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EFFICIENCY IN FRONTIER PRODUCTION FUNCTIONS

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RICHARD ALMY BARCLAY

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EFFICIENCY AND FRONTIER PRODUCTION FUNCTIONS

BY

Richard Almy Barclay

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Agricultural Economics

1986

Copyright by
Richard Almy Barclay
1986

ABSTRACT

EFFICIENCY AND FRONTIER PRODUCTION FUNCTIONS

BY

Richard Almy Barclay

This dissertation raises serious questions regarding
the validity of frontier production functions which are
considered better than traditional production functions
because they distinguish between "technical efficiency"
(TE) and "price efficiency" (PE).

In empirical work there are always variations in the
data. Frontier production functions represent a new
theory explaining the cause of these variations predicated
on the existence of an interior to an isoquant at the same
level of production. This dissertation first explores
unit isoquants and solid production possibilities sets.
It is shown that specification and/or aggregation error of
the sub-production function, due to a lack of attention to
the fixed inputs, account for the variations attributed to
TB and PE. It is also shown that the discrepancies between
observations within a unit isoquant attributed to differ-
ences in TE are due to the averaging process itself, so
that the unit isoquant represents the boundary between
Stages I and II of production.

Frontier production functions and distance functions

are frequently regarded as the same. It is demonstrated

Richard Almy Barclay

that the principles of duality implicit in distance
functions conflict with the distinguishing characteristics
of frontier functions. Specifically, duality proves that
TE and PE are always identically equal by definition.

The two appendices present background theory and
agruments that substantiate the thesis that "TE" and "PE"
:nepresent specification and/or aggregation error. A brief
review of thermodynamics shows that thermal efficiency and
economic efficiency are the same, and that "technical

inefficiency" as used in the frontier production function

literature is physically impossible.

TABLE OF CONTENTS

Page
LIST OF TABLES ..................................... vi
LIST OF FIGURES .................................... vii
CHAPTER
1. INTRODUCTION ................................. 1
1.1. Variation In Observations In Applied Work 1
1.2. Why Frontier Production Functions Are
"Best" .................................. 4
1.3. Why Is This Important? ................. 8
1.4. Outline Of Chapters .................... 10
1.4.1. Chapter Two: Unit Isoquants And
Production Solids ............... 10
1.4.2. Chapter Three: Frontier Functions
And Distance Functions .......... 11
1.4.3. Chapter Four: A New Case ........ 12
1.4.4. Chapter Five: The Frontier
Production Literature ........... 13
1.5. Outline of Appendices .................. 13
1.5.1. Appendix One: Efficiency And The
Laws Of Thermodynamics .......... 14
1.5.2. Appendix Two: Efficiency In
Economic Theory ................. 15
1.6. Summary ................................ 19
2. UNIT ISOQUANTS AND PRODUCTION SOLIDS ......... 21
2.1. Unit Isoquants ......................... 22
2.1.1. What Is "Technical Efficiency?" . 22
2.1.2. What Is "Price Efficiency?" ..... 25
2.1.3. What Is "Economic Efficiency?" .. 25
2.1.4. Summary of Unit Isoquants ....... 27
2.2 Production Solids ...................... 29
2.2.1. Case 1 .......................... 31
2. 2. 2. Case 2 .......................... 33
2.3. The Explanation For "Observing" Cases 1
And 2 .................................. 34
2.3.1 The Specification Problem ....... 35
2.3.2. The Aggregation Problem ......... 37
2.4. Summary ................................ 38

3. FRONTIER FUNCTIONS AND DISTANCE FUNCTIONS .... 41

3.1. Test Of The First Hypothesis ........... 43
3.1.1. Why Frontier Production Functions
Appear To Be Distance Functions . 42

ii

 

Page

CHAPTER
3.1.2. Frontier Production Functions Are
Not Distance Functions .......... 46
3.2. Test Of the Second Hypothesis .......... 49
3.2.1. Conditions For the Tests ........ 50
3.2.2. Why Frontier Production
Functions Are Not Compatible
With Duality Theory - First Part 56
3.2.2.1. The Intuitive Argument . 57
3.2.2.2. Proof Using Duality .... 59
3.2.2.3. Proof Using Free
Disposal ............... 60
3.3.3. Why Frontier Production Functions
Are Not Compatible With Duality
Theory - Second Part ............ 63
3.3.3.1. The Intuitive Argument . 70
3.3.3.2. Proof By Duality ....... 71
3 4 Summary ................................ 77
4 . A NEW CASE ................................... 82
4.1 Is This A New Definition of TE ? ....... 86
4.2 TE Not Separate From PE ................ 87
4.3 'Interior' And 'Exterior' Points ....... 88
4.4 Summary ................................ 89
53. THE FRONTIER PRODUCTION LITERATURE ............ 91
5.1. Linear Programming And Free Disposal .... 92
5.1.1. Koopmans ......................... 92
5.1.2. Boles ............................ 94
5.2 Frontier Production Function Theory ..... 95
5.2.1 Farrell .......................... 95
5.2.2. Bressler ......................... 100
5.2.3. Nerlove .......................... 105
5.2.4. Yotopoulous ...................... 110
5.3 Formal Microeconomic Theory ............. 113
5.3.1. Henderson And Quandt ............. 113
5.3.2. McFadden ......................... 116
5.4. Other Literature ........................ 117
5.5. Summary ................................. 120
5. CONCLUSION ............................ . ....... 122
6.1. Variations In Data In Applied Work ...... 122
6.2. Isoquants Do Not have Interiors ......... 124
6.3. The Interiors In Frontier Production

Function Theory And "Technical
Efficiency" ............................. 126

 

CHAPTER

A1.

A2.

Page
6.4. Interiors To Frontier Production
Functions Would Mean "TE" # "PE" ........ 128
6.6. There Is Only One Type Of Efficiency .... 130
6.6. Final Considerations .................... 130
APPENDIX ONE: EFFICIENCY AND THE LAWS OF
THERMODYNAMICS .................... . .......... 133
A1.1 Definitions From Thermodynamics ......... 134
A1.1.1. Systems ......................... 135
A1.1.1.1. States ................. 136
A1.1.1.2. Process ................ 137
A1.1.1.3. Property ............... 138
A1.1.1.4. Homogeneity ............ 139
A1.1.1.5. Reversibility .......... 140
A1.1.1.6. Cycle .................. 142
A1.1.2. Work And Heat ................... 142
A1.1.2.1. Work ................... 142
A1.1.2.2. Heat ................... 145
A1.1.3. Summary Of Definitions .......... 146
A1.2. The Laws Of Thermodynamics ............. 147
A1.2.1. The First Law ....... ... ........ . 147
A1.2.2. The Second Law .................. 148
A1.2.3. Summary Of The First And Second
Laws ............................ 150
A1.3. Technical Efficiency ............. . ..... 151
A1.4. What Is Wrong With Frontier Production
Functions From The Perspective Of
Thermodynamics ......................... 154
A1.4.1. The First Situation ............. 154
A1.4.2. The Second Situation ............ 155
A1.5. Summary ..... ........................... 156
EFFICIENCY IN ECONOMIC THEORY ................ 158
A2.1. Production Theory ..... . ........ . ....... 159
A2.1.1. What Are Inputs And Outputs ..... 159
A2.1.2. Production Sets ................. 161
A2.1.2.1. Production Possibilities
Sets ............. . ..... 162
A2.1.2.2. Technical Change .. ..... 162

A2.1.2.3. Production Functions ... 163
A2.1.2.3.1. Sub-Production
Functions .... 164
A2.1.2.3.2. Input

Requirement
Sets ......... 168
A2.1.2.3.3. Isoquants .... 169
A2.1.2.4. Distance Functions ..... 170

iv

 

Page
CHAPTER
A2.2. Assumptions About Production Sets ...... 174
A2.2.1. Consequences of Fixed Inputs .... 176
A2.2.1.1. Constant Returns To
Scale .................. 177
A2.2.1.2. The Law Of Diminishing
Returns ................ 178
A2.2.1.3. Stage III Of Production 180
A2.2.2. Consequences of Monotonicity .... 182
A2.2.2.1. Free Disposal .......... 182
A2.2.2.2. Duality Theory And Polar
Reciprocal Sets ... ..... 185
A2.3. Maximizing Behavior .................... 187

A2.4. What Is Wrong With Frontier Production
Functions From The Perspective Of

Economic Theory . ....................... 188

A2.5. Summary ................................ 192

BIBLIOGRAPHY ........................................ 194
v

 

“‘A

1L1.1.

