ENFEREOR {NPUTS AND EXIERNAL EFFECTS

must: for the Dawn of DE. D.
MECHIMN STATE UNIVERSITY

Roger Blair
1968

TH can.

 

0-169

This is to certify that the

thesis entitled
INFERIOR INPUTS AND EXTERNAL EFFECTS
presented by
Roger Blair
has been accepted towards fulfillment

of the requirements for

Ph 0 D . degree in ECODOmiCS

<::::;ég;::7;/U 4:7[L_2 f“y*\,///

M4 jor pro essor

Date May 8, 1968

ABSTRACT
INFERIOR INPUTS AND EXTERNAL EFFECTS
by Roger Blair

This thesis explores the implications of inferior
inputs on the solutions to the resource misallocation prob-
lem created by external effects. To this end, the concepts
of external effects and inferior inputs are systematically
developed in separate chapters. Then a situation is hypoth—
esized in which an inferior input is the cause of an exter-
nal diseconomy. The solutions for removal of the Pareto
relevance of this external diseconomy are subsequently
analyzed. This analysis reveals that the solutions are

unaffected by the influence of input inferiority.

INFERIOR INPUTS AND EXTERNAL EFFECTS

BY

\

Roger'Blair

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Economics

1968

Q;§/44#

ACKNOWLEDGMENTS

Many people have greatly assisted me in whatever
educational achievement this dissertation represents. It
would be unconscionable not to thank all my former teachers;
however, some deserve special mention: Professors Michael
De Prano and Josef Hadar for my original interest in eco—
nomics and Professors Walter Adams, Robert Lanzillotti, and
Victor Smith for sustaining this interest. In addition,
Professors Anthony K00 and John Moroney read this disserta-
tion and offered helpful suggestions. .Special appreciation
must be expressed to Professor Charles E. Ferguson for his
suggestion of the topic, assistance in research, and rapid
reading of the preliminary drafts. But, above all, I must
thank my mother, my late father, and my wife, all who have

done without that I might spend most of my life in school.

ii

TABLE OF CONTENTS

Chapter
I 0 INTRODUCTION 0 O O O O O O O O O O O O O O 0
II. THE CONCEPT OF EXTERNALITIES . . . . . . . .

The Definition of Externality . . . . . .
The Effect of an Externality . . . . . . .
Solutions for the Externality Problem . .
Bargaining . . . . . . . . . . . . . .
Taxes and Subsidies . . . . . . . . .
Mergers . . . . . . . . . . . . . . .
Cost Functions and Reciprocal
Externalities . . . . . . . . . . . . .
Unilateral Externalities . . . . . . .
Mutual Externalities . . . . . . . . .
Second-Order Conditions . . . . . . . . .

III. THE CONCEPT OF INFERIOR INPUTS . . . . . . .

A Basic Model of Input Demand . . . . . .
Inferior Inputs . . . . . . . . . . . . .
An Alternate View of Inferior Inputs . .
Inferior Inputs and the Level of Profit .
Input Inferiority and Restrictions on

the Production Function . . . . . . . .
Input Inferiority and the Cross-Elasticity

of Input Demand . . . . . . . . . . . .
Inferior Inputs and Output and

Substitution Effects . . . . . . . . . .

IV. THE INFLUENCE OF INFERIOR INPUTS UPON
EXTERNALITY SOLUTIONS . . . . . . . . . .

The General Problem . . . . . . . . . . .
Solutions . . . . . . . . . . . . . . . .
Bargaining . . . . . . . . . . . . . .
Taxes and Subsidies . . . . . . . . .
Mergers . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . .

APPENDICES . . . . . . . . . . . . . . . . . . . . .

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . .

iii

Page

18
20
22
27
31

33
34
37
43

48
49
52
55
56
57
61

62

68

69
71
72
78
81
82

85

94

LIST OF APPENDICES

Appendix Page

A. DERIVATION OF COST FUNCTIONS WITH
EXTERNALITIES PRESENT . . . . . . . . . . . 85

B. AN ALTERNATIVE VIEW OF THE EXTERNALITY—
CAUS ING INPU'r O O O O O O O O O C O O O O O 9 1

iv

CHAPTER I
INTRODUCTION

,Although the early discussions of external effects
were filled with errors, inconsistencies, and semantic
difficulties, as with most theoretical concepts, a clearer
picture finally emerged. In 1959, F. M. Batorl cleared
away most of the rubble and summarized the theory that had
been developed. Since that time, there has appeared a host
of articles on the subject. Some of these have represented
attempts to define externalities more)precisely. Some dis-
cussed various solutions to the resource misallocation prob-
lem caused by external effects, i.e., the achievement of
Pareto optimality. Still others have introduced further
complications into this resource misallocation problem.

This dissertation will also introduce a further complication:
the influence of inferior inputs.

While the notion of an inferior good, i.e., one
whose consumption decreases as an individual's income
increases, is certainly not new, the analogous concept of an

inferior input was not clearly defined until D. V. T. Bear's

 

1F. M. Bator, "Anatomy of Market Failure," Quarterly

Journal of Economics, LXXII (1959), 351-79.

 

article2 appeared. There have been but two subsequent
developments of this concept. While Charles Plott's contri-
bution3 was not very explicit, that of C. E. Ferguson4 was.

I shall combine this concept with that of external
effects by hypothesizing a case where one firm's use of an
inferior input causes another firm to suffer an external
diseconomy. The solutions that will "correct" the external-
ity problem, i.e., that make the attainment of Pareto opti-
mality possible, will then be analyzed. The purpose of this
analysis is to determine whether these solutions still apply
in this special case.

To accomplish this, I shall carefully develop the
definition and effects of externalities along with their
solutions in Chapter II. The concept of inferior inputs and
some of their consequences will be developed in Chapter III.
After these careful surveys we shall have all the tools at
hand to complete the investigation. This will be done in

the concluding Chapter IV.

 

”2D. V. T. Bear, "Inferior Inputs and the Theory of

the Firm," Journal of Political Economy, LXXIII (1965),
287-89.

3Charles Plott, "Externalities and Corrective Taxes,"
Economica, XXXIII (1966), 84-87.
4C. E. Ferguson, "'Inferior Factors' and the Theories
of Production and Input Demand," Economica, to appear May
. 1968.

 

 

 

CHAPTER II

THE CONCEPT OF EXTERNALITIES

l. The Definition of Externality

 

Mathematically, welfare maximization involves the
solution of a constrained extremum problem. Solving this
problem by Lagrange methods yields a set of first-order
optimality conditions that include the familiar Lagrange
multipliers. These multipliers represent the costs of the
constraints. Thus, as Davis and Whinston note,1 they also
represent the implied costs of the constraints on technology
and the market prices of factors and commodities. Under the
usual assumptions concerning tastes, technology, and profit
maximization, the equilibrium quantities of inputs and com-
modities that result from pure competitors responding to
these prices will satisfy the conditions for Pareto effi-
ciency. In other words, pure competition will put society
on its "bliss" frontier. If the competitively imputed

incomes could be redistributed without cost in some lump-sum

 

1O. A. Davis anth. B. Whinston, "Welfare Economics
and the Theory of Second Best," The Review of Economic
Studies, XXXII (1965), p. 4.

way to achieve the "correct" income distribution, the social
welfare function could then be maximized.

Achieving the "bliss" frontier can be thwarted by
many real world phenomena: imperfect information, inertia,
non-profit maximizing behavior, risk and uncertainty, etc.
But these foils can be ignored in this discussion as "they
have to do with the efficiency of 'real life' people in a
nonstationary world of uncertainty, miscalculation, etc."2
This thesis is concerned with the phenomena that disrupt
Pareto-efficient resource allocation under the assumptions
of individual profit- and utility—maximization in a station-
ary world. These phenomena are labeled externalities. The
discussion of these externalities will be largely in partial
equilibrium terms; therefore, it must be understood that the
rest of the economy is, and remains, organized so that the
Pareto optimum conditions are fulfilled. As will be empha-
Isized later, this does not involve a second-best situation.

As Bator pointed out,3 the modern formulation of
externalities is embedded in the idea of direct interaction.
This interaction is a result of nonindependence of some
utility and/or production functions, i.e., the nonindepen-

dence may be between producers, between consumers, or between

 

2F. M. Bator, "The Anatomy of Market Failure,"
' Quarterly Journal of Economics, LXXII (l958), p. 352.

3F. M. Bator, "Market Failure," p. 358.

 

producers and consumers. Such nonindependence causes some
Paretian costs and benefits to be omitted from the decen-
tralized, private cost-revenue calculations. In other words,
it causes social costs and social benefits to diverge.
Although Bator gives an example of this nonindependence con-
cept of externality, his verbal definition is not wholly
satisfactory. Buchanan and Stubblebine,4 however, have
developed a precise set of definitions formulated in utility
terms. These have been transformed into productivity terms
because this thesis is concerned with production.

Assume that there are two firms, i and j, with the

following twice differentiable production functions:

. = G x. ,x. ,...,x. 1.1

Q3 (31 32 3n) ( )
and

Qi = F(xil,xiz,...,xin,z) (1.2)
where xjk and xik are the amounts of input k used by firms j
and 1 respectively. We also assume that aG/dxjk > 0 over
the range considered as is aF/axik where k = l,2,...,n. The
element 2 is defined as

z = 9(le). (1.3)

 

, 4J. M. Buchanan and W. C. Stubblebine, "Externality,"

Economica, N.S. XXIX (1962), 371-384. The concepts developed
in equations (1.1) through (1.12) are simply re-statements in
productivity terms of their concepts.

 

i.e., j's use of input le yields some output 2 that enters
i's production function. To avoid confusion, I must make it
quite clear that z is some accidental by-product or condi-
tion that j creates. We must emphasize that 2 has no market
price. It is not the sort of by-product which is commonly
sold. To clarify this point, consider the classic case of
air pollution caused by a factory's operation where z repre-
sents the smoke emitted, g represents the burning process,
and le represents the fuel. Thus equation (1.2) implies
that the output of i is a function of the inputs under its
control and an output of j that is directly related to j's
use of input le. This condition constitutes the presence
of an externality.

Input le was chosen as the externality-causing
input merely for expositional convenience. Moreover, i's
production function could be written to include other inputs
under j's control; but without loss of generality, we may
direct our attention to the effects of the single input le.
In addition, since it serves no useful purpose to consider
mutual, or reciprocal, externalities when discussing defini-
tions, we assumed that j's production includes only inputs
under its own control. One further comment requires mention:
as a matter of notation we let dF/dxik = aF(xil,xiz,...,xin,z)/
axik’ i.e., dF/Bxik is, in general, a function of all the xi's
and z. The abbreviated form will be used throughout, but its

precise meaning should not be forgotten.

Since i has no control over the level of z = g(le),
z enters i's production function parametrically. Thus, we
assume that i attempts to maximize profits in the usual way,
subject to given values of 2. Whether i's optimum output
must be modified to account for various values of z is a
question considered later.

Assuming i and j maximize profits independently and

the product and factor markets are competitive, the problems

are to
n
Max Vi — Pi F(Xil'xi2”'°’xin’z) -k§l pk xik (1.4)
and
n 1
Max mj— Pj G(le,xj2,...,xjn) — kil pk-xjk ( .5)

where w is profit, Pi and Pj are the prices of the products
i and j produce, pk is the price of input k, and the xik and

x. are the quantities of the inputs used.

jk
The first-order conditions for profit maximization
are
Pi'dF/dxik - pk = 0 1f xik > O (k = l,2,...,n) (1.6)
and
Pj'dG/dxjk - pk = 0 1f xjk > O. (k = l,2,...,n) (1.7)

These are the usual Pareto conditions: in equilibrium the

value of the marginal product of each input must equal its

parametrically given market price. But i cannot take account
of 2 because there is no price attached to it even though it

enters i's production function. In all cases where
dF/dz = dF/dg(le)- dg(le)/dxjl # O, (1.8)

there exists a marginal externality. This concept can be

 

used to define explicitly external economies and diseconomies.

