

LM, the line of intersection of the production surface through L by a

vertical plane perpendicular to the other factor axis (so that the line MS

is horizontal, perpendicular to MQ, and is indicating increasing input of

AГ¬ toward S). The curved line QS is the projection of the isoquant seg-

ment KL onto the horizontal plane through M. To an entrepreneur weigh-

ing a productive process corresponding to the point K} the answer to the

question considered in the previous paragraph, insofar as it concerns the

possibility of point L, is that in order to offset a loss of the quantity MQ of

input of factor A2, it is necessary to expand the input of Ax by the increment

MS. An entrepreneur producing the quantity of output shown by the

point K can maintain the same level of output by withdrawing MQ of factor

A2 and adding MS of factor AlВ·. The relation between MQ and MS thus

1 58 MARKET THEORY AND THE PRICE SYSTEM

measures the rate at which factors can be substituted for one another at the

margin. From the diagram it is clear that the required relationship between

the increments of factor MQ and MS is defined by the condition that each

is associated with the same marginal increment of product (in our case shown

as being the quantity KQ, equal to LS). If one unit of factor Ax has a higher

marginal increment of product (at the relevant margin) than one unit of

factor A2, then the increment of A2 required to offset the withdrawal of a

unit of Ax will of course have to be larger than one unit.

THE SHAPE OF THE ISOQUANT

AND THE SUBSTITUTABILITY OF FACTORS

Thus, the shape of the isoquants is the graphical expression of the

degree of substitutability between the two factors used in production. The

slope of a straight line drawn connecting two points on an isoquant

measures the degree of substitutability over this range. Thus, if in Figure

8-4 the straight line KL had been drawn, its slope with respect to the A2

axis (like the slope of the straight line QS) would be MS/MГ‡Вї, showing

the quantity of the one factor required to offset a withdrawal of a given

quantity of the other. The steeper the slope of KL, the greater would be

MS in relation to MQ, showing that AВ± would be less good a substitute for

A2 at the margin. For a continuous isoquant line, with the points drawn

closer and closer together, the slope of the line joining them becomes

very nearly the slope of the isoquant itself at a point. This slope measures

the marginal rate of substitution of factor Ax for the factor A2; that is,

the increment of factor Ax necessary to keep output level unchanged when

a small reduction is made in the employment of factor A2.5

The importance of the slope of the isoquants in this regard can be

spotlighted by contemplating two extreme theoretical situations, one where

no substitution at all is possible between the factors, the second where the

factors are perfect substitutes for one another (so that there is no economic

justification for distinguishing between them).

In Figure 8-5(a) isoquants are drawn that require the cooperation of two

factors Au A2, in a fixed proportion. Thus the point K, for example,

yields a level of output 1, using OR of Ax and OS of A2. The point L,

corresponding to a level of output twice that of K, requires OT (which

is twice OR) of Alf and OU (which is twice OS) of A2. An increase in

the quantity of factor AВ± used, without the required proportional rise in

factor A2 used, yields no additional output whatsoever. This is indicated

by the shape of the isoquant family. At K, for example, increases in

either Alt or A2, separately, yield no increase in output so that the isoquant

3 For a continuous isoquant line, this marginal rate of substitution of At for A2 is

then mathematically equal to the ratio of the marginal product of A2 to that of Ax.

59

PRODUCTION THEORY

вЂў > 2\

9

M

L

u

K

s

R T

(b)

(a)

Figure 8-5

is perfectly horizontal to the right of K (showing that an increase in Al9

by itself, does not lift output at all) and is on the other hand perfectly

vertical above K (showing that an increase in A2, by itself, does not raise

output at all). A higher output is achieved only when both factors are

raised proportionately. An example of such a process inight be the bottling

of a beverage that can be sold only in a given-size bottle. Each additional

unit of output requires the employment of one additional bottle, plus

one additional unit of the beverage. Use of two or more empty bottles

does not yield any product; neither does the use of additional beverageвЂ”

in any amountвЂ”without bottles.