LIST OF TABLES

Summary Of Various Work Equations

vi

 

LIST OF FIGURES

Page
2.H1 A Unit Isoquant With A Budget Constraint ........ 23
2..2 Technically Efficient And Inefficient Points Of
Production Within A Production Possibilities Set 26
:3. 1 A Unit Isoquant Derived From A Frontier
Production Function (A) And An Input Requirement
Set Derived From A Distance Function (B) ........ 44
3 . 2 An Input Requirement Set V(y) From A Distance
Function (A) And A Factor—Price Requirement Set
R(y) From A Cost Function (B) ................... 52
3 - 3 An Input Requirement Set V(y) From A Distance
Function (A) And A Factor-Price Requirement Set
R(y) From A Cost Function (B) ................... 58
3..4» An Input Requirement Set V(y) From A Distance
Function (A) And A Factor-Price Requirement Set
R(y) From A Cost Function (B) ................... 72
E 4. 1 A Production Function (Surface) Showing Total
3 Physical Product (TPP), Average Physical Product
(APP), And Marginal Physical Product (MPP) Of v1
' And V2 In Least Cost Combination Given 23
f Equal To A Constant ............................. 83
4 - 2 A Unit Isoquant ................................. 85
5 . 1 A Shorter-Run Average Cost Curve (SRAC) , A
Longer-Run Average Cost Curve (LRAC) , And An
"Efficient' Envelope" (EE) . . . . .................. 102
A1.1. The Change In The State Of A System (S1 To 82)
By Means Of Either Process A Or B ............. 141
‘AJHZ Tflue Change In The State Of A System (81 To 82)
By Means or A Cycle ............................ 143
A2'1 A Distance Function ............................ 172
A2.1 ISoquant Showing Stage III ..................... 181

vii

 

CHAPTER ONE

INTRODUCTION

It is becoming increasingly fashionable to estimate
production functions in a 'new and better way.‘ Increas-
ingly researchers estimate a "frontier" production
function rather than a traditional production function.
The frontier production function is supposed to be a new
way to explain differences in the observed production
behavior of different firms [King, 1980]. The frontier
function is supposed to be 'better' because it represents
an estimate of the best performance observable.

A frontier production function
may be thought of as a "best practice"
production function (Forsund and Jan-
sen) or a function that expresses the
maximum product obtainable from various
combinations of factors given the exis-

ting state of technical knowledge.
[King, 1980, page 1]

1.1. VARIATION IN OBSERVATIONS IN APPLIED WORK

Undoubtedly the frontier production function approach
has some intuitive appeal, or it would not be gaining such
Wide acceptance and popularity. The intuitive appeal of
the frontier production function approach is probably due
t° the fact that in applied work it is not uncommon to

°bserVe two firms 0 and P, which appear to be using the

l

same technology with different amounts of the same inputs
to produce the same amount of the same output.
One should remember that all applied problems assume
a consistent theoretical basis for all the observations
included in the data. This theoretical basis is under-
stood to be an abstraction from reality which helps one
distinguish what is important from what is unimportant.
In firm level production analysis this frequently means
that all observed firms are assumed to use the same
technology, use the same inputs, pay the same prices for
their inputs, and receive the same price for their output.
These same assumptions are generally made for frontier
production functions, too. It is clear that the as-
sumptions do not strictly hold in reality; that firms do
not use the same technology, do not use the same inputs,
do not pay the same prices for their inputs, and do not
get the same price for their output. Traditionally, re—
Beachers have assumed that the real observations represent
a distribution around the theoretical points of "same"
technology, "same" inputs, "same" input prices, and "same"
output price. This distribution is caused by "noise," or
random uncontrollable factors affecting the real observa-
t-‘lons . Consequently, one uses the data to estimate para-
IIIeter‘s of the system at the mean of the data, i.e., at the
theoretical points of "sameness." That is, with careful
attention to one's assumptions, one in effect theoretical-
ly insures the variation does not exist. Dealing with the

variation that does exist in the data is an empirical

 

 

problem, rather than a theoretical one.

Traditionally, one deals with the variation by care-
fully specifying the sub-production function in terms of
the levels of the fixed inputs, so that all observations
are on the "same" sub-production function, and by aggrega—

ting inputs that are very close to the "same" for all

observations. How close is "very" close is a matter of
judgement. When the specifying and aggregating is not
"very close," one commits either a specification or an

aggregation error. That is, if one includes data from two
firms in estimating a function which are on two different
sub-production functions, one has a specificationjroblem.
()r', if one includes two firms using different inputs which
are treated as the "same," one has an aggregatiowoblem.
One expects that the actual variation left after speci—
fying the sub-production function and aggregating the
inputs will be variation that deviates from the mean of
the data points, the point of theoretical "sameness," and
not from one extreme. Thus, such observed differences
betWeen firms are not due to "technical" efficiencies or
inefficiencies but, rather, to either implicit specifica-
tion and/or aggregation errors.

Frontier production functions represent not only a
e”Pil'lical departure from the traditional approach but also
a Lhaoretical justification for this departure. Frontier
produ£>tion functions constitute a new theory of produc-

tion, not simply a new method for explaining variation in

data. No longer are observations simply deviations from
their means, they are deviations from their extremes.
Why not use a statistically estimated
average unit isoquant, rather than a
frontier isoquant? The answer is that
the frontier function, which determines
"best" practice in the industry and
which all firms are attempting to emu-
late, may not be a nuetral transforma—
tion of the average function. The
frontier production function may have
entirely different factor elasticities
from the average function. [Timmer,
1971, page 779]
Because the frontier function is the "best practice"
function, it reputedly allows the researcher to discrim-
irxate between firms on the basis of "technical" efficiency
(TE) and "price," or allocative efficiency (PE) to explain
'ttme difference between Q and P. That is, some of the
variation in the data, due to its lack of "sameness," is

exPlained as being due to differences in "technical"

a“Id/or "price" efficiency between observations.
1.2. WHY FRONTIER PRODUCTION FUNCTIONS ARE "BEST"

The theory of frontier production functions is still
evcl\ting. This means that in understanding frontier pro-
duction functions, one must accept the presence of two
Obstacles. These two obstacles result in problems of
inconsistency, ambiguity, and vagueness, and are the re-
sult (of the fact that frontier production functions prove.

in eVery case, to be misleading interpretations of real-
ity,

First, there is more than one definition of a fron-
tier production function and hence a lack of consensus
among authors as to what represents a frontier production
function. There might be disagreement or inconsistencies
between authors regarding some aspects of what constitutes
frontier production function theory.