A marginal external diseconomy exists when

5

dF/dz < O, (1.8a)

i.e., a small change in the quantity of le used by j will
change 2, which in turn will change the output level of i in
the opposite direction. Similarly, a marginal external

economy exists when
dF/dz > O. (l.8b)

An infra-marginal externality exists at all points

where
dF/Bz = O and equation (1.2) holds. (1.9)

An infra-marginal external diseconomy exists when, for any

g1ven set of values of xil’xi2""’xin’

 

5Although I shall use dF/dz to represent the longer
'expression dF/ag(x. ).dg(x.l)/dx.l, its precise meaning
should not be forgdéten. 3 j

Z
dF/dz = 0, f0 dF/dz dz < O, and equation (1.2) holds. (1.9a)

This means that although small changes in 2 do not affect the
total output of i, the total effect of j's use of le is to
decrease i's output. Analogously, an infra-marginal external
economy exists when, for any given set of values of

X0 X0 0.. X.
11' 12’ ’ 1n’

Z
dF/dz = 0, f0 dF/dz dz > O, and equation (1.2) holds. (1.9b)
Now small changes in z = g(le) do not change i's total out—

put, but the total effect of j's using le increases i's

output.6

 

6The meaning of an infra—marginal externality can be
clarified by considering an example. Let i's production
function be 2 1/2

oi = (a -(x-1)2 - (z-l)2)

Then the first-order conditions require

in/dx = (l-x)/Qi = 0

where a > 0.

and
in/dz.= (1-z)/Qi = O.
This implies that an extremum is found where x = l and z = l.
S1nce
szi/dx2 = - az/a3 < 0,
8203/822 = - az/a3 < O,
and

sz-/dxdz = O at the point where x = l and
z = 1, then 1
(azoi/axz) (azoi/azz> > (szi/dxazlz-

Therefore, the necessary and sufficient conditions for a
maximum of Q. are fulfilled. It is clear that infinitesimal
changes in 2 will not change i's output. But evaluation of

10

The classifications so far introduced resulted from
evaluating the partial derivatives of i's production function
with respect to 2 over the whole range of 2. Further con—
cepts of relevance and irrelevance require considering the
extent to which the externality-causing factor is used by
the firm that has control over it, i.e., j. For an external-

ity to be potentially relevant, the use of the externality-

 

causing factor must create a desire on the part of i to
change j's level of use. If an externality creates no such

desire, it may be termed irrelevant. Formally, a potentially

 

relevant marginal external diseconomy exists when

 

dF/dz < O. (1.10)

In this case, i would like j to decrease its use of le
because that would decrease z and, consequently, increase
i's output. Similarly, a pgtentially relevant marginal

external economy exists when

 

dF/dz > O. (1.11)

 

I; aF/dz dz will show that j's use of x.l does have an effect
on i's output: 3

f; (1—2)/Qi dz = (az-(x-1)2) (aZ-(x-l)2-l)l/2 > 0.

Thus we have an example of an infra-marginal external econ-
omy. The quantity of 2 would not normally be equal to one
if j ignored its effect on 1. Hence we must suppose that
either j made a mistake that resulted in z = l or that it
'was done purposely to increase social welfare. If the
latter is the case, j was misguided because, as we shall see
later, increasing 2 until dF/dz = 0 will not, in general,
maximize social welfare.

1/2

ll

Firm i would like j to increase its use of le for analogous
reasons.

The concepts of relevance and irrelevance may also
be applied to infra—marginal externalities. Infra-marginal
externalities are clearly irrelevant for small changes in
the quantity of 2. But when discrete changes are introduced,
i will want to alter the quantity j employs in all cases
except when

dF/dz = O and

(1.12)

F(Xil’xi2’°°"xin’le) Z_F(xil,xiz,...,xin,le)

l
jl' When equality holds in (1.12), i is getting the most

for all le # 25 where E51 is the equilibrium quantity of

x

"good" or the least "bad" from j's use of input le.
Although potential relevance depends upon i's desire

to alter j's behavior, this does not imply that it is pos-

sible to do so. But Pareto relevance of an externality does

 

depend upon this possibility. Specifically, an externality
is Pareto relevant when the quantity of z can be changed
such that i is better off without making j worse off. In
other words, if the externality is Pareto relevant, there
are mutual benefits available. More formally, a marginal

externality is Pareto relevant whenever

lPi-dF/dzl > le-dG/dle - pl . (1.13)

12

This means that for i to be in a position to alter the quan-
tity of le used, the value of the effect on i's operations
must exceed the value of the benefit j receives less the
cost of purchasing the input, i.e., the net increased bene-
fit to i must exceed the net cost to j consequent upon j's
reducing le from the present employment level. This dif-
ference is available to i and j: so a change can be made
that will make at least one better off without making the
other worse off.

From equation (1.7), if j is maximizing profits,

Pj-dG/dle = pl when x. .Clearly then, when j is

31 = 3Ejl'
maximizing profits, a potentially relevant marginal exter-
nality must also be Pareto relevant because the right-hand
side of inequality (1.13) vanishes. Thus there must be room
for mutual benefits, i.e., both i and j can gain from some
adjustment on j's part.

From condition (1.13), it follows that for the

production sector the condition for Pareto equilibrium when

externalities are present is

lPi°dF/dz

 

= le°8G/dle — pl . (1.14)

An extremely important implication of equation (1.14) is

that Pareto equilibrium does not require the removal of the

13

externality.7

The opportunity for mutual benefit, however,
is removed, i.e., the marginal externality is no longer
Pareto relevant; the interests of the two firms are exactly
offsetting.

Instead of dealing with only two firms, we can also
include situations where j's action affects a group of other
firms. This modification really does not change anything
except the conditions for Pareto relevance and Pareto equi-
librium. In this case, Pareto relevance requires

5

z Pi°dFi/dz

.- . - 1.1
i=1 > IPJ dG/dle pl ( 5)

 

where Fi is the production function of the i-th firm. of
course, we are still assuming that all firms are pure com-
petitors in both the product and factor markets.

Again, if j is in equilibrium, all marginal external-
ities must be Pareto relevant since the right-hand side of
inequality (1.15) will vanish. Under these circumstances,
the condition for Pareto equilibrium is

s .
Z Pi-dFl/dzl = le°dG/5le - pl . (1.16)
1

 

7This demonstrates that the policy—maker cannot
merely focus on the existence of an externality. He must
determine whether it is Pareto relevant before he can make
any decision. ‘

14

The same comments that applied to equations (1.13) and
(1.14) apply to these conditions.

An additional point worth emphasizing is the joint-
supply nature of externalities.8 As noted previously, 2 is
supplied to 1 without charge. The reason no price is
attached to z is that z is an accidental by-product of pro-
ducing Qj' In many cases, by-products are sold by the firm
that produces them; but in this instance, such is not the
case because independent operation precludes j from knowing
of its effect on 1.

Since joint-supply characterizes j's operation, its

production function should be re—stated:

. = . + z = G x. ,x. ,...,x. . 1.17
Q3 Q3 (31 32 gm) ( )

In general, j could produce Qj independently; but when joint-
supply occurs with a zero price for 2, we may assume that
this is simply because it is more efficient than separate
supply. To demonstrate this, define j's alternative cost

functions as functions of output:
C = h l where i = . + z, 1.18
1 1(0)) Q] Q] ( )

or

 

8This relation was pointed out by J. M. Buchanan in
"Joint Supply, Externality and Optimality," Economica, N.S.
XXXIII (1966), 404-415. I have merely adapted his develop—
ments for my purposes.

15

c2 = h2(Qj) and c3 = h3(z). (1.19)

Then the condition for the efficiency of joint-supply is

given by
acl/aoj < acz/an + ac3/az, (1.20)

i.e., the marginal cost of producing Qj and 2 together is
less than the sum of the marginal costs of producing them

separately. Since equation (1.20) is not inconsistent with
BCl/BQj > dCz/de, (1.21)

2 will be supplied without charge if, and only if, j finds

it more efficient to produce 03 than Qj’ i.e., when
acl/an g_ac2/aoj. (1.22)

This follows from the necessary marginal conditions for

exchange equilibrium under joint-supply:

acl/an = Pj + Pz. (1.23)

Clearly, when P2 = O and inequality (1.21) holds,
there will be no joint-supply. In addition, we may note
that any situation satisfying condition (1.22) automatically
satisfied inequality (1.20). That is to say, the existence
of an externality implies joint—supply. But we should also
note that joint-supply does not necessarily imply the

existence of an externality because joint-supply may exist

16

when (1.20) holds and the price of z is non-zero, even if
(1.21) also holds.

Finally, it is worthy of mention that when j sup-
plies z to 1 without charge, input le becomes collective in

a sense.9 When j buys and employs input x the output 2

jl'

that is directly related to x. becomes available to i. Of

31
course, in the case of externalities, the factor 2 is
imposed upon 1. This is analogous to some of the standard
examples of collective goods, e.g., society decides it wants
a certain amount of National Defense and whether or not I
want any of it I am forced to consume it. The equilibrium

condition for a collective input is the same as that which

results from joint-profit maximization, viz.,
Pi°dF/dz + Pj°dG/dle = pl. (1.24)

This condition merely states that the sum of the values of
marginal products must equal the price of the input. Inspec-
tion of equation (1.14), the condition for Pareto equilibrium
when externalities are present, reveals that these two condi-
tions are the same.

These definitions are not strictly in accord with

all the literature. For example, Bator would find them too

 

9The collective nature of such a factor was men—
tioned by Charles Plott in "Externalities and Corrective
Taxes," Economica, N.S. XXXIII (1966), 86.

 

l7
restricted.lo He defines externality in a much broader
sense as the existence of any phenomenon that precludes
decentralized pricing from sustaining Pareto optimal outputs.
While this discussion is limited to his first type, owner-
ship externalities, Bator also includes technical external-
ities and public good externalities. Technical external-
ities are a consequence of indivisibilities or smoothly
increasing returns to scale, which cause non-convexity of
the set of feasible input-output points. The result is the
natural monopoly case where decentralized competitive pric-
ing cannot sustain Pareto optimal outputs because perpetual
losses would be incurred. On the other hand, public goods
supposedly preclude the existence of a set of prices asso-
ciated with the point of maximum social welfare that would
sustain the Pareto optimal output configuration, i.e., the
exclusion principle fails to be operative. We have seen
that this concept is not wholly at variance with our own.
In fact, there must be some element of "publicness" in any
instance of direct interaction in production because of the

externality-causing input's collective nature.

 

1OF. M. Bator, "Market Failure."

18

2. The Effect of an Externality

 

In the absence of externalities, the transformation
relation between inputs and outputs for society may be given

in implicit form as

X ,...,X) =0, (2.1)

T(Yl,Y 2 m

2,...,Yn;Xl,

where the Yi represent the total amounts of society's n out—
puts at full employment and the Xj represent the total
amounts of society's m inputs. This transformation function
is a surface in n-space that shows the maximum amount of any
one YR given the values of the other Yi's and the Xj's.
Thus to increase Y1, for example, we must decrease some
other output, or outputs, if we hold constant the amounts of
inputs employed.ll

Let us now introduce the externality discussed in
the previous section, viz., the one specified in equation
(1.2). The externality will change the transformation func-

tion to

X X

* .
T (Yl’Y n+1! l'l 2I'°°I

Yn’Y Xm) = O, (2.2)

2,...,

where Yn is the output 2. Assuming that i produces only

+1
Y1, that the externality is a diseconomy, and that its

effects are confined to i, the transformation surface will

 

llSee P. Samuelson, Foundations of Economic Analysis
(Cambridge: Harvard University Press, 1963), p. 230.

19

be lowered in the Yl—direction. In the case of an external
economy, it will be raised in the Yl-direction. Thus when
there is a change in the technical relation between inputs
and outputs, our frame of reference concerning Pareto effi-
ciency is different.

After introducing the externality, Pareto efficiency
in production requires that society operate on surface T*:
and we can ignore surface T as it is no longer relevant.
When the condition for Pareto equilibrium is fulfilled, we
shall be on surface T* even though a marginal externality
exists; but if a Pareto relevant marginal externality exists,
we shall be operating below the surface T*. Clearly,
resources are not allocated properly when there exists a
Pareto relevant marginal externality. It should be stressed
that whatever the optimal adjustment is for an external dis—
economy, its existence implies that society is worse off
than it would be without the diseconomy in the sense of
there being less output.12

We should also note that introducing an externality
does not also introduce a second-best situation. The theory

of the second-best involves cases where society is operating

below the relevant transformation surface and for some

12The effect of an externality on the production
.possibility surface was noted by E. J. Mishan in "Reflec-
tions on Recent Developments in the Concept of External
Effects," The Canadian Journal of Economics and Political
Science, XXXI (1965), 105, 113, 114.

20

reason cannot remove the impediment to Pareto optimality.
Satisfaction of equation (1.14), the condition for Pareto
equilibrium, implies that some method has been found for
removing the obstacle from society's path to Pareto effi-
ciency; society is doing the best that it can. Simply
because surface T* is not the same as surface T does not
mean that we are in the foggy realm of the second-best.