Such a case is one where there is no substitutability between factors.

This is expressed in the L-shaped pattern of the isoquant family. The

marginal rate of substitution of Ax for A2 in the vertical portion of the

isoquants is zero, since the slope of the isoquant with respect to the A2

axis is zero. No additional units at all of Ax are needed to offset the with-

drawal of units of A2 (because the quantity of A2 available, compared with

that of Alf had been greater than that required by the fixed proportion).

On the other hand, in the horizontal portion of the isoquants, the marginal

rate of substitution of A1 for A2 is infinitely large (as is the slope of the

isoquant with respect to the A2 axis) showing that no matter how much

additional A-В± might be used, it would be insufficient to offset the loss

of even a small quantity of A,В· The level of output depends, not on the

quantity of either input by itself, but on the number of "units" each of

which is compounded of a fixed quantity of the one factor together with

a fixed quantity of the other factor. An entrepreneur, in making his

decisions as to the quantities of input that he should purchase, will in

fact treat units of the two inputs as component parts of a single unit of a

composite factorвЂ”in the same way as he would treat the two blades of

a pair of scissors.

The diagram in Figure 8-5(b), on the other hand, depicts the diametri-

160 MARKET THEORY AND THE PRICE SYSTEM

cally opposed situation where the factors used in production are perfect

substitutes. Here the isoquants are downward-sloping parallel straight-

lines throughout their extensions, showing that the same additional quan-

tity of any one of the factors can always be used instead of a given quantity

of the other factor. The marginal rate of substitution of one factor for

the other is thus constant at all points on the diagram and is neither zero

nor infinite.

However, the two cases shown in Figure 8-5(a) and in Figure 8-5(b) are

extreme, limiting cases. In the real world the proportions between inputs

seldom are technologically completely fixed. Usually there is room for

some alteration in input proportions without altogether wasting any input.

On the other hand, we have already seen that if two factors were perfect

substitutes in production, then they would be classed together as units

of an economically homogeneous group of goods. Typical isoquants, there-

fore, will be neither parallel to the factor axis nor straight lines throughout

their length. They will express the fact that inputs are partial substitutes

for one another; that within limits, a withdrawal of one input can be

offset by additional use of the other input, but that such substitution be-

comes more and more impractical. The marginal rate of substitution of

one factor for the other becomes greater and greater as the substitution is

carried forward. Greater and greater quantities of a factor are needed

to replace given withdrawn quantities of the other factor as the replace-

ment goes on. The typical situation is thus one where the proportion

in which the factors will be used, while not fixed absolutely by technological

considerations, is yet by no means a matter of complete indifference.6

These possibilities are sometimes described with the assistance of the

concept of the elasticity of substitution. The elasticity of substitution

between two factors measures the degree to which it is possible to substitute

one of the factors for the other, without bringing about more than a given

increase in the marginal rate of substitution of the first factor for the

second.7 A high elasticity of substitution characterizes two factors sub-

stitution can take place freely between, without causing more than a mod-

6 These considerations governing the substitutability of factors have their counterparts

(in the theory of consumer demand) with respect to substitutability between commodities

in consumption. We saw in earlier chapters that as a consumer gives up quantities of

one good in order to acquire additional units of a second, he tends to be willing to

continue such exchange only on increasingly attractive terms.

7 Mathematically the elasticity of substitution between two factors Ax and A., is defined

Г— MRS ГЃlAs/вЂ”> w h e r e MRS

as d(вЂ”Г¬/d(MRS AlA2) ^A2 is the marginal rate of substitution of

` Г….n' . . A%

/ Вїl \

At tor A2. The di вЂ” ) term denotes the change in the use of Ax as compared to that

of A2. The d(MRS AlAi,) term denotes the change in the marginal rate of substitution.

The remaining terms are introduced to make the result independent of the size of units

used.