Second, for a particular representation of a frontier
production function, not all the underlying preconditions
(assumptions) are necessarily clear. Nor are all the
implications or ramifications of a particular characteris—
tic stated, or even clearly implied.

Despite these two obstacles all frontier production
functions share two concepts: (1) TE 76 PE due to an in—
trinsic separation between physical decisions and value
deoisions, and therefore (2), there is a technical effi-
c: iency and a technical inefficiency that are exclusively
different.

The first characteristic shared by all representa—
t dons of frontier production functions is that TE is
separate from PE; that technical efficiency and allocative
e fficiency are mutually exclusive phenomenon. Recent
e<.‘{uivocation, that allocative efficiency means only get-
t 1119 inputs in the proper ratios but not necessarily in
the proper quantites [Kopp, 1981a, Schmidt and Lin, 1983],
has not altered the basic premise that one can observe
technical efficiency (or inefficiency) in isolation from

Q‘Dserving price efficiency.

Contrary to price efficiency which is a
purely behavioral concept, technical
efficiency is purely an engineering
concept. It entirely abstracts from
the effect of prices. [Lau and
Yotopoulous, 1971, page 95]

If technical efficiency "entirely abstracts from prices,"

111 so doing,

vvjnthout value (price) are considered to be "free"

and therefore are of no economic consequenc

e.

it ignores the fact that physical commodities

goods,

Quite

Simply, some frontier production theory suggests that one

may get

increasing technical efficiency. Since technical

something for nothing (more output)

by

simply

effi-

c1 ency is separate from price efficiency the increase in

ef ficiency should be costless.

These results also show, however,
that on average there is 22.82 (=1.0 -
0.7718) percent and 23.27 (= 1.0 —
0.7673) percent technical inefficiency
in the cases of crop and mixed farm
samples, respectively. This means that
actual (observed) output is about 23
percent less than maximal output which
can potentially be achieved from the
existing level of inputs. In other
words, through the efficient use of
existing inputs the farm output can be
increased by almost 23 percent without
any additional cost to the farmers.
[Bagi and Huang, 1983, page 255]

This quote not only exemplifies that TE and PE are re—

gasleded as mutually exclusive, but also exemplifies the

Eg“~‘9c:ond concept that all frontier production function in-

t § rpretat

This

ions share.

second concept is that technical efficiency

theans one might get more than one quantity of output given

 

the same technology (the same set of identical fixed and
variable inputs) and that the failure to achieve the same

level of output cannot be corrected using economic theory

as a guide.

The economic decision-making process

can fail in two different ways. The
whole core of economic theory is con—
cerned with the first of these - the

marginal revenue products of some or
all factors might be unequal to their

marginal costs. If this is true the
allocative decision is said to be in-
efficient. The second source of
failure is the technical production
function - a failure to produce the

greatest possible output from a given

set of inputs means the technical deci-

sion is inefficient. [Timmer, 1971,

page 776]
The suggestion that one might get two or more possible
o\JLtcomes when using homogeneous inputs in identically
3pacified production processes has lead some to suggest
prlicitly that the production set is a solid rather than
‘31 surface; that an isoquant is a plane rather than a line

[ Jamison and Lau, 1982] .

As can be seen from the quotes above, both these
QOncepts are so interdependent that there is virtually an
1 f~and-only-if connection between them. This is because
both of them rely entirely on the existence and interpre—
tation of interiors £9 isoguants. Being on the isoquant
mQans being technically efficient, while being within the
1Interior of the isoquant means being technically ineffi-

Qllent. Being at the point where the budget constraint is

 

 

tangent to the isoquant, or being on a ray from the origin
through this point, means being price efficient. Being
away from the tangency point, or the ray through it, means

being allocatively inefficient.

1.3. WHY IS THIS IMPORTANT?

The frontier production function is an attempt to

explain the cause of variation in data in empirical work.
This is a practical problem rather than a theoretical one.
Unfortunately, frontier production functions do not merely
JFGEJpresent a method for dealing with an empirical problem.
They also represent a theory of production which is physi-
C=ialllry and logically questionable. This confusion of theo—
zt‘ifie'tical issues with empirical issues has resulted in an
aJn'biguous and misleading frontier production function
1 i terature.

Eliminating this confusion is important because the
SalziLstinction between technical and price efficiency reduces
the credibility of economic theory in explaining reality.
Utrltle distinction places economists at odds with engineers
is‘lrld all physical scientists. Frontier production function
theory encourages researchers to believe that efficiency
‘2“E311 be solely concerned with physical production relation—
et‘lfips (TE), or solely concerned with value relationships

(IEEIE). This suggests that economic efficency is not merely
minimizing opportunity cost, but that there is a purely

physical aspect that is unassociated with value.

COH-

current production literature increasingly
'two

The

tains examples of research that have measured these

of efficiency [Bagi and Huang, 1983, Bravo-Ureta,
Cooper,

types'
and Rhodes, 1978, Charnes,

1983,

Charnes, Cooper,
and Schmidt, 1979,

Lovell,
Forsund and Hjalmarsson,
1980,

Forsund,
1974,

and Rhodes, 1981,

Forsund and NJ almarsson ,

Hall and Le Veen,

and Lovell, 1977,
is a fundamental difficulty in the logic

or more, types

1978, Lesser and Greene,

Schmidt and Lovell, 1978].

15379,
ESCflhmidt

of

of

S ince there

the proposition that there are two,
are fundamentally flawed and

efficiency, these studies
Researchers have measured

I'each erroneous conclusions.
What is in reality variation in the data due either to (1)
from two or more different sub-

c: Onbining observations
production functions (the specification problem) or (2)
aQgregating heterogeneous inputs (the aggregation prob—
differences are then attributed to differ-
There is a

Jl-isem). These
‘EPJraces in "technical" efficiency between firms.

‘5lnianger that the conclusions of these studies will be used
deci—

1 In formulating public policy or in entreprenuerial
More importantly, the credibility and le-
jeo—

Qdon making.
gitimate development of the science of economics is
and misleading

pIl‘odized by institutionalizing illogical

t lleory .
Frontier production functions are both a theory and
iaur‘ explanation of variation in data. If the theory is in-
functions, properly inter-

valid , frontier production

lIEJPeted, can still be an empirically valid way of detecting

 

 

O
I
(I.

Ll

3 I-
“A

...;

IO:

+4»

‘1:

CU

H
H

'l
75
‘I

10

and explaining variation in the output of firms. The
methods currently used in frontier production function

analysis can be used diagnostically to detect aggregation

and/or specification errors.

1.4. OUTLINE OF CHAPTERS
1 - 4.1. CHAPTER TWO: UNIT ISOQUANTS AND PRODUCTION SOLIDS

Two cases of frontier production function theory and
‘vac3 mutually exclusive types of efficiency are discussed
1111. this chapter. First, the unit isoquant is defined.
This concept is used in the original and fundamental
de finition of a frontier production function. It compares
f 5- :rms on the basis of input used per unit of output ob-
‘t:*El.ined. The next section considers two cases of a solid
production set. A solid production set implies that two
identical sets of the same inputs can produce different
01:. tput using the same technology.
These cases preserve the essential frontier produc—
“:‘:i.on function characteristic that there is an interior to
aatlrl isoquant at the same level of production. The last
aection explains why such an interior can appear to be
Q"Deerved in reality gm if a specification or aggregation
errors exist. Specification error means one is making a
Qanparison across sub-production functions or input re-
quirement sets, rather that within them. Aggregation
$1":Ir'or means one has aggregated heterogeneous inputs and

identified them as homogeneous.

ll

Apparent differences in TE are attributed to firms
having different amounts of fixed inputs, or aggregating
heterogeneous inputs. Frontier production function analy—
sis often ignores the presence of fixed inputs and the
economies of investment/disinvestment necessary to change
the amount of fixed input used in production (see Edwards,
.1958) .
1 -4.2. CHAPTER THREE: FRONTIER FUNCTIONS AND DISTANCE
FUNCTIONS

Frontier production functions are often assumed to

c=<3nform to the same underlying assumptions as distance
functions; concavity and monotonicity. Therefore, it
111:1 ght be inferred that frontier production functions are a
8E>ecial case of distance functions. First, it will be
s1'10wn why frontier production functions and distance
f‘I.:lnctions may mistakenly be interpreted as being the same.
Tc many, the frontier production function appears to be a
b3.7-product of the attempt to provide a rigorous mathemati-
Qa1 proof to duality theory. The third and fourth sec-
t lions prove by contradiction that frontier production
functions and distance functions are incompatible.