What it does mean is that there has been a change in
the technical conditions underlying production. With such a
change, it no longer makes any sense to talk about Pareto
optimality in the absence of externalities because the
marginal externality does not disappear in equilibrium as
equation (1.14) shows. The externality-free transformation
surface T is no longer relevant. Society must live with the
technically feasible surface T*. On the other hand, if, for
some reason, the Pareto relevance of the marginal externality
cannot be removed, then society will operate below surface T*,

and we have a second-best problem.
3. Solutions for the Externalitnyroblem

We know that an efficient allocation of resources
requires operation on the transformation surface T*. What
prevents the attainment of surface T* is the Pareto rele-
vance of the externality caused by 2. Any solution to this
'problem must involve removing the Pareto relevance found in

equation (1.13), i.e., a solution must result in Pareto

21

equilibrium. As the previous sections pointed out, this
does not require removal of the externality. In fact, we
must live with it. But this does not mean that simply
ignoring the externality constitutes a solution because that
would not lead us to a Pareto optimum. .Such a second-best
approach is appropriate only when the costs of achieving
efficiency exceed the gains.13
Assume that the problem is defined by equations
(1.1), (1.2), (l.8a), and (1.13), i.e., there exists an
external diseconomy and independent profit maximization
demonstrates that it is Pareto relevant. We can discuss

the following types of solutions: bargaining, taxes and

subsidies, and mergers.

 

131 might point out here that we can avoid intro—
ducing the costs of adjusting to the presence of the exter-
nality only if we assume that the pricing system works
smoothly, i.e., without cost. This assumption is not
strictly legitimate as it is clear that some resources must
be expended in making the adjustments. If the costs of
adjustment vary with output, we must add them to the social
marginal cost. Their effect will be to reduce optimal out—
put further. On the other hand, if they are lump sum, the
decrease in social loss from removing the Pareto relevance
of the external diseconomy must exceed the lump sum cost for
the adjustment to be worthwhile. (On the costs of solution,
see Mishan, "Reflections," p. 111, and R. H. Coase, "The
Problem of Social Cost," Journal of Law and Economics,
October 1960, pp. 2, 15—19.) In the subsequent discussion
of external diseconomies we shall assume that the costs are
lump sum and that the adjustment is worthwhile. Further,
the discussion will primarily deal with external disecon-
omies as the treatment of external economies is quite
.symmetrical.

22

Bargaining

 

One approach to solving the problem is through
direct bargaining between i and j. Three comments on bar-
gaining are in order: first, bargaining is most feasible
when the number of firms involved is not too large. In our
case of two firms this presents no difficulty, but the
results of this analysis cannot be taken to apply in all
cases without recognizing the problems inherent in large-
group decisions. .Second, the form that the bargaining
process will take depends upon the property rights defined
exogenously by law, i.e., the direction of payment depends
upon who is liable to whom. We will take the law as given
and discuss the problem around it. Third, throughout this
entire thesis, I shall exclusively deal with the production
sector. Because bargaining involves confrontations of pro—
ducers, we must assume that these discussions concerning the
employment of resources in no way affects the markets for
final output.

Since i is suffering from an external diseconomy
imposed upon it by j, from i's point of view, it will appear
that j has chosen the "incorrect" quantity of le. Recall
that fulfilling condition (1.13) means i desires a change in
the behavior of j and it is possible to induce such a change,
possible in the sense that there is a mutually advantageous

alternative.

23

We can see the effect the law will have on the
bargaining process by considering several legal arrangements
separately. Let us begin with the assumptions that the law
imposes no legal constraints on j, the firm that creates the
externality, and that we are concerned with an external dis-
economy, i.e., dF/dz < 0. The presumption is that i offers
to pay j $B for each unit of le that j does not use, i.e.,
is the amount of

j will receive $B(X§ -le) from i where'fi.

1 31
input le that j would otherwise use. Since 1 cannot know
the precise form of j's production function, the offer must
only be a tentative one. If the offer of $B per unit does
not result in an optimum for i, the offer will be withdrawn
and further offers will be made until an optimum is reached.

Formally, the offer of a bribe changes the profit

functions for i and j to

n
. = P.°G x. ,x. ,...,x. + B X1 -x. - 'x. 3.1
and
n
”1 = Pi°F(xil,xi2,...,xin,z) - B(le-le) -k:lpk°xik.(3.2)

Because of the change to conditions (3.1) and (3.2» the
first-order conditions for a profit maximum also change.

These conditions for j are now

Pj'dG/Bxjk - pk = 0 1f xjk > 0 (k = 2,3,...,n) (3.3)

and

24

Pj-dG/ale - (pl+B) = 0 if x > 0. (3.4)

jl
Since pl < (pl+B), le < le, i.e., Since the bribe offer
increases the "effective" market price of le,

that j employs will decrease. This is the direction of

the quantity

change i desired, but the magnitude of the change may not be
sufficient for the attainment of surface T*.
To determine this we must investigate the effect on

i's first-order conditions:

Pi'dF/Bxik — p = 0 if x. > 0 (k

k 1k l,2,...,n) (3.5)

and
. S.
Pi aF/az + B {5.) 0. (3.6)

If "<" holds, a further decrease in j's use of le is
desired by i and a new offer B will be made such that B'> B.
If ">" holds, the first offer made was too large and j
decreased its use of le

A A
Firm i will then make a new offer B such that B < B.

by more than the optimum amount.

Finally, if "=" holds, there is no incentive for i
to make a new offer because he has already made the offer
most profitable to him. We should recognize that the first
offer may not be the correct one because, although i may
know the exact value of Pi'dF/dz, it may not know the exact
Achange that will occur in the quantity of 2 as a result of
a decrease in the usage of le. In other words, i may not

know the precise form of the relation 2 = g(le). Thus this

25

bargaining process will continue until equality holds in
equation (3.6) and equilibrium is attained. When equations
(3.3), (3.4), and (3.5) hold and there is equality in equa—
tion (3.6), we have a Pareto optimum. Neither firm can be
made better off without making the other worse off.

More formally, we can solve equation (3.6) for B and

substitute into equation (3.4) to get
Pj°dG/ale + Pi-dF/dz - p1 = 0. (3.7)

This will be recognized as exactly the same as equation
(1.24) and essentially the same as equation (1.14). Thus
bargaining has led us to Pareto equilibrium in the presence
of a marginal externality. Note, however, that the exter-
nality has not been removed.

Now let us consider the same problem, except that
the law does not allow j to impose an externality upon 1 in
the absence of i's consent. That is, the law prescribes

le = O. For a problem to exist under these circumstances

we must have

A
Pj-dG/dle - p1 > 0 for le < le (3.8)
/\
where le is the quantity of le that would make an equality
hold in expression (3.8). Consequently, j has an incentive
to bribe i for permission to use le. The offer of a bribe
to i changes the profit functions to

n
”i = Pi°F(xil,xiz,...,xin,z) + ijl -kilpk-xik (3.9)

26

and
n

) -(B+pl) le - Z pk°Xjk. (3.10)

w. = P.-G(x. ,x. ,...,x.
3 3]- 32 k=2

3 3n

Since i will suffer the inconvenience of j's use of
le, i will presumably specify indirectly the quantity of le
that j may use for a payment of B per unit. Firm i's speci-
fication must be indirect because the precise relation
2 = g(le) may not be known to i. At any rate, offers will
be made and rejected until the first—order conditions are

satisfied:

Pi'dF/dxik - p = 0 if x. > 0, (k = l,2,...,n) (3.11)

k 1k
Pi-dF/dz + B = 0, (3.12)
Pj°dG/dxjk — pk = 0 1f xjk > 0, (k = 2,...,n) (3.13)
and
Pj-dG/dle - (B+pl) = 0 1f le > 0. (3.14)

Since all we changed was the legal constraint, we can compare
this equilibrium with the previous one. Interestingly, equa-
tions (3.3), (3.4), (3.5), and (3.6) correspond to equations
(3.13), (3.14), (3.11), and (3.12). Moreover, we can solve
equation (3.12) for B, substitute into equation (3.14), and
derive a relation exactly like equation (3.7). Of course,
the same comments that applied to equation (3.7) also apply

to this condition.

27

An intermediate case could be analyzed, one in which
the law permits j some use of le and, therefore, some out—
put 2. Davis and Whinston handle this case and find that
although both firms may attempt bribes initially, there will
come a stage in the bargaining process when both will real-
ize which firm must pay in order to reach an equilibrium.

In conclusion, when bargaining is feasible, i.e.,
when the number of parties is not too large, a Pareto
optimum can be attained without outside interference. The
question of what the legal constraints ought to be is a
question of equity and does not have a bearing on the ques-
tion of efficiency. But once the legal constraint for lia-
bility is specified, no further legal constraints should be
imposed because they might prevent the firms from reaching
an optimal solution. In other words, all the law should do
is make clear the liability for externalities since bargain-

ing can then move society to a Pareto optimum position.l4

Taxes and Subsidies

 

An alternative to bargaining is the tax-subsidy
approach. In general, this solution involves taxing the

firm that causes the externality and compensating the firm

 

14This entire discussion of bargaining depends
heavily on O. A. Davis and A. B. Whinston, "Some Notes on
- Equating Private and Social Cost," The Southern Economic
Journal, October 1965, pp. 113-126.—IR. H. Coase in
"Social Cost" also deals with bargaining and the effect of
legal constraints on income distribution.

28

suffering the externality in like amounts. But as Plott
points out, it is important to levy the tax on the correct
thing.15 In our example, j causes the externality when it
produces Qj’ but Qj is not the culprit. The source of dif-
ficulty is 2, which is a joint- or by-product of Qj' Since
2 is a function of input le, the tax should be placed on 2
or on the use of input le. In fact, levying the tax on Qj
will result in an increase in 2 when le is an inferior
input. Plott demonstrated this result graphically for a
two-factor production function, but it can be generalized to
n inputs. Input inferiority will be dealt with in detail in
the next chapter.

Obviously, taxing the offending firm inherently
presumes a legal constraint which places the burden on that
firm. The result of such a tax—subsidy scheme is the iden-
tical resource allocation and income distribution that pri—
vate bargaining yields when the law specifies?jl = 0 in the
absence of i's permission for it to be otherwise. To show
this formally, all we must do is let B represent both the
tax on j and the payment to i in equations (3.9) through
(3.14). Now equation (3.12) states that the compensation is
exactly equal to the damage done by j to i. In addition, by
solving equation (3.12) for B and substituting into equation

(3.14) we get condition (3.7):

 

15
pp. 84—86.

C- Plott, "Externalities and Corrective Taxes,"

29

Pj°dG/dle + Pi-dF/dz — p1 = 0.

Rearranging this equation we get

Pj'dG/ale pl — Pi-BF/Bz. (3.15)

In words, j must equate the value of the marginal product of
le with its price plus the value of the damage it does to
i's operation. Now the social value of the marginal product
of le is equated with its price.

In addition to the factor inferiority objection to
levying the tax on Qj' we can now see another objection.
Levying the tax on Qj so that condition (3.15) is satisfied
will render j's choices for all other inputs non-optimal.
Taxing Qj changes j's profit function to

n

w. = P.’G x. ,x. ,...,x. — 2 -x. - tQ., 3.16
3 J (31 32 3n) k=lpk 3k 3 ( )

where t is the per unit tax. The first—order conditions now

become
a - OX. = P.°OG X- " _ t = 00

At the margin, the tax is levied on each input; therefore, j
does not equate the social values of the marginal products

with their respective input prices except for factor le.

30

Thus equation (3.15) is fulfilled, but all the other input
conditions are violated.16

In the absence of a good reason for supposing the
government has some special knowledge of the precise forms
of the production functions, we may assume that it must
arrive at the appropriate tax or subsidy throngh some iter-
ative procedure much like that used in the bargaining solu-
tion. Thus the difference between the private bargaining
and the tax-subsidy approaches is that the government is an
intermediary.

One very important point should be stressed: if j
is to be taxed, an amount equal to the tax must be paid to 1.
When i is not so compensated, there remains a Pareto relevant
marginal externality, i.e., there is room for further bargain-
ing.l7 We can easily show this by supposing that a tax is
levied on j and no compensation is made to i. Firm j's

decision calculus changes because its profit function is

altered to

n

w. = P.-G x. ,x. ,...,x. — B+ x. — x. 3.17

J J (31 32 gm) < P1) 31 kizpk 3k < )
where B is the tax on j's use of le. Now for each value

that B takes on, the marginal condition for le is

 

l6C. Plott, "Corrective Taxes." Plott touched on
this point, but did not demonstrate it explicitly.

17This was most clearly shown by Buchanan and Stubble-
bine, "Externality" and R. Turvey, "On Divergences between
Social Cost and Private Cost," Economica, N.S. XXX (1963)
309-313.