61

PRODUCTION THEORY

erate worsening of the rate further substitution can be made at. In the

special case of perfect substitutes, the elasticity of substitution is infinite.

No matter how far substitution has been carried, it is always possible to

carry it still further at the same rate of substitution. There is in such

a case no "optimal" proportion, deviation from which makes further

substitution more and more disadvantageous.

A low elasticity of substitution, on the other hand, characterizes two

factors from which best results can be obtained only by combining them

in rather definite proportions. A significant deviation from these pro-

portions brings about a very sharp drop in efficiency, so that the more the

one factor has been substituted for the other (thereby departing from the

best proportions) the more disadvantageous are the terms on which still

further units of the first factor can be substituted for the second. In the

special case of factors, the proportions between which are technologically

fixed with complete rigidity, the elasticity of substitution is zero at the

point of fixed proportions. When the quantity used of one of the factors,

relative to the quantity used of the second factor, is slightly less than is

required by the fixed proportion, then its marginal rate of substitution for

the second is, we have seen, zero. As soon as the quantity of the first

factor has been raised to meet the required proportion, its marginal rate

of substitution for the second has risen to infinity (no amount of it can

offset the slightest reduction in the amount used of the second factor).

Such an abrupt rise in the marginal rate of substitution, brought about

by only the slightest alteration in the relative employments of the factors,

constitutes zero elasticity of substitution.

O Rs Ax

Figure 8-6

The typical processes of production lie somewhere in between these

two extremes. The isoquant family will show a pattern that is exem-

plified, at least for a portion of the production surface, in Figure 8-6. In

the diagram the isoquants are drawn convex to the origin. An entre-

preneur who has been operating at point K can maintain the same level

1 62 MARKET THEORY AND THE PRICE SYSTEM

of output by withdrawing the quantity KT of input A2 and increasing by

quantity TLВ· the input of factor At. By moving from the production

situation at K to that at L, the entrepreneur increases the proportion in

which input Ax is employed relatively to A2, from the proportion RO/KR

to SO/LS. This is shown graphically by the reduction in slope from that

of the line OK to that of the line OLВ·. The convexity of the isoquant

means that a further withdrawal ofLВ·V(drawn to be equal to KT) from the

quantity employed of factor A2 will require, for the maintenance of the

output level, an additional quantity VM of A1 that is greater than TLВ·

(which had been previously required). The extension of a straight line

joining KLВ· to N (that is, continued substitution on the same terms), would

bring it into the neighborhood of lower isoquants. The convexity of the

isoquant means that substitution of either factor for the other, if carried

on at a constant rate of substitution, would bring about progressively lower

output yields.

The elasticity of substitution at any point on one of these "typical"

isoquants depends on the convexity of the curves. If the isoquants are

only slightly convex (or, at any rate, in that portion of an isoquant where

the curvature is slight), the marginal rate of substitution (shown by the

slope of the isoquant) changes only slowly so that the elasticity of substitu-

tion over the relevant range is high. This is the case for the central por-

tion of the isoquants. Thus, in the region of KLВ· in the diagram, a given

percentage change in the ratio of Ax/A2 used does not alter the slope of

the isoquant as considerably, for example, as it does in the neighborhood

of MC. The elasticity of substitution is thus quite high in the central

portion of an isoquant (corresponding to efficiently proportioned combina-

tions of factors) but drops rapidly at the outer portions of the isoquants

where a small amount of substitution brings about a rather sharp deteri-

oration in the terms on which further substitution can take place. Thus,

at the point C, the isoquant is parallel to the Ax axis. This means that

the marginal rate of substitution of A1 for A2 has reached an infinite level:

no amount of additional A1 can maintain output should the input of A2

be cut slightly. From a point slightly to the left of C, to the point C, this

marginal rate of substitution has jumped from a finite (high) level to a

level greater than any assignable valueвЂ”this corresponds to an elasticity

of substitution very close indeed to zero.