The proofs in the third and fourth sections rely
liQavily on an interpretation of duality theory presented
by McFadden (1978) . McFadden's duality theory excludes
S‘tage III of production because he postulates that mar-
ginal physical products are always non—negative. His

qIlality theory implies that TE and PE are identical; i.e.,

t...

0'. b

3;“

C33
9!:

in;

the

1.4

 

12

that all points on a production function that are TE are
also PE by definition. This is demonstrated using polar
reciprocal sets. Polar reciprocal sets make a one to one
mapping from physical to value relationships (excluding
Stage III); the physical aspect of production (TE) and the
*value aspect of production (PE) are inseparable. There-
fore, the third section demonstrates that there can be no
difference between TE and PE.
The fourth section demonstrates that an isoquant
Cannot have an interior at the same level of production
\Jssing duality theory and its inherent principle of free
disposal. Free disposal in production space means the
<=C>rresponding cost is also freely disposed. Free disposal
ll"earns that 92th input and output are freely disposed and
completely removed from accounting, i.e., it is as if the
eJ-ittra input and output never existed; consequently "extra"
‘ifituput/output cannot affect technical efficiency because

“:lney can not be included in production or cost.
1 .4.3. CHAPTER roan: A m casz

The last case presented is a new case that has not

Iaeen considered in the frontier production literature.
firhis case presents a valid interpretation of the unit
.isoquant where there is neither specification nor aggrega-

tion error. This case is fundementally different from a

frontier production function because there is no dis—

tinction between TE and PE, and the unit isoquant may have

 

9‘:
\‘

 

 

 

13

both interior and exterior points. It defines "technical
efficiency" as obtaining maximum average physical output.
Therefore, the point of "technical efficiency" can be
viewed as roughly equivalent to the boundary between Stage
I and II in production and is 33211 efficient only if the
firm is in long run equilibrium under perfect competition.
1.4.4. CHAPTER FIVE: THE FRONTIER PRODUCTION FUNCTION
LITERATURE
Chapter Five deals with the frontier function litera—
‘ture itself. Careful reading shows that TE is due to
especification error and/or aggregation error. Only the
asaiient theoretical literature is dealt with since much of
‘tlae literature is reptitious. The "other literature" is
also briefly described. This literature falls into one of
“twat: categories, (1) it develops a method for measurement
(:1? TE and PE, or (2) it develops related concepts that

encounter the same basic difficulties.
1.5. OUTLINE OF APPENDICES

The Appendices present two mutually exclusive argu—
men-ts, each of which raises serious doubts about frontier

Production functions and all of their ramifications; (1)

that a firm can be "technically inefficient" given a set

Of lilaputs (system). an output (useful work and any change

it! ‘the state of the system and its environment), and a

sula“1;roduction function (processes), and (2), that one may

s
Sparate physical efficiency (TE) from value efficiency

 

 

14

(PE). Both appendices demonstrate that there can be no
relevant logical interior to an isoquant.
1.5.1. APPENDIX ONE: EFFICIENCY AND THE LAWS OF
THERMODYNAMICS
There is no single term for "technical efficiency" in
engineering. Rather there are a number of different types
of efficiency which describe aspects of what some eco—
nomists call technical efficiency. Thermal efficiency is
an example of one such technical efficiency. The focus of
this appendix is on thermodynamics, the laws of which
determine thermal efficiency. Thermal efficiency is de—
fined as the ratio of useful output to costly input
(Dixon, 1975], so that in thermodynamics the physical
aspect of production is not separated from the value
aspects of production.

The first part of the appendix defines the important
terms and concepts from thermodynamics.

The second section presents the first and second laws
01‘ thermodynamics. This establishes the one to one rela-
tionship between total input and total output, i.e., es-
tab; ishes that one cannot get more of the same output with
less of the same input using the same processes.

Finally, thermal efficiency, as a case of technical
efficiency found in thermodynamics, is described. In the
appendix it is shown that, (1) thermal efficiency is an
evaluation of output which means TE and PE are insepar-

ab
13 . and (2) , thermal efficiency is not the TE of fron—

15

tier production literature. It is also shown that dif-
ferences in thermal efficiency can be found only in situa-
tions where one is comparing two different sub-production
functions (cycles) or two different sets of inputs (sys-
tems). Differences in technical efficiency using the same
bundle of inputs, and the same production processes, are

xaot possible in thermodynamics.
11.5.2. APPENDIX TWO: EFFICIENCY IN ECONOMIC THEORY

Microeconomic theory, simply but clearly stated. has

110 logical place for the frontier production ’function

(zoncept. Using set theory notation, the first section of

‘tiie appendix carefully defines terms, including input and

catltput, production functions and sub-production functions,

input requirement sets and isoquants, and distance
functions.

The second section of this appendix states the funda-
mental assumptions of production sets. The implications
of these assumptions are critical to understanding, (1)
"I13? frontier production functions are invalid, and (2) the
real ity they misrepresent.

The assumption of concavity implies the importance of
f1"“341 inputs. In order to evaluate efficiency, something
must always be fixed. Therefore, the consequences of
f1’{‘3<i inputs are critical to evaluating efficiency.

Each of the relevant consequences of fixed inputs is
taut‘ila up in turn in this appendix. Without fixed inputs.

t
he law of diminishing returns implies constant returns to

 

 

 

 

16

scale. Even within this context one may conclude that
total input or output must be fixed or constrained in
order to discuss efficiency. The law of diminishing re—
turns states that fixed inputs affect the productivity of
the variable inputs. The law of diminishing returns is
important if one is going to freely dispose of the
input/output of one firm in order to make it "comparable"
to another firm; precisely what is implicitly done in the
frontier production function theory. The last, often over
looked, consequence of a fixed input is the existence of
Stage III of production. While it is true that additional
input always results in additional output (monotonicity) ,
it does not always result in additional useful output, or
product. The idea that one can get more or the same
useful output with fewer of the same inputs, is only true
for a move from Stage III towards Stage II. Stage III
also means that marginal physical products can be nega—
tive. Thus, producing- in Stage III is not economically
rational, not because it is not technically efficient nor
because it allows one to distinguish between TE and PE,
but because it does not pay to be there, which is simul-
ta~11eously a physical and an economic phenomenon. There-
fc’lsia, being in Stage III is inefficient.
The assumption of monotonicity means there can be
free disposal in production. Free disposal has little if
any counterpart in the real world. Theoretically, it

exiSts for perfect complements generally assumed in linear

 

wit':

"fic

6“.

u
I"

‘
I

th

N‘

l7

programming. It is used in the conventional theory simply
as a means to an end [McFadden, 1978]. In order to pro-
vide a rigorous mathematical proof for duality theory,
excluding Stage III of production, one ordinarily deals
with convex sets rather than convex functions.
The theory establishing the dual
relation between cost functions and
production functions was introduced
into economics by Shepard (1953), who
drew heavily on properties of convex
sets discovered by Fenchel (1953).
[McFadden, 1978, page 5]
{this means one hs an 'interior' to the production (dis—
tance) function, since it is a convex set, rather than
just a convex function.
Even though the identical word convex
appears in both the term convex set and
the term convex function, it has a
widely different connotation in each
context. In describing a set, the word
convex is concerned with whether the
set has any holes in it, whereas, in
describing a function, the word has to
do with how a curve or surface bends.
[Chiang, 1974, page 643]
Since an interior is clearly a violation of nature and
e‘=<>tmomic theory, free disposal is used to reconcile being
0:1 iand off the isoquant at the same time. This is how one
j‘ls3‘tifies an input requirement set, which has an
I
interior' of higher order isoquants, as a description of
the sub-production function or distance function.