31

dwj/dle = Pj-dG/dle — (B+pl) = O. (3.18)

Recall the definition of Pareto relevance and modify

it to account for the tax:
‘Pi-dF/dzl > lpj~ae/ale — pl — B . (3.19)

Firm j will select quantities of le to fulfill condition

(3.18), but i cannot maximize its profits because dF/dz < 0
for all x.
jl

until the externality is completely removed. Of course,

> 0, i.e., condition (3.19) will always hold

removing the externality requires levying a prohibitive tax
on the use of le. Until le = 0, room for bargaining will
exist; therefore, Pareto equilibrium will not be reached

18

unless x. = 0 so long as i is not compensated.

jl

Mergers

A third solution is the merger alternative. So far,
the desired change in resource allocation has been accom—
plished through market transactions: directly via bargain-
ing and indirectly via government tax-subsidy decisions. A
merger will accomplish the same result by substituting an

entrepreneurial decision for a market transaction. That the

 

18This tax-subsidy alternative is dealt with or
touched upon by Davis and Whinston, "Some Notes"; Coase,
"Social Cost"; 0. A. Davis and A. B. Whinston, "External-
ities, Welfare, and the Theory of Games," Journal of Polit-
ical Economy, June 1962, pp. 241—262; Buchanan and Stubble-
bine, "Externality"; Turvey, "On Divergences"; and Mishan,
"Reflections."

 

32

result will be the same is fairly easy to demonstrate. We
have seen that Pareto relevant marginal externalities permit
gains from trade to exist. And we have also seen that reap-
ing these gains through either the proper imposition of
taxes and subsidies or bargaining yields a Pareto optimum
resource allocation. Since this final equilibrium is
exactly the same as the joint-profit maximizing solution, a
merger clearly offers the same resource allocation. But
recall that bargaining resulted in the two firms' sharing
the gains from trade. In this important respect, division
of the spoils, the merger approach more closely resembles
the bargaining solution than the tax-subsidy solution. Of
course, this will appear in the terms of the merger agree-
ment.

The effect of the merger is to internalize the
externality so that account is taken of its existence when
output decisions are made. Therefore, whenever the adminis—
trative costs of the new, single firm are less than the
costs of bargaining that it replaces, we should expect a
merger to occur.

Continuing to deal with a unilateral, or non-
reciprocal, externality, we must realize that where the
incentive to merge lies depends upon the legal framework.
‘When the law specifies that the offending firm is liable for
damages, the offending firm will be the one interested in a

merger. On the other hand, if the law specifies no such

33

liability, the damaged firm will be anxious to merge. If

0 I I 19
we con31der a mutual, or rec1procal, externality,

i.e., i
imposes an externality on j and j imposes an externality on
i, the likelihood of a merger is increased. The reason for
the increased likelihood is obvious: a merger is now bene-
ficial to both firms. And so long as the market structure

remains competitive, the merger is beneficial to society

0 I I 20
because optimal resource allocation is ensured.

4. Cost Functions and Reciprocal Externalities

The existence of externalities can be represented by
including an output of another firm in the cost function of
the affected firm.21 This alternative view is worth discuss-
ing because it sheds some light on a few difficulties that
have yet to be mentioned. The amount of difficulty we shall

encounter depends primarily upon whether the externalities

are unilateral or reciprocal and whether the cost functions

 

19Mutual externalities are explored in more detail
in the next section.

20The merger solution was suggested by Coase,
"Social Cost": Davis and Whinston, "Theory of Games"; and
Mishan, "Reflections."

210n the derivation of cost functions, see J. M.
Henderson and R. E. Quandt, Microeconomic Theory (New York:
NmGraw-Hill Book Company, 1958), pp. 55-62, 66-67.

34

are separable or non-separable.22 Let us begin with the

easier cases and proceed to the more difficult.

Unilateral Externalities

 

When the externality is unilateral the cost function
of 1 includes an output of j, but j's cost function depends
only upon its own output. This is essentially the case that
has been discussed so far; however, by inspecting the effect
on the cost function, we can see a little more clearly the
effects of an externality on the firm's operation.

Consider an external diseconomy. Firm i‘s cost

function is

Ci = Ci(Qi.Z), (4-1)

and j's cost function is
C. = C. E 4.2
3 3(93) ( )

where Qi is the output of i, z is the externality-causing by-
product of firm j, and Q5 represents the joint products Qj
and 2. Since i cannot control the quantity of z, the first-

order conditions for independent profit maximization are
Pi = Sci/SQi and Pj = acj/an. (4.3)
2Reciprocal externalities and the distinction

ibetween separable and non-separable functions are handled
in detail by Davis and Whinston, "Theory of Games."

35

For Pareto equilibrium, this resource allocation
must correspond to that of joint-profit maximization. It is
quite obvious that this is not the case since the joint

profit function in the competitive case is
77- = Pi'Qi + Ppoj " Ci(Qir Z) " Cj (Qj)! (404')
and the first—order conditions are
aw/aoi = Pi — Sci/SQi = 0, (4.5)
and
a i = P. - 5c. 5 1 — ac. 52 = 0.
57/ Q3 3 3/ Q3 1/

If Sci/dz # 0, conditions (4.5) are not the same as condi-
tions (4.3), and non-optimal output decisions are made
because the deleterious effect of z is ignored.

Any function, f(yl,y2,...,ynL is termed "separable"

if and only if
f(yl.y2.....yn) = fl(y1) + f2(y2) + ..- + fn(yn). (4.6)
In our case, Ci = Ci(Qi’z) is separable if and only if
Ci(Qi’z) = Cil(Qi) + Ci2(z). (4.7)

VVhen this is the case, the marginal cost of producing Qi is
Ilnambiguously defined as dCil/in, i.e., as a function
Shalely of its own output. The consequence of separability

iis that, although the total cost of i is a function of two

36

variables, the marginal cost for i is unaffected by changes
in 2. In graphical terms, the height of the total cost
curve varies with changes in 2, but the slope is the same at
all levels of output. In other words, if TCi is the total
cost curve for Q; = 0, Tc: = TCi + k for some Q; > 0 where

k is constant. Of course, as Q; varies, the value of k will
vary. Since the marginal costs are not affected in this
case, there exists a unique output which will maximize i's
profit regardless of the level of z. The only relevance
that the level of 2 has lies in its effect on i's total
profit. The greater is z, the smaller is vi. But all

this does not mean that the externality has no allocative
effects.23 Clearly, if j takes account of the effect that

2 has on the profits of i there will be a different resource
allocation.

When the externality enters i's cost function in a
multiplicative way the separability condition (4.7) is not
satisfied and the cost function is said to be "non—separa—

24

ble." The effect of non-separability is that when 2

changes the total cost curve is not vertically displaced

 

23O. A. Davis and A. B. Whinston, "On Externalities,
Information and the Government-Assisted Invisible Hand,"
_§conomica, N.S. XXXIII (1966), 304-305.

24In Appendix A, I derived the cost function of a
firm with a Cobb—Douglas production function and of one with
a CBS production function. Neither production function
yields a separable cost function.

37

by a constant amount. The total cost curve will be altered
in some way such that the marginal cost will be affected.
Firm i can no longer define its marginal cost unambiguously
without first knowing the value of 2 since marginal cost is
now a function of its own output and the output of j. There-
fore, i must know the value of 2 before it can make the
correct allocative decisions. But once it knows the value
of z, i can proceed to maximize its profits as best it can.
So far, the cost function alternative has no real
impact on anything done before this section. But this exer-
cise has pointed out that when the cost functions are separa—
ble, the external effects do not affect i marginally. On
the other hand, non-separable cost functions plus external-
ities will give rise to changes at the margin. As long as
we retain the assumption of unilateral external effects, no

new problem arises and all the externality solutions apply.

Mutual Externalities
When externalities are mutual, j's output affects

i's cost function and vice versa, i.e., the cost functions

become
C- = Ci(Qi.Z) (4.8)
and

C- =Cj(Qj.y) (4.9)

38

where y is an externality—causing by-product of 1. Here,

again, independent profit maximization yields
Pi = dCi/in and Pj = de/de (4.10)

as first-order conditions because each firm can only maximize
profits with respect to the variables under its control.
Comparing conditions (4.10) with the joint profit maximizing

conditions

"U
ll

aci/BQi + acj/ay (4.11)

and

'1?!
ll

acj/aoj + aci/az

reveals that non-optimal decisions are made.

The effect of separability in the mutual external-
ities case is to leave the marginal costs of both firms
unchanged regardless of the quantities of y and 2. Thus i
and j can unambiguously define their respective marginal
costs in terms of their own outputs, and, therefore, there
exist unique outputs which will maximize the profits of i
and j individually. This means that one firm's output deci—
sion is wholly independent of the other firm's decision.

Since non-optimal decisions are made, we are inter-
ested in solutions to these problems. For a tax-subsidy
scheme to constitute a solution, the government must be able

'to find the correct outputs for i and j. It can do this by

39

solving the necessary conditions (4.11) for Qiand OS. If we
let t represent the per unit tax and?)i and 65 the optimal

outputs, the correct tax is given by

"d
l
”-
ll

BCi/in (4.12)

P ' t 0 EEC 0 a 3 .

Inspection of equations (4.11) makes it clear that the tax
on each firm should equal the damage done to the other firm.
As before, the tax collected must be paid to the damaged
firm. In a similar manner, the bargaining procedure can be
carried out. Optimal output decisions will now follow
because each firm is made aware of its effects on the prof-
its of the other firm. We should recognize that, when the
cost functions are separable and there are mutual external-
ities, the optimal solution simply involves removing the
Pareto relevance of each externality separately. In other
words, we can deal with one external effect at a time.

When the cost functions are non-separable and the
externalities are reciprocal we encounter a bit of a problem.
Since the marginal cost of each firm depends upon the output
decision of the other firm, each firm would like to wait for
‘the other firm to commit itself before making its own deci-
Sion. It is fairly easy to appreciate this fact when one

<20nsiders that i's output decision changes whenever j's

40

output decision changes and the same is true for j. Such a
situation introduces a measure of uncertainty into each
firm's decision calculus. This certainly can be removed by
internalizing the externalities through a merger of the
firms. So long as the post-merger market structure remains
competitive, a merger would prove to be mutually beneficial
to the two firms and socially desirable since optimal output
decisions would be ensured. Moreover, this solution may
very well be the most practical.

A tax-subsidy scheme might also be devised if the
government knows the cost functions of the two firms. The
solution involves finding new cost functions ci and cj for
the firms such that these new cost functions are single—
valued functions of "own" output and they account for all
social costs.25 Then the use of these new cost functions is
supposed to solve all the problems. But Davis and Whinston
have pointed out some rather serious difficulties with this

method.26

For the new cost functions to account for all social

costs,

dci/in = dCi/in + acj/ay (4.13)

 

25This proposal was suggested in S. Wellisz, "On
External Diseconomies and the Government-Assisted Invisible
IHand," Economica, N.S. XXXI (1964), 358-359.

26

 

Davis and Whinston, "On Externalities," pp. 307-312.

41

and
dcj/de = aci/az + acj/aoj (4.14)

must be satisfied. But if the cost functions are non-separa-
ble, the terms on the right-hand side of (4.13) are func-
tions of Qi and Qj' Therefore, the domain of Qj would have
to be restricted for the function to be single-valued.
There is, however, no a priori reason why the domain of Qj
should be restricted.

If we ignore this problem, what remains is to find
Ql as a function of Qj and Qj as a function of Qi' This
involves solving partial differential equations where no
truly general method of solution exists. But assuming the

equations can be solved, we would have

01 = hj(Qj) and Qj = hi(Qi)' (4.15)

Each of these functions involves a constant of integration.
Since hi and hj represent the taxes or subsidies, correct
values for these constants must be found. Finding these
values requires solving the joint profit maximization prob-
lem for the optimal outputs'—Q-i and'Oj and substituting into
equations (4.15). Then the new cost functions are, by

substitution,

ci = Ci(Qi.hi(Qi)) and cj = Cj(Qj.hj(Qj)). (4.16)

42

Supposing that we have assumed away all the problems
or have overcome them in some way and the new cost functions
(4.16) have been found, the coup de grace may now be applied:
there is no way that the government can force the firms to
use these new cost functions. In all the previous cases it
was in each firm's interest to move to the socially optimal
output, but in this case the firm may not believe that using
the prescribed cost function will not hurt it. If each firm
was previously aware of the other firm's influence on its
cost function, they may not believe that the costs are truly
independent now. Moreover, if, e.g., i does not believe
this and produces some Qi fi'ai, j will experience costs that
the new function, c., does not reflect. This experience
would certainly lend credence to any skepticism j previously
held with respect to the efficacy of using the new cost
function.