It is now quite easy to perceive the relation between what we have

called the "typical" isoquant, and the two special cases between which it

is intermediate. The case of rigid, technically fixed proportions is one

where the central portion of the typical isoquant has become shrunk to

a single point. It is as if points C and D coincided; the range where some

substitution is possible (and where the elasticity of substitution is not zero)

has become narrowed to the vanishing point. On the other hand, the

163

PRODUCTION THEORY

case of perfectly substitutable factors is one where the central portion of

the typical isoquant extends throughout the production surface. The

range of high (in fact, infinite) elasticity of substitution is not bounded

by any limits whatsoever.

CHANGES IN FACTOR PROPORTIONS,

AND CHANGES IN THE SCALE OF FACTOR EMPLOYMENT

The insights gained in the preceding section should make it easy to

distinguish between the effects of two quite different kinds of changes that

can be made in the input of productive factors. The first kind of change is

alteration in the proportions in which the various factors are combined.

The second kind of change is alteration in the scale in which inputs com-

bined in a given proportion are applied. Here too the isoquant map

provides useful graphic aid in showing the two kinds of input changes.

Figure 8-7

In Figure 8-7 a number of dotted straight lines are superimposed upon

an isoquant map. OP and OQ are straight lines meeting the origin, differ-

ing from one another in their slopes; SR is parallel to the A2 axis, and

TV is parallel to the A1 axis. Any two points on a straight line passing

through the origin (such as W, P on the line OP) represent two combina-

tions of the factors A1 and A2, in both of which the factors are combined

in the same proportions. The difference between inputs at the two points

is one purely of scale. Just as an architect may construct a scale model

of a building (retaining the relative proportions of all lengths while re-

ducing all absolute lengths by a constant scale factor), so too the point W',

for example, is a "model" of the input situation at the point P (retaining

relative proportions but with absolute measurements of factor input mul-

tiplied by the scale factor, in this case OW/OP). An increase in the scale

of input, of course, may take place with any given proportions of factor

164 MARKET THEORY AND THE PRICE SYSTEM

combination; that is, along any straight line passing through the origin.

Points on different straight lines passing through the origin correspond

to combinations of factors between which there is a difference in the pro-

portions of the factors employed. Thus, for example, the point W differs

from the point V, and the point Вѕ in that W is characterized by a ratio

of the quantity employed of A2 to that of AГt which is equal to the fraction

WR/OR (the tangent of the angle WOR), while both V and Q have a ratio

of A2 to A1 equal to VR/OR (the tangent of the angle VOR).

RETURNS TO SCALE

The problem of defining the consequences upon output of a change

in the scale of input is the source of the concept of returns to scale. If a

given percentage change in the scale of inputs brings about the same per-

centage change in output, then the production process is said to yield

constant returns to scale. If one hour's employment of a typist's services,

together with the use of given typing facilities, can produce 10 typed pages,

and the employment of two typists, each similarly equipped, yields 20

pages in the same time, then constant returns have prevailed. On the

isoquant map this would be expressed by the condition that intercepts

(marked off along a straight line passing through the origin) between

pairs of isoquants have lengths proportional to the differences between

the output levels represented by the respective isoquants. Equal incre-

ments of output should mark off equal distances along any straight line

passing through the origin.

Output

Output Output

of of of

X X X

/

c Q

QS y /

B B

R/ B

/

A

A A A

/

l\

/

Ai

1. Вїr Inputs 0 MA *\ E Inputs

0 0 Kz ВЈВЇlnp

tK

Mz LВЇ2

(a) (b) (c)

Figure 8-8\

If a vertical section were made of a production surface characterized

by constant returns to scale, along any horizontal straight line passing

through the origin, we would obtain a situation shown in Figure 8-8(a).

Output is measured along the vertical axis; AB, BC represent equal incre-

PRODUCTION THEORY 1 65

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