A misunderstanding of free disposal leads to a defi—

n
ition of TE that is in conflict with the laws of dimin-

18

ishing returns. Free disposal does not mean that a pro-
duction function is a solid rather that a surface, i.e.,
it does not mean that what one usually conceives of as an
isoquant has an interior at the same level of production.
The 'interior' of an input requirement set is a collection
of true isoquants which is consistent with the fact that
given identical sets of inputs one always gets identical
sets of outputs. Frontier production theory misrepresents
free disposal by applying the concept to only one side of
the inputzoutput equation. It suggests one may retain all
the input bundles within an input requirement set and
equate them to the output represented by the "bounding"
lowest isoquant of the set by ignoring ("freely dispo—
sing") the additional output of each of the higher iso-
quants within the set. "Technical inefficiency" implies
one does not ignore the additional inputs themselves that
are consumed in producing the additional output that is
freely disposed, or ignored. That is, the extra inputs
are included in the accounting of what is required to
achi eve the given lower level of output.

It can be shown that a distance function (a gen—
era-1 ized production function excluding Stage III that
disE>lays free disposal) is from the same set as its cost
fur1'12tion. Therefore, there is a one to one mapping from
distance functions to cost functions. Factor-price re-
quirement sets in cost space are analogous to input re-
quirement sets in production space, and are a direct

c
onsequence of duality theory. Together they are polar

19

reciprocal sets. Polar reciprocal sets mean that free
disposal in production space is always accompanied by free
disposal in cost space.

The third section briefly describes profit maximizing
behavior which constitutes a valid definition of efficien—
cy. The ratio that determines the efficient point of
production in economic theory is the same ratio that
determines efficient production in thermodynamics; i.e.,
equating the ratios of the marginal physical products with
their respective prices.

The last section explains how and why frontier pro—
(duction functions violate the tenets of microeconomic

theory .
1.6. SUMMARY

First and last this thesis is a discussion of effi-
c::iency. It is predicated on the fundamental fact of
jg>luysical reality that there is a 1:1 physical relationship
batween input and output (product, waste, and pollutants)
5113» production; a one to one accounting between everything
that goes in to everything that comes out. Given two sets
of identical inputs, using the same sub-production
function, one cannot get two different amounts of iden-
tical output. Therefore, isoquants cannot have interiors
at the same level of production. One cannot observe tech-
I-l~1|-¢::al inefficiency without simultaneously observing price

1rIefficiency. One cannot distinguish between them or

 

Ir. :

..u

it!
”0'

uh.
I"

20

discuss them separately. One cannot measure technical
inefficiency except with respect to price(s) or value.
Therefore, there is only one type of economic efficiency,
involving the physical and value attributes.

Frontier functions are illegitimate since, (1) they
postulate that one may get something for nothing, i.e.,
more output with the same inputs and nothing else of value

in addition, and (2), they attempt to separate technical

efficiency from price efficiency.

CHAPTER TWO

UNIT ISOQUANTS AND PRODUCTION SOLIDS

In this chapter, frontier production theory, inclu-
ding its distinction between technical and price efficien-
cy will be summarized briefly. The first section, Section
2.1, will develop the concepts of technical efficiency

(TE) and price efficiency (PE) by describing the unit iso—

 

guant. The frontier production function originated as a
unit isoquant in Farrell (1957), and is still often so
described. Much of the first section follows the presen-
tation of frontier production functions made by Bressler
( 3966) .
The second section, Section 2.2, will describe two
c:aases of frontier production functions that are not expli-
citly sets of unit isoquants, but retain the essential

characteristic of the unit isoquant: there is an interior

 

to the production surface (frontier) that is made up of
" technically inefficient" points of production. Without
an interior so characterised, there can be no frontier
Iz’t‘thiuction functions. The laws of diminishing returns and
111: :llity indicate that the interiors described by frontier

pt‘Oduction functions cannot exist (see Appendices One and

TWO) .

21

 

‘uufi
..‘A'

H:

be“

£- 1
0“

SE?

a!
it

2.1

:3

22

Section 2.3 explains why the frontier production
function interpretation of these interiors arose from the
misinterpretion of observed phenomena. Apparent observed
differences in firm performance attributed to differences
in technical efficiency, are in fact due to comparing two
separate sub-production functions (specification error),
or aggregating heterogeneous inputs (aggregation error).
In neither of these cases do identical sets of inputs

produce different levels of identical output.
2.1. UNIT ISOQUANTS
231.1 WHAT IS "TECHNICAL EFFICIENCY?"

In much of the frontier literature, technical effi-
ciency (TE) means being on a "unit isoquant [Farrell,
.1957, Bressler, 1966, King, 1980, Nerlove, 1965, Timer,
1 9 71] . "

Consider an input requirement set with two variable
inputs V1 and v2, and one fixed input 23, that together
EDIFthuce some output Y. Since there is only one input

PetI"-lirement set, one is implicitly considering iden-
tical sub-production functions. The inputs and outputs
31“: assumed to be homogeneous. These conditions assume
that technology is fixed and identical for all observa-
tith. The technology of this production process is
reEiresented by the unit isoquant yo in Figure 2.1. The
unit isoquant is found by dividing the input quantities by

t
‘r‘fié output quantities they produce, i.e. the unit isoquant

n<|H<\

 

 

23

 

FIGURE 2.1

A UNIT ISOQUANT WITH BUDGET A CONSTRAINT

\/

24

maps the average variable input required per unit of
output (not the marginal input bundle that produces one
unit of output at the margin). Such a mapping can produce
observations located at widely varying positions in the
quadrant. The points lying closest to the axes are con-
nected to produce the unit isoquant, or the "technically
efficient" isoquant. That is, all points on this unit
isoquant are regarded as "technically efficient," while
all points within it are regarded as "technically ineffi—
cient." Consider an observation on firm P. The firm uses

the inputs V1 and v2 in the same relative proportions as

the firm Q, but uses more of both v1 and v2 on average in
[aroducing a unit of output as Q; P gets less output per
Izriit of variable input on average than Q.
The distance OP relative to OQ measures
the extent to which the same amount of
output could be produced with fewer
inputs used in the same proportion....
[Nerlove, 1965, page 88]
The ratio OQ/OP is the measure of technical inefficiency.
This means that P can produce the same quantity of output,
"1 th less of the same homogeneous inputs, using the same
I:’I'°cesses or sub-production function, merely by becoming
teehnically efficient. TE means getting different output
g““Ven the same inputs and the same production processes.
art‘SE firms on the unit isoquant are technically (physi-
c=iifilly) efficient since they produce on average the most

Q‘u-tput per unit of input, or conversely, they use on

a"erage the least input per unit of output. The interior

25

points are technically inefficient because

they use

greater physical quantities of both v1 and v2 on the

average to produce a unit of output.