From this discussion, one can readily appreciate the
difficulties inherent in the case of mutual externalities
with non-separable cost functions. Davis and Whinston go on
to discuss an iterative procedure which purports to take
care of this case. But the authors admit that they know
hnothing about the speed with which the procedure will con-
Verge to an equilibrium. Although I do not intend to go
into this problem any further, it is worth recognizing the

difficulties that this case presents; especially when a

43

merger is not permitted by the requirement of maintaining a

competitive market structure.27

5. Second-Order Conditions28

To this point, we have focused our attention on the
first-order conditions for a profit maximum and have neglect-
ed the second-order conditions. But these second-order con-
ditions can be ignored no longer since satisfaction of the
first—order conditions does not ensure a maximum. We can
approach this problem through the firm's cost function.
Because profit is the difference between total revenue (R)

and total cost (C), the profit function for i is

Tri = R(Qi) - Ci(Qinle21 ...,Pn) . (5.1)

Of course, optimum output occurs when profit is maximized.
Assuming the functions are twice differentiable, this

requires
awi/aoi = aR/aoi - aci/aoi = 0 (5.2)
and

2 2 _ 2 2 2 2
a wi/aoi — a R/aQi — a ci/BQi < 0. (5.3)

27Mishan pointed out in "Reflections" that the crux
of this problem lies as much in the assumption of reciprocal
externalities as in the assumption of non-separable cost
functions.

28Most of this discussion is based on Samuelson,
QEQundations, pp. 57-62, 76-78 and Appendix A, pp. 357-379.

44

Equation (5.2) says that marginal revenue equals marginal
cost at equilibrium. But this is not enough because equat-
ing marginal revenue and marginal cost will yield minimum
profits when the slope of the marginal revenue curve exceeds
the slope of the marginal cost curve. This event is ruled
out by requiring that condition (5.3) be satisfied. Then,
the optimum output, 6:, is found by solving equation (5.2).

Using cost functions assumes that they express the
least total cost for each level of output. For the total
costs of producingai to be a minimum the marginal produc-
tivity of the last dollar spent must be equal in all uses,
29

i.e., that

dF/dxik 8F/5(-z)

l/lx = T — T— . (k = l,2,...,n) (5.4)

These are the first—order conditions found in the con—
strained cost minimization problem. Since we assume that
costs are a minimum for each output level, these equations
must hold when a; is produced. Samuelson3O has shown that
A = MC, i.e.,
Pk B

MC=7\ =W=W (k= l,2,...,n) (5.5)

It seems that all we must add to the assumption of perfectly

competitive pricing is diminishing marginal productivity and

 

29See Samuelson, Foundations, p. 60. On the sign of
2, see my Appendix B.

 

30Samuelson, Foundations, pp. 65-66.

 

45

we will then ensure satisfaction of the second-order condi-
tion (5.3). This, however, is not true. We must have dimin-
ishing marginal productivity, but we need something more.
The "something more" did not show up because we assumed that
the second—order conditions were satisfied in equations
(5.4) and (5.5), but these equations are not sufficient to
ensure a minimum.

Consider i's profit function where all the inputs

are considered independent variables. Let

Vi = Pi-F(xll,xiz,...,xin,(z—z))
n (5.6)
31
- Z P 'X B(X -x ).
k=1 k k 1 1
As before, the first-order conditions are
dwi/dxik = Pi°dF/8xik - pk = 0 (k = l,2,...,n) (5.7)

and

0.

dwi/d(-z) = PidF/d(—z) - B

Since Pi is marginal revenue, this again says that marginal
revenue equals marginal cost in equilibrium. But it is well
known that "a regular relative maximum requires that the

quadratic form whose coefficients are the second partial

 

31For this slight re-formulation of i's profit
function, see Appendix B.

46
derivatives be negative definite."32 It can be shown that
the negative definiteness of this quadratic form implies
that the principal minors of the Hessian determinant of the
profit function must alternate in sign beginning with nega-
tive. Since Pi > O and it appears in every term, we may

ignore it and write the Hessian as

 

 

F11 F12 '°° Fln F12
F12 F22 °'° an F22
2 2 s 2
F1n F2n °°' an Fnz
Flz F22 "' Fnz Fzz
where Fhk = sz/dxihdxik (h,k = 1,2,...,n,z). Because the

first principal minor, F11, must be negative, we can con-
clude that the marginal productivity of factor xi1 must be
decreasing. Since the numbering of inputs is wholly arbi-
trary, this condition must be invariant under any renumber-
ing of inputs. Therefore, all Fkk must be negative. Thus,
we have diminishing marginal productivity again. But this
is still not enough. There could be a situation where
increases in all factors will yield an increase in profits.
Therefore, we also require that the decrease in a factor's
"own" marginal productivity outweighs the positive effects

on the marginal products of the other factors. For clarity,

consider a two-input case. The Hessian will then be

 

32Samuelson,§oundations, p. 360.

 

47

F11 F12
H = . (5.9)
F12 F22
. 2
For a max1mum, we need Fll < 0 and F11 F22 — (F12) > 0.
The second inequality implies Fll-F22 > (F12)2. We can see

that the "cross" effects of increasing both factors xil and
xi2 must be outweighed by the "own" effects. When these
conditions are satisfied the marginal cost curve will inter-
sect the marginal revenue curve from below and the solution
will represent a maximum of profit. For our purposes, we
will assume that these second-order conditions are satisfied
in all final equilibria. This is not an unreasonable assump-
tion since we are just assuming away saddle points and per-
petual losses. Saddle points are assumed away by convention

and perpetual losses by common sense.

CHAPTER III

THE CONCEPT OF INFERIOR INPUTS

In the preceding chapter we encountered the concept
of inferior inputs. We shall examine this concept in more
detail in the present chapter. Although Professor Hicks
alluded to something akin to inferior inputs, D. V. T. Bear
first defined inferior inputs in a formal manner. Bear
developed his definition under the assumption of competition
in the commodity and factor markets. But he did not explore
the concept much beyond the definition. In an article soon

to appear in Economica, C. E. Ferguson generalized Bear's

 

work to include imperfect competition in the commodity mar—
ket. In addition, Ferguson extended the concept by investi—
gating the consequences that input inferiority has on other
relations. Charles Plott's contribution was to demonstrate
graphically the significance of inferior inputs with respect
to measures taken to correct externality-caused resource

misallocation.l

 

1J. R. Hicks, Value and Capital (2d ed.; Oxford:
Clarendon Press, 1962), pp. 93-94. D. V. T. Bear, "Inferior
Inputs and the Theory of the Firm," Journal of Political
Economy, LXXIII (1965), 287—89. C. E. Ferguson, "'Inferior
Factors' and the Theories of Production and Input Demand,"
Economica, to appear May 1968. Charles Plott, "Externalities
and Corrective Taxes," Economica, XXXIII (1966), 84—87.

 

 

 

48

49

To explore these developments, we shall first set
out a model of jointly-derived demand functions for inputs
under general competitive conditions. Then we shall intro-
duce the definition of inferior inputs and investigate some

of the consequences of input inferiority.
l. A Basic Model of Input Demand

Assume that a firm sells its output under competi-
tive conditions and produces its output according to the

twice differentiable production function

Q = f(xl.x2....,xn). (1.1)

where Q is total output and x1.. is the quantity of input 1.
Competition in the commodity market implies that the firm's

total revenue is

R P°Q = P°f(xl,x2,...,xn), (1.2)

where P is the commodity price.

Competition in the factor markets implies that the
firm accepts the input prices as given. Defining profit as
the difference between total cost and total revenue, the
firm's profit function may be written as

n

w = P~f(xl,x2,...,xn) -i:lpixi, (1-3)

where pi is the price of input 1. The firm attempts to

50

maximize profit by selecting appropriate quantities of the
n inputs.
The first-order conditions for a profit maximum are

obtained by differentiating (1.3) with respect to xi:
dW/dxi = P-f. — p. = O, (i = l,2,...,n) (1.4)

where fi = df/dxi. That is, in equilibrium the price of
each input must equal the value of its marginal product.
Equations (1.4) represent the n jointly-derived input demand
functions in implicit form.

The second-order conditions for a regular relative
maximum to exist at the point in n-space where the n first

partial derivatives vanish require
d w = Z Z (BZTr/dxi-dxj)dxi dxj < 0, (1.5)
13'

i.e., the quadratic form must be negative definite. The

determinant of (1.5) is

 

 

Pfll Pf12 ... Pfln
Pfl2 Pf22 ... Pf2n
N = = Pn F*, (1.6)
Pf1n Pf2n ... ann
where fij = azf/axi-ij. Since P appears in every term, we

may ignore P and consider F*. It is well known that the
negative definiteness of (1.5) implies that the principal

minors of F* must alternate in sign beginning with

51

f11 < 0. Without loss of generality, we may assume n is
even. Thus F* > 0.

Since the numbering of the inputs obviously should
not matter, assume that the price of input 1 changes while
the prices of the other n - 1 inputs are held constant. To
observe the effect of changes in pl, differentiate the first-
order conditions (1.4) with respect to pl. In matrix form,

the result is

 

r' r ' — 7

£11 £12 ... fln axl/apil l/P

f12 f22 ... f2n dxz/dpl 0

. . . . = . . (1.7)
hfln f2n ... fnn hpxn/dp¥_ ._0

 

 

 

 

 

The determinant of the coefficient matrix in (1.7) is pre-
cisely the F* of equation (1.6).

Equations (1.7) can be solved by Cramer's Rule:
= . *
axl/apl Fil/P F , (1.8)

where F11 is the cofactor of the l - 1 element in F*. Since

F* and P are positive,
. = . *
Sign dxl/dpl Sign Fll' (1.9)
But
sign F11 = — sign F* (1.10)

because F* is negative definite. Thus

52

axl/apl = Ffl/P-F* < 0, (1.11)

i.e., the quantity of input 1 demanded is inversely related
to its price. Moreover, because the selection of input 1
was wholly arbitrary we may conclude that this holds for all

inputs:
ax./ap. = F#./P°F* < 0. (j = l,2,...,n) (1.12)
3 J 33
Similarly, solving for dxj/dpl yields
= * o *
axj/apl Flj/P F . (1.13)
Further,
° *
dxj/dpl Z 0 according as Flj Z 0. (1.14)

Thus, a priori we cannot say anything about the Sign of

2
dxj/dpl.

 

2. Inferior Inputs

 

In words, an input is termed inferior if and only if
an increase in its price leads to an increase in the optimal
output of the firm.3 From the production function (1.1),

input 1 is inferior if and only if

 

2This entire section is simply an adaptation of a
model developed by C. E. Ferguson in a manuscript entitled
"Neo-Classical Theory of Production and Distribution," to be
published by Cambridge University Press, January 1969.

3Bear, "Inferior Inputs . . .," p. 287.

53

ao/apl =.
3

“MS

lfj-axj/apl > o. (2.1)

This result can be expressed in more familiar terms, however.
First, differentiate the first-order conditions (1.4) with

respect to P, the commodity price. The result is

fi + P i fij dxj/BP = 0. (j = l,2,...,n) (2.2)

Equations (2.2) may be written in matrix form as

 

 

 

 

 

f11 f12 ... fln OXl/OP ‘ fl
f12 f22 ... f2n dxz/BP f2
. o ' o : = — l/P . (2 o 3)
_f In f2n . . . fnn— laXn/apr _an

 

The negative definite determinant of the coefficient matrix

in (2.3) is

F*=f >0. (i,j

.1,2,...,rn (2.4)

 

 

13'
Clearly, the F* of (2.4) is identically the F* of (1.6).

Using (2.4) and Cramer's Rule, the solutions of

equations (2.3) are

 

2 f. F*.
i 1 ji . . 2
dxj/BP — - P-F* (1,3 — l,2,...,n) ( .5)
where F31 is the cofactor of the j - 1 element in F*.

We have seen that differentiation of the first-order

conditions (1.4) with respect to pl resulted in

54

n
jilfij axj/apl = l/P 511' (2.6)

where 511 is the Kronecker delta. Further, the solutions to

(2.6) are
axj/apl = Fij/P-F*. (j = l,2,...,n) (2.7)

Substitute equation (2.7) into definition (2.1):

 

n
f.-F*.
_ jil 3 13
aQ/apl — P.F* . (2.8)

Since P and F* are positive, we may conclude that input 1 is
inferior if and only if
n

f.-F*. > o. 2.9
3 13 ( )

j 1

By using equation (2.5) in equation (2.8), we shall

find
dQ/dpl = - dxl/dP. (2.10)

In words, the change in optimal output resulting from a
change in the price of input 1 equals the negative of the
change in the optimal usage of input 1 that results from a
change in the commodity price. In the case of inferior
inputs, the change in optimal output due to a change in pl
is positive by definition. Consequently, the change in the

quantity of X1 when commodity price changes is negative, i.e.,

55

when commodity price increases, the quantity of input 1

employed decreases.4
3. An Alternate View of Inferior Inputs

Equation (2.10) provides an equivalent definition of
inferior inputs, viz., one whose use declines as commodity
price, and hence output under perfect competition, increases.
Consideration of this alternative makes the analogy between
an inferior good in consumer demand theory and an inferior
input in the theory of the firm easier to appreciate. But
it will also point out that the analogy is not quite com—
plete. .In the theory of consumer behavior, a commodity is
inferior if the quantity demanded varies inversely with the
consumer's income level. But the consumer is solving a
constrained maximum problem. Thus changing his money income
and re-computing his optimal expenditure pattern will reveal
successive positions of equilibrium.

In the theory of the firm, however, the firm deter-
mines optimal inputs by solving an unconstrained maximum
problem rather than a constrained maximum problem. Total
cost is found by evaluating the cost equation after substi-
tuting this vector of optimal inputs. The firm simply
spends this amount in order to maximize profits. Although

the firm's expansion path is analogous to the consumer's

 

4This section depended heavily upon work done by
C. E. Ferguson in an unpublished manuscript.

 

56

income-consumption curve, the expansion path is not a locus
of profit-maximizing points. But it is a locus of cost-
minimization points and, therefore, every profit—maximizing
point must also lie on the expansion path. Thus we can
direct our attention to these specific profit-maximizing
points for analytical purposes. But the slope of the expan-
sion path can be examined for conceptual purposes.

The firm's response to a demand led increase in i

 

commodity price, ceteris paribus, is an expansion of output.

 

Since the firm was in a profit-maximizing position before

the price change, its iso-output surface was tangent to an
iso-cost surface. To increase its output, the firm must

move to a higher iso—cost surface. When one compares the

two profit-maximizing positions in n-space, one will see

that dxl/BC < 0 where C is total cost, i.e., the tangency

of the new, higher iso-cost surface to the new, higher iso—
output surface involves a diminished employment of the
inferior input, x1. Thus the two concepts of inferiority are
more closely related than the first definition of an inferior

input might have indicated.

4. Inferior Inputs and the Level of Profits

 

We have seen that an increase in the price of input
1 calls for an expansion of output when input 1 is inferior.
But this expansion of output does not affect profits favor-

ably. In fact, the increase in optimum output is accompanied

57

by a decrease in profits. This can readily be demonstrated

by differentiating the profit function (1.3) with respect

to pl:
5W _ . . ' _
5p]. - is (P fi-pi) an/apl "" X1. (1 _ 1’2, ...,n) (4.1)

From the first-order conditions, i.e., in equilibrium, the

terms in parentheses are identically zero. Thus we have

Since xl cannot be negative, - xl must be negative. There-
fore, we may conclude that profit always varies inversely
with input prices regardless of whether the input is infe-
rior or not.5

5. Ipput Inferiority_and Restrictions
on the Production Function

 

 

Input inferiority places certain restrictions on the
form that the production function can take. It can easily
be shown that the production function cannot be homogeneous
of degree one. To this end, suppose it is homogeneous of
degree one. The bordered Hessian determinant of the produc-

tion function is

 

5Bear proved this result in an elegant fashion. But
he asserted that this proved that inputs could not be infe-
rior at all levels of output because if they were, profits
would vary directly with input price. Since he proved the
opposite was the case in gay event, I find his argument
rather unconvincing.

58

0 fl f2 ... fn
fl £11 £12 ... fln
f2 f12 f22 "' f2n

F = . . . . . (5.1)
fn fln f2n ... fnn

 

 

As a matter of notation, number the rows and columns
0,1,2,...,n and let Fi represent the cofactor of the i—th
element in column 0, Fij the cofactor of the i - j element

in the body of F, and F the cofactor of the i - j element

Oij
in F0.

Expansion of F1’ the cofactor of the 1 element in

column 0 yields

n
F =->_‘,f.'F .. (5.2)
l j=l j 013

But linear homogeneity of the production function implies

n

n
2
-'Zlfj Folj = (xl'F/Q )'21 j xj. (5.3)
J: 3:

Since Euler's Theorem applies, we may write (5.3) as
n 2 n
-j:lfj FOlj = (xlF/Q )jilfj xj = xlF/Q (5.4)

Noting that x1, F, Q > 0, linear homogeneity implies

f. F
13

n
= Olj > 0. (5.5)

3

Inspection of F and F* reveals

F0 = F* (5.6)

59

and
FOlj - Flj' (5.7)
Thus
n n
_j:lfj FOlj = -jilfj Flj' (5.8)
Inequality (5.5) then implies that linear homogeneity
requires
n
—j:1fj Flj > 0, (5.9)
or
n
jilfj Flj < 0. (5.10)

Therefore, by (2.9) linear homogeneity precludes input
inferiority.6
Moreover, if the production function is such that an
increase in any input increases the marginal products of all
other inputs, input inferiority is also precluded. To
demonstrate this proposition, suppose that the production
function satisfies this condition and that input 1 is infe-

rior. Differentiating the first-order conditions for a

profit maximum (1.4) with respect to P yields

[fij] [dxi/BP] = - l/P [fi] (i,j = l,2,...,n) (5.11)

 

6This point was mentioned as being obvious by Bear
and was proved by Ferguson.

60

in matrix form. Under the hypothesis,

fij2> o for i s j. (5.12)

3
Equation (5.11) may be solved by matrix methods

It has been proven that all the elements of [fi ]-1

are
negative.
to obtain

[dxi/BP] = - 1/p [fij]-l [fi]- (5.13)

Since fi and P are positive, the right-hand side of (5.13)

must be positive and this implies that

dxi/BP > 0. (5.14)

But substitution of (5.14) into (2.10) implies

dQ/dpl < 0. (5.15)

We clearly have a contradiction of definition (2.1). Thus a

production function in which all marginal products are

increasing functions of all other inputs effectively pre-

cludes input inferiority.7

 

This result was proved by Bear, "Inferior Inputs
. . .," p. 288.

61

6. Ipput Inferioripy and the Cross-Elasticipy
of Input Demand
We can derive the cross-elasticities of input demand
functions from inequality (1.14) and relate this to the con-
cept of inferior inputs. From the first-order conditions,

fj = pj/P. Multiply both inequalities (1.14) by fj and sum

over j:
> . >
l/P Z pj dxj/dpl < 0 according as 2 ijlj < O. (6.1)
3 j

The proportion of total cost spent on input j can be

expressed as

. = . x. .x.. i = l,2,...,n 6.2
x] pJ J/i p1 l ( ) ( )

The price cross-elasticity of input demand between inputs i

and j is defined as

eij = axj/Bpi°pi/xj- (6.3)

To introduce these definitions, multiply the left-hand
inequality in (6.1) by

ZPiXi '§;

 

.9 (.4

Then condition (6.1) can be expressed as

E3121 - z x. e . Z 0 according as 2 f. F*. 2 0- (6-4)
P'pi j :1 13 j J 13

62

ZPiXi
P-pi

Since > 0, the weighted sum of the cross-elasticities

of input demand is negative if input 1 is normal. But when

input 1 is inferior, this sum will be positive.8

7. Inferior Inputs and Output and
Substitution Effects

 

So far, we have been dealing with a profit—maximizing in
firm. This, of course, requires taking cognizance of total
revenue. But prior to any revenue considerations, it is
implicitly, if not explicitly, assumed that each competitive
firm has chosen optimal quantities of inputs such that total

output is maximized for each level of total cost, i.e.,

Q = f(xl,x2,...,xn) (7.1)

is maximized subject to a given cost constraint
2 p.x. = E’ (i = l,2,...,n) (7.2)

where it is still assumed that input prices are paramet—Q
rically given by the market.
To find these optimal input quantities, form the

Lagrange expression

L = f(Xl,X2, '001Xn) _7\(Z pixi - C)! (7'3)

 

8This entire section and the following section are
mere adaptations or restatements of work done by Ferguson in
his unpublished manuscript.

63

where A is an undetermined Lagrange multiplier. The first-

order conditions are

OL/dxi = f- " 7\P- = OI

i i
(7.4)
aL/ax = z pixi - 6': 0. (i = l,2,...,n)
i
A regular relative maximum requires
2 n n
d f = Z 2 f.. dx. dx. < 0 (7.5)
i=lj=113 1 3
for
n
df =.z fi dxi = O, (7.6)
i=1

where not all dxi are zero. This is equivalent to requiring

the bordered Hessian determinant,

0 pl p2 "'p
p1 f11 12 °°' fln
f

p f f ...
D = 2 12 22 Zn ’ (7.7)

pn f1n f2n "° fnn

 

 

to have principal minors that alternate in sign, the first

being positive. From the first-order conditions (7.4)

Pi = fi/A- (1 = l,2,...,n) (7.8)

Thus

 

64

0 f1 f2 ... fn
f1 f11 f12 "' fln
f f f ... f
D = 1/12 2 12 22 2n = l/AZ F. (7.9)
fn fln f2n "° fnn

 

 

Since it can be shown that 1/1 represents marginal

cost, l/x and consequently l/7\2 are positive. Thus
Sign D = Sign F (7.10)

and the signs of the corresponding principal minors of D and
F must also be the same. Further, assuming that n is even
renders the sign of F positive.

In order to find the output and substitution effects,

introduce a change in pl. The result may be expressed in

matrix form:

 

'0 pl p2 ... pn" "Lax/dplf 'in“

pl fll £12 ... f1n dxl/dpl A

p2 £12 £22 ... f2n de/dpl = 0 (7 ll)
pn fln f2n ... fnn dxn/dpl O

 

 

 

 

 

The solutions to equations (7.11) may be found by using

Cramer's Rule and (7.8):

2
"A X1 Fo + i F1

-OA/Opl = 4——ir———- F , (7.12)

 

65

 

and
2 2
-A x F. A F .
_ l J 1:1

These solutions can be expressed in a more meaningful form
by introducing the definition of the partial elasticity of

substitution,

 

.-=- l - —1-l ' (i,j = l,2,...,n) (7.14)
and the identities

F. = (-1)3 2: fk Fojk =(—1)3 >3 fk ng. (k=l,2,...,n) (7.15)

3 k k

Using these in equations (7.12) and (7.13) reveals that

 

 

—-A2>3 fk ka 12 x1 F*
-aA/apl -— F - ——?———, (7.16)
and
Ale z fk Fik A2 x12
a"1/5‘31 = F + m 011 (7'17)
1

Because A and F are positive,

dA/dpl Z 0 according as 2 fk ka + x1 F* 2 O. (7.18)

If input 1 is inferior, z fk Flk is positive while xl-F* is
also positive. But we know that A is the reciprocal of
marginal cost. Thus marginal cost varies inversely with the
price of input 1. Now it should be much easier to appreciate

why the optimal output is higher when the price of an infe-

rior input rises. One certainly should expect a firm to

66

increase its output when marginal cost falls. On the other

hand, if input 1 were normal, Z:f *would be negative

*
kFlk

while x -F* is again positive. But when the price of a

1

normal input rises, optimal output decreases. Therefore
marginal cost must vary directly with pl. Hence xl-F* must

be less than the absolute value of Z fk Flk'

Because F is negative definite, F1 is negative when

1
n is even. Thus 011 is negative. The second term on the
right-hand side of equation (7.17) may be called the substi-

tution effect, which is clearly negative in all cases. This

 

simply means that the quantity of input 1 demanded varies
inversely with changes in input l's price when output is

held constant. If input 1 is normal, the output effect,

 

which is the first term on the right-hand side of (7.17),
will also be negative. In this case of normal inputs, when
input price changes both the output and the substitution
effects operate to move quantity demanded in the opposite
direction.

_On the other hand, if input 1 is inferior, we shall
have a positive output effect. Whether the total effect
will result in an increase or decrease in the employment of
input 1 depends upon the relative Sizes of the two effects.
It is possible for dxl/dpl to be positive if the positive
output effect is larger (in absolute value) than the nega—
tive substitution effect. It must be emphasized that this

does not imply that the slope of the firm's input demand

67

function is positive. We have seen in equations (1.11) and
(1.12) that the slope of the true demand function is always
negative regardless of the type of input considered. Yet we
now find that Bxl/apl may be positive. This apparent incon-
sistency can easily be resolved: in equation (7.17) we are
22E_considering an input demand function. The demand func—
tions for inputs are derived from the profit maximizing
conditions. Since revenue considerations never entered the
calculations underlying (7.17), an input demand function
cannot be derived from the calculus of maximizing output for

given levels of total cost.

CHAPTER IV

THE INFLUENCE OF INFERIOR INPUTS

UPON EXTERNALITY SOLUTIONS

In the previous chapters, the concepts of external
effects and inferior inputs were developed in some detail.
We are now in a position to examine the effects of inferior
inputs on the solutions to the resource misallocation prob-
lem caused by externalities. As previously seen, an exter-
nality in production is caused by one firm's use of an input
having an effect on another firm's production function.