2.1.2. WHAT IS "PRICE EFFICIENCY?"

Technical efficiency does not imply price
in frontier production theory since they are

exclusive phenomena. Price efficiency (PE) impl

efficiency

mutually

ies inputs

being used in their least cost combination ratios. It

reflects the proportions of v1 to v2 but not necessarily

the quantities. In Figure 2.1 Q and Q' are both

technical-

.1y efficient, but only Q' is price efficient since it is

tangent with aa' , the budget constraint. The ratio that

Jreeflects the degree of price efficiency for both P and Q

i s OA/OQ.

The distance [OA] relative to CO mea-
sures the fraction of costs for which
the output could be produced if the
relative use of inputs were altered.
[Nerlove, 1965, page 88, underlining
added]

2 ' 1.3. WHAT IS "ECONOMIC EFFICIENCY?"

Economic efficiency (EE) in the frontier
function literature is defined as the product of

ef ficiency and price efficiency:

(2.1) EE TE * PE = OQ/OP * OA/OQ

OA/OP

production

technical

26

 

 

V/

Input

FIGURE 2.2

'r
EQHNICALLY EFFICIENT AND INEFFICIENT pomtrs or PRODUCTION

WITHIN A PRODUCTION POSSIBILITY SET

27

While this formulation makes a distinction between tech—
nical and price efficiency, economic efficiency remains
connected with prices or values.

Note that price and economic efficiency

measures are in fact cost comparisons:

The "price line" aa' represents total

expenditure per unit of output, with

slope representing the inverse ratio of

the given factor prices; lines parallel

to aa' through Q and P represent the

higher unit expenditures. Thus, the

economic or overall efficiency ratio is

equivalent to the ratio of the average

cost of producing at Q' to the average

cost of producing at P. [Bressler,

1966, page 130]

In order to apply this theory to "reality," one must
(mallect observations on firms producing the same amount of
'time same output (or assume constant returns to scale),

tissing the same technology (same sub-production function),
Ettjd different amounts of the same variable inputs. If one
c:<:uld find such observations, the next step would be to
plot this unit requirement in the positive quadrant and
connect the observations closest to the axes. These con-
he3<=ted observations define the unit isoquant from which
a1 3- efficiency comparisons are made. A set of such unit

18°quants, for different levels of production, define what

is called the frontierirmction function.

2 ‘ 1 .4. SUMMARY OF UNIT ISOQUANTS:

The essential and critical aspect of the unit iso-

c;‘1ant as described above is the existence and interpreta-

 

 

28

tion of the interior points. Consequently, frontier pro—
duction functions are valid only if one, or both, of two
things are true about reality.

First, given two sets of variable inputs that differ

gnly in quantity, e.g., one set is "a" times larger, and
using the same fixed inputs, one may produce the same
quantity of identical output. That is, different quanti—
ties of given inputs produce the same output using the
same technology. It is producing a different quantity of
output from identical inputs and technology that explains
the existence of the interior to a unit isoquant for a
given level of output. Without an interior to the 'unit
.isoquant for firms producing the same output there is no
tweed for the concept of a frontier production function.
Consequently, the existence and the nature of this
.jgnterior is critical to the interpretation of the frontier
production function.

Secondly, one must be able to separate TE from PE.

In the frontier production formulation, TE is purely a
JEllxysical concept, PE is purely a gglgg concept, and TE and
IF'EE are observable mutually exclusively of each other.
ntfilmerefore, measurements of physical quantities have no
zi-Jratrinsic relationship with their corresponding prices or
“’"ialues. In Figure 2.1 for firm P, TE is measured at the
:IEPCDint P, while PE is measured at point Q.

Though TE and PE are regarded as mutually exclusive,

2:313 and total cost may not be. That is, the measurement of

PE is the same for firms P' and Q'. Both firms P' and 0'

‘

 

‘3

O.

53!;

Sim

\ 29
are price efficient since the measurement of PE = OQ'/OQ'
are the same for both. Since P' uses more of both v1 and

V2 in total and on average to produce one unit of output,

the total cost for P' must be higher than for Q'.
Current frontier production function literature often

deals with a frontier production function that is not

 

explicitly defined as a collection of unit isoquants since
there is no averaging process. This type of frontier
;pmoduction function clearly represents a production set
‘that is solid, rather than the surface that one ordinari-
.1y associates with a production function in traditional
zproduction theory.

2.2. PRODUCTION SOLIDS

It should be evident from the description of the unit
isoquant given above that the production set represented
EDEV' a frontier production function is characterized as a
Surface (the traditional production function) and its
iiJEJterior which is a set of "technically inefficient"

points of production. The two cases of production sets
t1Plat are solids, rather than shells, presented in this
£3‘E’tction retain this critical aspect of unit isoquants,
“'lblmile ignoring the average aspect of unit isoquants.
Therefore these two cases highlight the critical aspect of
6‘ frontier production function that its interior is a set
(3315‘ technically inefficient points. In so doing, it demon—

&trates the deficiency of frontier production functions by

$11Owing that this interior is either physically impos—

a

 

 

4
h;
to!

n
In!

Ova
n

’3

VIC

.
.-
...

30

ble, or a misinterpretation of what is observed. Both
ses fundamentally revolve around specifying the appro-
iate sub—production functions, and the economic ad—
stment of investment and disinvestment implicit in
ving from one sub-production function to another. Both
ses arise from either committing a specification error
d/or an aggregation error in identifying the sub—produc—
on function described by the set of observed production
ints. These are empirical issues rather than theoreti-
l ones.
Consider the production possibility set represented
Figure 2.2. In Figure 2.2, points Q and P are
... technically efficient in the sense
that they are in the production possi-
bility set, and there is no way to
obtain more output than depicted by
these points without using more of the
input. Point [P'] is technically inef-
ficient, in that more output could be
obtained with no more input. [Jamison
and Lau, 1982, page 54].
"frontier" of this frontier production function is

E‘esented by the surface upon which points Q and P lie.

Flt P' represents a point within the solid interior to

frontier. P' is technically inefficient since one
It: get the same amount of output, Y1! "with no more
311:;" i.e., by using less input and producing at point

l§dditionally, point P' is technically inefficient, "in

more output," Y2, "could be obtained with no more

3‘ same] input," by producing at point P. In both cases

 

 

31

it is implicit that neither does the total input bundle
nor does the technology change, only the degree to which

the technology is used technically efficiently. That is,

the input VP can produce either y1 or y2 using the same

technology merely as a matter of technical efficiency.
2.2.1. CASE 1:

Case 1 is the obvious case of a solid production

 

possibility set for the variable inputs given the same
technology and the same fixed inputs. That is, the
technology used and the quantities of the same fixed
inputs are identically equal at points Q, P, and P'. An
obvious difficulty with this is that both points P and P'
use the same input, VP, but get different outputs. Since
the variable input is the same, the fixed input is the
same, and the technology is the same, there is nothing to
account for the marginal physical product being positive
at P and zero between P and P' . Jamison and Lau (1982)
ex£31a1n the difference as follows:

Technical inefficiency results from

combining available inputs poorly; for

example, by plowing the insecticide

into the ground or spraying fertilizer

on the plant leaves. [Jamison and Lau,
1982, page 54]