When this situation is further complicated by the external-
ity-causing input's being inferior, we Should like to know
whether the remedies analyzed in Chapter II must be altered.
In order to accomplish this, we shall pose a general problem
of externality in which the input causing the trouble is
inferior. The various solutions to the problem will then be
examined to determine the influence, if any, of input infe—
riority. Since the analysis will be conducted in partial
equilibrium terms, it must be emphasized that the rest of
the economy is, and remains, organized so that the Pareto

optimum conditions are fulfilled.

68

 

69

l. The General Problem

 

Suppose there are two firms, i and j, that produce

according to the production functions:

0' = F(Xillxi21 °°°Ixinl Z), (1'1)
and

Q. = G(x.

j jl'xj2’°'°’xjn)’ (1.2)

where xik and xjk are the amounts of input k used by i and j
respectively. We assume that these production functions are
continuous and at least twice differentiable. The argument

2 in (1.1) is defined by

z = g<le) (1.3)

as in equation (1.3), Chapter II. This means that j's use
of input le yields some accidental by-product or condition
that enters i's production function parametrically.

Further, we shall postulate that i and j operate
in competitive factor and commodity markets and that each
attempts to maximize profits independently. Thus each firm
accepts commodity and input prices as given. Moreover, the
behavioral assumption of independent profit maximization
indicates that each tries to maximize the following profit
functions:

n

7r. = Pi-F(xil,xi2,...,xin,z) -E

1 k pk Xikl (1.4)

l

70

and

w. = P.oG x. ,x. ,...,x. — x. . ,
3 3 ( 31 32 3n) 2 pk 3k (1 5)

While there is no problem for j, i cannot satisfy all its
first-order conditions so long as dF/dz # 0.
Employing the definitions established in Chapter II,

we shall assume that 2 creates a marginal externality, i.e.,
aF/az < 0. (1.6)

Further, we shall assume that the external diseconomy is

Pareto relevant, i.e.,
Pi-dF/dzl > le-dG/dle - pl , (1.7)

where Pi and Pj are product prices and p1 is the price of
input le. This simply means that not only would i like to
change j's use of input le, but that it is also possible for
i to do so. A change is desirable because j's use of le has
a deleterious effect on i's operations. Moreover, a change
is possible because the damage done to i exceeds the net
marginal benefit that j receives from its profit-maximizing
usage of le.

So long as inequality (1.7) holds, i.e., so long as

the externality is Pareto relevant, there is a misallocation

 

1We must keep in mind that dF/dz =
5F/59(le)°d9(le)/dle.

71

of resources that prevents society from being on its trans—
formation surface. Only by removing the Pareto relevance of

the externality, i.e., by changing inequality (1.7) to

lPi-dF/dz

 

= l j as/ale - pl , (1.8)

can society attain its transformation surface. We might
note again that this does not require removal of the mar-

ginal externality, but only the removal of its Pareto rele- ,

 

vance.
Finally, assume that le is an inferior input. As

we have seen, this requires

n
an/apl = : Gk axjk/apl > 0, (1.9)

k l
where Gk = dG/dxjk. By introducing this additional influ-

ence into the externality problem, analysis of the solutions
set out in Chapter II, "Solutions for the Externality Prob-

lem," will reveal whether-any complications result.2

2. Solutions

 

As in Chapter II, we shall only consider procedures
that permit society to attain its transformation surface.

In particular, we shall again examine the bargaining,

 

2We shall retain the assumptions made when we first
dealt with these solutions, viz., the costs of adjustment
are lump sum and the adjustment is worthwhile. See footnote
13 p. 21 for a discussion of these costs and assumptions.

72

tax-subsidy, and merger solutions under the additional

influence of inferior inputs.

Bargaining_

 

As in Chapter II, we must consider the legal pre-
scription for liability as given and analyze the problem of
external effects within this context. We shall begin with

the case where the law prescribes no liability for j, the

a.

source of the externality, and then proceed to the case
where j suffers full liability.

No liability foryj.—-In this situation i is inter-
ested in j's reducing the quantity of le employed. Because
of the inequality in condition (1.7), i.e., because the
external diseconomy is Pareto relevant, it is possible for
i to induce j to decrease its use of le with a resultant
mutual advantage. Since there is no legal restriction on
j's actions, 1 must make a monetary appeal to j. Assuming
that j's optimal use of le is 251, the presumption is that
i will offer an amount B(§jl—le) to j.3 In other words, B
is the side payment or bribe that 1 offers j per unit
decrease in j's employment of le. Recall that le is the
input that is directly related to z, the externality-causing
by-product or condition, which enters i's production func-

tion parametrically.

 

3Of course, no payment will occur if the term in
parentheses is negative, i.e., if j increases its use of le.

73

The fact that le is inferior has no effect on 1
because i is merely interested in removing the deleterious
influence that le creates. If the inferiority of le has
any influence at all it will appear in j's response to i's
offer. We can examine the profit function that j now
attempts to maximize:

n
w. = P..G x. ,x. ,...,x. + B i1 -x. — x. .
J 3 ( 31 32 3n) ( :11 31) kilpk 3k

(2.1)

The first-order conditions for a regular relative maximum

are

Pj-dG/dxjk - pk = 0 if xjk > O, (k = 2,3,...,n) (2.2)
and
Pj°dG/dle - (pl+B) = 0 if le > 0. (2.3)
Since B > 0,
191 < (pl+B) . (2.4)

Thus i's offer is equivalent to an increase in the
price of le. This is what one should expect. The side
payment from i to j is exactly equivalent, from j's point of
view, to an increase in the market price of le. That is
just to say,when externalities exist, some nonmarket mech-
anism must accomplish what the market would otherwise do.
Whether such a price increase will induce the desired change

in j's employment of le is now the relevant question.

 

74

But the answer to this question has already been
obtained in Chapter III, equation (1.11). There it was
found that the quantity demanded of any input is inversely
related to its price. This relation holds whether the
input is inferior or not. Therefore, we may conclude that
through an iterative procedure, the correct B can be found
such that not only will equations (2.2) and (2.3) hold, but

the first—order conditions for i will also be fulfilled,

i.e.,
Pi-dF/dxik - pk = 0 if xik > 0, (k = l,2,...,n) (2.5)
and
_ 4
Pi-aF/az + B — o. (2.6)

By solving equation (2.6) for B and substituting
into (2.3), we shall find that the bargaining scheme will
result in satisfaction of the requirement for Pareto opti—
mality. Thus the inferiority of input le has no effect on
the bargaining solution in regard to this policy for achiev-
ing Pareto optimality when the law prescribes no liability
for j's actions. But this inferiority does have an effect

upon the resultant configuration of final prices and output.

 

4These first-order conditions follow from i's

altered profit function: n

”1 = Pi°F(xil’xi2”°°’xin’z) - B(le—le) _k:1 pk Xik'

 

75

From equation (2.6),
B = - Pi-dF/dz. (2.7)
Substituting into (2.3) and rewriting yields
Pj°dG/dle = pl - Pi-dF/dz. (2.8)
Since dF/dz < 0 by condition (1.6),

pj-ae/ale = pl + lpian/az . (2.9)

 

Thus j must adjust its usage of le such that the value of
le's marginal product exceeds its market price by the value
of the damage done to i. In other words, when the side pay-
ment or bribe is offered, the "effective market price" of
le, from j's point of View, becomes pl + Pi'dF/dz . Firm
j will then reduce its employment of le. Normally, reduc-
ing le would cause a reduction of output and, if all entre—
preneurs followed suit, the market price of commodity j
would increase. But in the special case of le being an
inferior input, output varies directly with the price of the
input by definition. Thus the increase in the "effective
market price" of le causes an expansion of j's output even
though the employment of le is still reduced. Now, if all
entrepreneurs follow suit, the market price of commodity j
will fall.

Full liability for j,--Now assume that the law

prohibits j's imposing any externality upon 1 unless i is

76

willing to accept it. It is quite obvious that i will
require some compensation in return for its permission to
suffer the existence of 2. It is also quite obvious that

A
if le is the optimal quantity of le and

/\

Pj°8G/8le - p1 > 0 for le < le, (2.10)

j will find it profitable to compensate i for permission to
use some positive amount of le.
Suppose j offers 1 a bribe of B per unit of le for

i's permission to employ le. Firm j's offer will change

their profit functions to

n
Hi 2 Pi°F(Xil’Xi2"°"Xin’z) + B le -k§1 pk xik’ (2.11)
and
n
w. = P.-G x. ,x. ,...,x. - B+ x. - x. . 2.12
3 j(3132 3n) (P1)31k§29k3k( )

Since these profit functions are essentially the same as
those where no liability existed, we shall find that the

first-order conditions are the same, i.e.,

Pj-dG/dxjk - pk = 0 if xjk > O, (k = 2,3,...,n) (2.13)
Pj-oG/dle - (B+pl) = 0 1f le > 0, (2.14)
Pi°8F/dxik — pk = 0 if xik > 0, (k = l,2,...,n) (2.15)

and

Pi°5F/dz + B = O. (2.16)

 

 

77

In light of the results (2.13) - (2.16), we can see that the
effect of j's offer of a bribe is equivalent to an increase

in the price of input le. Because we have shown that,

0
O

irrespective of the type of input under consideration, an
increase in input price causes a decrease in the quantity
demanded, this legal arrangement will also lead to Pareto
optimal input combinations. The direction of payment is
reversed in this instance; but this is irrelevant to ques-
tions of efficiency.5

Here again, however, the inferiority of input le
will affect final prices and output. In order for j to
employ le, j must make a side payment to i in addition
to Paying the usual market price for the input. Thus the
"effective market price" is increased from j's point of view.
Of course, this results in j's decreasing its use of le
below the level that would prevail in the absence of any
legal liability. But, because le is inferior, output and
input price vary directly. Therefore, the increase in the
"effective market price" causes an expansion of j's output.
Again, if all entrepreneurs follow suit, the market price of

commodity j will fall.

 

5The intermediate case mentioned in Chapter II,
"Bargaining," could also be analyzed as these polar cases
have been. .Clearly, the result would be the same: Pareto
optimality would ensue.

 

78

Taxes and Subsidies

 

As we have seen, the tax-subsidy approach represents
an alternative to the bargaining scheme.6 Since j's use of
input le is causing the external diseconomy, this alterna-
tive solution requires the levy of an appropriate tax on j's
use of le and an equal subsidy payment to i. Although the
result of this approach is fairly obvious, it is worth
demonstrating.

Let T represent both the tax on j's use of le and

the subsidy to i. The profit functions of i and j now

become
n
W1 — P.-F(xil,xi2,...,x.n,z) + ijl —k:lpk Xik’ (2.17)
and
n
Wj = PjG(Xj1'Xj2’°'°’Xjn) - (T+Pl) le -kizpk Xjk' (2.18)

Thus the first-order conditions are

= 0 if x > 0, (k l,2,...,n) (2.19)

Pi°5F/axik ‘ pk ik
Pi-dF/dz + T = 0, (2.20)

2,3,...,n) (2.21)

Pj-dG/dxjk - pk = 0 if xjk > 0, (k

and

pj-aG/ale — (T+pl) = 0 1f le > 0. (2.22)

 

6See Chapter II, "Taxes and Subsidies."

 

79

Equation (2.20) shows that the appropriate tax to be levied
on j is one that equals the value of the damage done to 1.
Clearly, these first-order conditions are identical
to those obtained by private bargaining. If we solve equa-
tion (2.20) for T and substitute into equation (2.22), we

obtain
pjoaG/ale - pll = lPi-dF/dzl, (2.23)

i.e., we obtain the condition for Pareto equilibrium.

Rearranging this equation to
rj-aG/ale = pl + lPi-dF/dzl (2.24)

shows that the effect of the tax is to induce j into equating
the value of le's marginal product with the price of le
plus the value of the damage its use does to i's operation.
Again, input inferiority has no effect on this policy for
achieving Pareto optimality because the price of le is
raised by the tax, and this causes a decrease in the quan—
tity of le demanded.

But from equation (2.20L

T = - Pi-dF/dz (2.25)

Thus the tax and subsidy are precisely the same as the bribe
or side payment in the bargaining case. As one might expect,

the results will be the same: the increased "effective mar-

ket price" of le will cause a reduction in the employment

 

80

of le and a consequent increase in j's output. As before,
if all entrepreneurs follow suit, this will lead to a reduc-
tion in the market price of commodity j.

The result of taxing j's output when le is inferior
can now be demonstrated.7 If, despite all objections by
economists that taxing output will render j's choices for
all inputs but le non-optimal, the government decides to
tax output to remove the Pareto relevance of the external
diseconomy caused by j's operation, the direct opposite of
the desired result will occur when le is an inferior input.
To demonstrate this, suppose a per unit tax t is levied on

j's OUtPUt Qj' Firm j's profit function will then be

n
w. = P.°G x. ,x. ,...,x. - x. — t Q.. 2.26
J J (31 32 3n) k=lpk 3k 3 ( )
Because

. = G x. ,x. ,...,x. , 2.27
Q3 (31 )2 3n) ( )

equation (2.26) can be re-written as

n

v. = P.-t 'G X. ,x. ,...,x. - z x. . 2.28
J (J ) (31 32 JH) k=lpk 3k ( )

Clearly, the effect of the per unit tax is to decrease the
"effective market price" of j's output rather than to
increase the "effective market price" of the input that
causes the externality. We have seen in equation (2.10),

Chapter III,that

 

7This was mentioned in Chapter II, "Taxes and Sub-
sidies," without explanation. The result was demonstrated
graphically by C. Plott in "Externalities and Corrective
Taxes," Economica, XXXIII, 84-86.

 

81

de/dpl = - dle/dpj (2.29)

in all cases. But the definition of input inferiority
requires that de/dpl be positive. Thus expression (2.29)
implies that the quantity of the inferior input le must
vary inversely with commodity price. Since le causes the
externality and the tax is supposed to remove this effect,
we can now see the perverse result that Plott's graph
demonstrated, viz., a tax on Qj' by decreasing its effective
market price, causes j to increase its employment of le.

This is obviously the opposite of the desired result.

Mergers

We can now turn to the third solution: merger of
the firms involved. The effect of the merger will be to
internalize the external diseconomy. This occurs because

the new firm's profit function is

n
W = Vi + Wj = Pi-F(xil,xiz,...,xin,z) -k§1pk xik
n (2.30)
+P..G X. ,X. '00.,X. -2 X. o
3 ( 31 32 3n) kzlpk 3k

The first-order conditions for profit maximization are

k — 0 if xik > O, (k = l,2,...,n) (2.31)

pk 0 1f xjk > 0, (k

Pi-dF/dxik - p

2,3,...,rn (2.32)

P.’8G dx. -
J / 3k
and

Pi°dF/dz + Pj-dG/ale — p1 = 0 if le > 0. (2.33)

82

It is obvious that condition (2.33) can be rearranged to

read

lPi-dF/dz

 

= le-dG/dle — pl , (2.34)

which is the condition for Pareto equilibrium.

While the other proper solutions involved raising
the price of the externality-causing input, the merger solu-
tion yields the desired result for a slightly different
reason. The new, merged firm fully appreciates all the
costs of employing le, i.e., the decrease in Oi for any
given vector of inputs xil’xi2"°"xin as well as the price
of le. Thus the optimal amount of le will be used. Since
there are no longer any external effects, the fact that le
is inferior can have no bearing on questions of Pareto
optimality. However, as in the bargaining and tax-subsidy

cases above, the fact that le is inferior leads to a differ-

ent output configuration than would otherwise Obtain.

3. Conclusion

 

In Chapter III, we saw the effects of input inferi-
ority on the theory of production and the theory of derived
demand. But in this chapter we found that input inferiority
does not alter any of the solutions to the resource misallo-
cation problem created by external effects. Since the possi-
bility of the existence of inferior inputs cannot be ignored

and their empirical Significance is quite difficult to

83

assess, this is a rather encouraging result. Moreover,
since external effects cause enough complications by them-
selves, it may be a blessing that inferior inputs do not
further complicate matters.

In the case of a non-inferior input, which is the
cause of a Pareto relevant externality, full employment of
all resources leaves society short of the transformation
surface because the full utilization of the externality-
causing input has a deleterious effect upon the operation of
some firm or firms. When the Pareto relevance of the exter-
nality is removed by bargaining, tax-subsidy manipulations,
or internalization through merger, the offending firm
employs less of the externality-causing input and conse-
quently produces less output. The inputs released by the
offending firm will then be employed by other firms whose
usage of the inputs does not lead to external effects. This
will expand their outputs and permit society to attain the
transformation surface.

But if the externality-causing input is inferior to
j and non—inferior to other firms, the result is somewhat
different. Removal of the externality's Pareto relevance
through bargaining, tax-subsidy manipulations, or internal-
ization will still cause the offending firm to decrease its
usage of the inferior, externality-causing input; however,
the firm's output will now increase as we have seen above.

The released inputs will now be transferred to firms whose

84

usage of the inputs does not give rise to external effects.
Since the externality—causing input is not inferior to these
firms, their outputs will also increase. Thus, in this case,
removal of the externality's Pareto relevance actually

causes an outward shift of the transformation surface.

APPENDIX A

DERIVATION OF COST FUNCTIONS

WITH EXTERNALITIES PRESENT

While the term cost equation refers to cost expressed

 

in terms of quantities of inputs and their respective prices,
the term cost function denotes cost as a function of output.
Cost functions can be used in the profit maximization problem
by assuming that the firm employs optimum input combinations

for all levels of output. Then the profit function becomes:

W P-Q - f(Q) (A.l)

where P is the product price, Q is the quantity of output,
and f(Q) is the cost function. The first-order condition

for profit maximization is now

dW/dQ = P - df(Q)/dQ = 0. (A.2)

Since df(Q)/dQ is obviously marginal cost, we have the usual
condition for profit maximization: price equals marginal
cost. The derivation of the cost functions, assuming the
presence of an externality, can be revealing for our purposes.
Therefore, I shall derive the cost function for a firm oper-

ating in the presence of an externality under the assumptions

85

86

of (1) a Cobb-Douglas-type production function and (2) a

CBS—type production function.

Cobb-Douglas

 

Let the production function be

a b c

Q = A X1 x2 2 (A.3)

where xl,x2 are factors of production under i's control and
z is the externality-causing output of j that enters i's
production function parametrically. Differentiating the

function partially with respect to the inputs yields the

following marginal products:

MP of X1 dQ/dxl = aQ/xl, (A.4)

MP of x2 - dQ/dx2 bQ/x2,

and
MP of z = 50/52 = cQ/z.

In accord with the definitions set out in Chapter II, "The
Definition of Externality," we have an external diseconomy
if dQ/dz < O which requires c < 0. On the other hand, c > 0
implies 80/82 > O, and we would have an external economy.
Since the value of z is given parametrically, the

necessary conditions for profit maximization are

a°X2/b’xl = pl/pZI (A05)

87

where pl,p2 are the factor prices. Solving equation

for x2 and substituting into equation (A.3) yields

Q = A~kbozcoxi+b,
where k = pl-b/pza. Then

x1 = (Q/Akb.zc)l/a+b
and similarly

x2 = (Q/A.k-a.zc)l/a+b.

(A.5)

(A.6)

(A.7)

(A-B)

Substituting these results into the cost equation

 

gives us the cost function:

 

b c)1/a+b

C = pl.(Q/A.k .z C)l/a+b

+ pZ-(Q/A-k-a-z

(A.9)

An interesting point to note is that this cost function is

not separable. Let us next consider a CBS—type production

function.

CBS-Type

Let the production function be

+ a ~zd l/g.

_ b b
Q — A(al-xl + a2 x2 3

In this case, the marginal products are

b—l
Qal'b°xl /g(Z):

MP of X1 dQ/dxl

b-l
MP of x2 — aQ/ax2 Q~a2-b'X2 /9(Z),

(A.10)

(A.11)

88

and
\ d-l
MP of z = oQ/dz = Q-a3-d-z /g(Z)

where Z represents the term in parentheses in equation
(A.10). .Again, if d,< 0, we have an external diseconomy:
if d > 0, we have an external economy. Introducing z

parametrically, we have as necessary conditions

b—l b-l _
l.xl /a2.x2 — pl/pz. (A.12)

a
Solving (A.12) for X1 yields

_ b-l 1/b-l
xl - (pl.a2.x2 /p2.al) . (A.13)

Substituting into equation (A.10) and letting W =

(plaZ/p2a1)b/b'l, we have
0 = A(al.w.x§ + a2.x§ + a3-zd 1/9. (A.14)
Then
x2 = ((Q/A)g-a3°zd l/b (al-W+a2)-l/b (A.15)
and similarly
x1 = ((Q/A)g-a3'zd l/b - (al+a2-W-l)-l/b. (A.16)

Now, substituting these results into the cost equation gives

us the cost function:

l/b l/b
(Q/A)g - a3-zd (Q/A)g — a3 zd
C : pl. 1 "l' p2“ ' (A'l7)

a1 + a2-W al-W + a2

 

 

89

Note that in this case we also do not have separable cost
functions. Further, we can see the effect of the external-
ity in equations (A.4) and (A.11): the marginal products of
the firm's own inputs are decreased in the diseconomy case
and increased in the external economy case.

An Example of a Separable

Cost Function

Let the production function be

Q = x? + x; + ZB‘ (A.18)
The marginal products are
_ _ o-l
MP of X1 — dQ/dxl — oxl ,
MP of x2 = aQ/ax2 = oxza-l, (A.19)

MP of z = 80/82 = BZfi-l.
If dQ/dz < 0, 2 causes a marginal external diseconomy.

The necessary conditions for a profit maximum are

 

x101-1 pl
x d—l =-—- . (A.20)
2 p2
Solving (A.20) for X1 yields
p x o—l l/d-l
x = —l——3——— . (A.21)
1 p2

Substituting into the production function gives

 

 

p x a-l a/o-l
l 2 a 8
Q _ p2 + x2 + Z . (A.22)
Thus
5 l/o
+Z
x2 = 479* o/o l , (A.23)
(pl/p2) +1
and,similarly,
B l/d
x = 9+3 .
l —1
l + (pZ/pl)a/a
These values for X1 and x2 can be substituted into

the cost equation to obtain the cost function:

 

 

 

 

 

 

B 1/a B l/o
C = pl + Zo/o—l + p2 g+0501-1 °(A’24)
l + (92/91) (pl/p2) + 1
Since this cost function can be written as
l/o l/o
C=p Q? +p 25
l -l 1 -l
1 + (pz/pl)a/a . 1 + (pz/pl)a/a ( 25)
A.
l/d 1/d
+ P Q—T‘ + P 26
2 -1 2 -l ’
(pl/p2)“/“ +1 (pl/p2)a/a + 1

this cost function is separable. That this result occurs

is apparent from the form the marginal products take, i.e.,
they are independent. It appears that so long as the exter—
nality-causing input enters the production function in a
purely additive way, the resultant cost function will be

separable.

 

APPENDIX B

AN ALTERNATIVE VIEW OF THE

EXTERNALITY—CAUSING INPUT

I have been assuming that successive increases in
2 will cause increasing diminutions of output for i. The
inclusion of z in i's production function makes it difficult
to visualize the corresponding isoquants, or more correctly,
the corresponding iso-surfaces. We may be able to gain
greater insight by altering some of our concepts. First,

let us consider i's production function:

Qi = F(Xil'xi2"'°’xin’z)' (B.l)

Note that I have replaced z with E'to indicate that j has
decided upon its own optimum output and, therefore, upon the
quantity of its inputs and, in particular, the quantity of
le. Recall that z = g(le). When the production function
has a specific form and a definite value is assigned to 2 we
know the output possibilities that confront i. Any of these
possibilities may be attained by employing certain quanti-
ties of the n inputs. Naturally, the particular output and
the corresponding vector of inputs decided upon will depend
upon the price of the product and the prices of the inputs.

The conceptual problem arises when we allow 2 to vary

91

92

because a positive 2 imposes a negative benefit on 1.

Although it is natural for one to recognize that the xik

l,2,...,n) represent positive changes from zero, it is

(k
somewhat unusual to think of E'as the origin for changes in
z from i's point of view. But I am suggesting we do this so

that firm i's production function becomes

12-2) 0 (B.2)

All the constructs that were previously developed
could easily be reworked in these terms. The advantage of
using this formulation is that when i undertakes to change
the value of 2, we can differentiate equation (B.2) with
respect to -z. The marginal product of i's increased employ-
ment of negative units of z is then positive. This altera-
tion will render convex to the origin any isoquant's projec-
tion onto the relevant plane in (n+l)-space. In a three—
dimensional example, all we need do is label the axes

correctly: y

 

 

-Z

93

Note, in particular, the labeling of the —z axis and
the definition of the z origin. Units of negative 2 are now
"proper" inputs and the firm's iso-surfaces are concave from
above. In addition, the assumption that increases in 2
yield increasing diminutions of i's output implies that the
marginal productivity of negative units of z is decreasing,
i.e., 82F/8(—z)2 < 0. It is now clear why we previously
treated 2 as a nearly "normal" input: negative units of z

are normal inputs.

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