There are two problems with this explanation. The first

18' cane can increase TE without additional cost, since one

81‘WE>:1y uses less of the same variable input to produce the

3
ani‘* output, i.e. moving from P' to Q [Bagi and Huang,

 

32

1983]. In frontier production theory the only physical
(technical) difference between Q and P' is the quantity of
variable input used. In fact, improving TE should result
in a savings since one is using less costly variable
input. Therefore, one can increase profit at no cost; get
something for nothing. Therefore, either there is no

opportunity cost to instituting the changes that will

 

bring about the improvement in TE, or the net opportunity
cost is always positive by definition and no marginal
analysis is needed to consider changes to improve TE. In
the examples quoted above: there is no cost in taking care
not to plow any insecticide into the ground nor in taking
care not to spray any fertilizer on the leaves.
The second is that in traditional theory, inputs are
defined with respect to time, location, and quality (see
APpendix Two), so that one cannot combine inputs "poorly"
01' "better" since any particular combination of inputs is
tinnee, location, and quality specific. Changing any of
these aspects of the combination means changing the inputs
by definition. How inputs are combined in practice is an
appl ied problem and not a theoretical one. The frontier
production literature confuses theory and application.
Ines*=—"<:ticide plowed into the ground is not the same homo-
ger‘fietaus input as insecticide not plowed into the ground
beczii\ase they are not in the same location. Similarly,
fel“::ilizer sprayed on the plant leaves is not the same as
fez?1t.1lizer sprayed on the ground. The problem of treating

he’tezlrogeneous inputs as "homogeneous" groups for the pur-

.

 

33

poses of analysis is a specification or aggregation prob—
lem. It is a problem in application that is assumed away
in theory by the homogeneity conditions.

The theory of frontier production functions maintains
that the same homogeneous inputs (systems) can be used
with the same sub-production function (processes) to get
different output (actual work and/or final states). This
is clearly in violation of the laws of thermodynamics (see
Appendix One) and diminishing returns (see Appendix Two).
Therefore, Case 1 is physically impossible; i.e., in—
teriors to unit isoquants for the same levels of total

output.
2.2.2. CASE 2:

The second case is the case of a traditional produc—
txic>n possibility set, where all the points are also within
tilee feasible production set. Therefore, if a point P'
“Ses more of the same homogeneous variable inputs, and the

same technology as point Q, and only gets the same amount

of. identical output, then P' and Q must be on different

s“"5>-—production functions; i.e., be using different amounts

°r' ltinds of fixed inputs. That is, point P' represents

131E, 'interior' to the production set, but only because it

ii; on a different sub-production function than Q and P.

T
11E: interior in Figure 2.2. is in reality a collection of

suit:

‘~production functions, for each of which the level or

n
a‘t'tilre of the fixed input is different. Because P' uses

 

 

 

 

34

less of the fixed input than Q, output of the variable
input is less by the law of diminishing marginal returns.
Consequently, for Figure 2.2 there are two problems.

The first is an indexing problem, due to the fact
that the different levels of the fixed input are not
clearly specified for Q and P, and P'. The input bundle
VP at P is different than at P'. Therefore, P' is not
"technically inefficient." At P' one cannot get more
output with the same ("no more") input, i.e., move to P,
because P' does not have enough fixed input. Nor can P'
get the same output by using less ("no more") variable
input, i.e., move to Q, for the same reason.

The second problem with Figure 2.2 is related to the
first. Figure 2.2 represents a case of not properly
treating the economics of adjusting the use of the
"fixed" inputs. The problem of making an adjustment with
resPect to fixed inputs, the analysis of investment and

disinvestment, has been treated by Edwards (1958) . Clear-
IY' there is a cost in moving from P' to either Q or P
girl<==e P' must invest in more of the fixed input to make
thfia change. The economics of this adjustment is totally
ignored in the frontier production function literature
"hi eh treats such adjustments as costless technical

Chair‘smes.
2.3. THE EXPLANATION FOR "OBSERVING" CASES 1 AND 2

In reality, when one makes observations on firms Q,

P.
and P', one is observing differences between the firms

 

 

35

that are due to either specification error or aggrega-
tion_gg£or.

The specification problem, involving the amounts of
which variables are included and which ones are not, is
almost indistinguishable from the aggregation problem.

When inputs are defined as homogeneous with respect to

 

time, form, and space, production systems and processes
become interdependent. It is also due to output being
both useful output (work) and the changes in the final
states from the initial states, one a function of the
processes (work) and the other a function of the systems'

properties (see Appendix One).
2 . 3 . 1 . TEE SPECIFICATION PROBLEM:

Ordinarily, a comparison of firms assumes all firms
are using the "same" sub—production function. Speci-
fication means identifying the sub-production function
Cleeéirly and accurately. Specification error means one has
1na.<:curately identified two different sub-production
functions as being the same. That is, one commits a
specification error in classifying the the firms as repre-
8e:t1":atives of the same sub—production function. A com—
parison between firms is valid only if their sub-produc-
t1c311 function(s) are properly identified, so that differ-
eanS between firms can be properly explained. If dif-
fEEI‘Qtnces are due to the firms having different sub-produc—

t
1c>t1 functions, then those differences must be properly

ill-Illl--__

 

36

identified. The specification problem is an empirical
problem not a theoretical one. In applied work specifica-
tion problems arise in several ways.

One way one creates a specification problem is by
aggregating observations across sub—production functions
in specifying the sub—production function to be analyzed.
This means that inputs that are fixed at the observation
level, e.g. tillable acres for crop production, are gg;

sumed variable within the sub—production function, i.e.,

 

across all observations.

Another way is to ignore the Egg; impact of some
unspecified input(s). One generally avoids this problem
precisely by assuming that the uncontrollable random
variables (the unspecified variables) are equal to some
"average" value across all observations when in fact they
Eire not, so they can be treated like fixed inputs, i.e.

flaving the "same" impact on all observations.

A third way of mis-specifying a sub-production
function is to ignore an input entirely, or to aggregate
‘Iwwo inputs inconsistently (this latter situation is also
clcnsidered an aggregation problem, which is considered in
't3lfie following section). If one omits a relevant variable
input from the specification of the sub-production
function, a variable input that is present in the produc-
t ion processes in different amounts across observations,
§~t‘id this omitted variable input is correlated with one or
.m‘tire included variable inputs, then the marginal physical

Ig’lroducts for identical quantities of the included variable

L_

 

37

inputs will appear to be different due to their including
the additional marginal product of the unspecified vari-
able input. This means that the TE observed is due only
to the differences in the marginal physical products for
the unspecified variable input. Note that the differences
in the quantities of the random uncontrollable variables
also has this effect. If the unspecified input is a fixed
input, and it is present in different amounts across
observations, then one is measuring and comparing marginal
physical products for the variable inputs across different
sub-production functions. The "technical inefficiency" is
not inefficiency, but a difference in the productivity of
the variable inputs due to the point of diminishing mar-
ginal returns for the variable inputs starting at dif-
ferent points for the different levels of the fixed

1 nput .
:2..3,2, THE AGGREGATION PROBLEM:

The aggregation problem is also a real world problem
t'ather than a theoretical one. In order for analysis to
be legitimate, one need only insure that this error is
‘ghjithin some recognized tolerable bound prior to conducting
the analysis, i.e., the inputs aggregated are "very
C::llose."

Aggregation error is suspect when one finds differ-
§rlees in marginal physical products for "homogeneous"

1klputs. One reason may be specification error as des-

 

 

 

 

38

cribed above. The other reason the same amounts of two
"like" inputs may have different marginal physical pro-
ducts, when measured with respect to the identical amounts

of fixed, or unspecified, input, is they are not in fact

homogeneous; they are not "very close" to the "same"
input. Aggregation error means inaccurately identifying
heterogeneous inputs as homogeneous inputs. Economists

recognize that labor is not homogeneous across laborers,
but in applied work this labor is aggregated and specified
as one input. This results in error being introduced into
the analysis due to the differences in quality (form),
time, or space of the inputs aggregated.
In either case, specification or aggregation, the
assumptions one makes about the error term (the unspe-
czified variables), the inputs which are fixed (which sub-
Erroduction functions are included), and/or how inputs are
<=<meined into "same" inputs implicitly means that the dif—
ferences in marginal physical products that do in fact

emtist are "unimportant" to the analysis.
2.4. SUMMARY

This chapter has outlined the theory of frontier
E’11"’<3duction functions as unit isoquants and production
SQlids. Frontier production function theory suggests
pub'coduc’cion within the interior of an isoquant at the same
lQVel of production is possible so that an isoquant is a
Iz):lane rather than a line. That is, one can produce dif-

erent amounts of the same output using identical amounts

 

 

 

39

of the same inputs and the same technology. Interiors are
critical since they allow one to separate TE and PE as
mutually exclusive phenomena which implies that TE é PE.
It also means that technical efficiency is purely physi-
cal, which is contrary to the laws of nature, e.g., the
laws of thermodynamics. This means that the physical
quantites of inputs used in production, and the value of
those inputs in production do not have an intrinsic one—
to—one correspondence, since they cannot have the same
marginal value products if they are not producing the same
output.

If one uses more inputs in one situation to produce
the same amount of work as in another situation, those
additional inputs have an opportunity cost. Yet the total
\halue of the bundles must remain the same if they produce

equal output.

If one is measuring a "frontier production function,"
sElbecification error or aggregation error accounts for the
discrepancies in observations attributed to "technical
inefficiency." That is, "technical inefficiency" is a
ll‘EP-‘iasurement of specification error or aggregation error.
II15 the two sets of "identical" inputs produce different
an"lounts of work, and if one tries to compare them by
insisting that they are from the same input requirement
E3‘3’1t, using the same processes or sub—production function,
then one has committed either an specification error, they

(1‘:* not belong to the same input requirement set, or an

 

4O
Jgregation (measurement) error, they are not the same
1puts used in the same proportions. It is a logical

asurdity to conduct an analysis where one creates or
atroduces discrepancies between observations by mis—
pecifying the function or by aggregating heterogeneous
iputs, measures the discrepancy, calls this discrepancy
lgnificant, and attributes it to some inherent differ—
1C8 (TE) between the observations. Yet, this is exactly
1at is done in order to obtain measurements of "technical

iefficiency" between firms.

 

ml

 

 

CHAPTER THREE

PRONTIER PRODUCTION FUNCTIONS AND DISTANCE FUNCTIONS

In this chapter two hypotheses will be tested. The
first hypothesis is that frontier production functions and
distance functions are 3h; game. The second hypothesis is
that technical efficiency (TE) and price, or allocative,
efficiency (PE) are not the same. These two hypotheses
are both tested in this chapter because each has a bearing
on the other. If it is true that frontier production
functions are distance functions, then the property of
frontier production functions that TE and PE are not the

Same will be true by transitivity. Conversely, if TE and
PE are not the same, as frontier production function
theory maintains, then this condition will be consistent
"1 th duality theory as demonstrated by distance functions.
The reason it is important to test the first hypothe-

sis, that frontier production functions and distance
E"-ila'ictions are the same, is that distance functions appear
to provide a legitimate theoretical basis for the exis-
hel'lce of frontier production functions within the current
set theoretic approach to microeconomic theory [Malinvaud,
1972, McFadden, 1978, Quirk and Saposnick, 1968, Varian,
13'28]. It might appear that a frontier production

f“Auction is simply a special case of a distance function;

41

 

 

0"“.
”.I
I

’uo“

...--

-

u-OI
I

II .4

n

.t
I ‘:
IOII

0..

01-.

Q.

Cl: u
....

FA.
5'6.

.0-

.3.‘

53'

t.

42

that frontier production functions and distance functions
arise from the same theoretical basis and display the same
properties. If frontier production functions are distance
functions, then they are legitimate functions and all
their conditions, notably that TE and PE are not the same,
are valid.

The reason it is important to test the second hy-
pothesis, that TE and PE are not the same, is two fold.
First, the apparent difference between TE and PE is a
consequence of frontier production functions. If TE and
PE are not mutually exclusive, the two terms as they are
now used in the frontier production function theory are
cmnfusing, inconsistent, and misleading because they sug—
gest that there is more than one type of efficiency, when
.111 fact there is only one type of efficiency [Knight,
15333].

In order to test the first hypothesis, it will be
8blown that frontier production functions and distance
f1llrictions are not the same because they have different
theoretical origins.

In order to test the second hypothesis it will be
demonstrated that the theory of frontier production
titllnctions is incompatible with duality theory and free
‘11 Sposal (or monotonicity) . The duality principles impli—
‘=11 ‘: within a distance function prove that TE 5 PE, while
:Elb‘fiee disposal eliminates the relevance of any 'interior'
9Q ints to the lowest bounding isoquant of an input re—

q‘l irement set .

 

 

-
OI.

,
but
.0.

. -
(L)

"i

43

This chapter will rely heavily on duality theory as
asented by McFadden (1978). The purpose of McFadden's
978) discussion of microeconomic theory is to provide a
>rma1 mathematical duality between [distance] and cost
xctions...." A modified presentation of this theory is
:lined in Appendix Two. An important consequence of
[lity theory is that a distance function and a cost
action have a unique one to one correspondence with each
[er which can be conceptually and graphically represen—
l by what are termed "polar reciprocal sets [McFadden,

78] . II
3.1. TEST OF THE FIRST HYPOTHESIS

L.1. WHY FRONTIER PRODUCTION FUNCTIONS APPEAR TO BE
DISTANCE FUNCTIONS:

A superfical interpretation of a distance function
[gests a similiarity with a frontier production
nction. Mathematically, a distance function is defined
McFadden (1978) as:

(3.1) F(y, v) = Max[ a > O | 1/a * v €.V(y)]

a difference between this definition and the one found
equation (A2.8) in Appendix Two is that (3.1) neglects
distinguish between variable and fixed inputs. In

rure 3.1A, TE = OQ/OP. In Figure 3.13, a = A/B, so that

(3.2) l/a = B/A = ov'/ov

b°th cases Yo looks like the "frontier" of a frontier

>duction function so that TE = 1 when P = Q while "a" =

 

 

 

 

44

‘l~<:.

 

 

 

(A)

..-

v' I l/A V - v

_L_
F(y,v)
1/

 

(B)

~<\L

FIGURE 3 . 1

A UNIT ISOQUANT DERIVED FROM A FRONTIER PRODUCTION
FUNCTION (A) AND AN INPUT REQUIREMENT SET
DERIVED FROM A DISTANCE FUNCTION (8)

 

45

1 when v = v'. In addition, consider the

quotes:

While reformulation of duality in terms
of distance functions is potentially

 

useful in application, its primary
appeal comes from the fact that it
allows us to establish a full, formal
mathematical duality between [distance]
and cost functions, .... [McFadden,

1978, page 24, underlining added]

It is sometimes useful to extend
the definition of the distance function
to all non-negative input bundles v by
applying the formula [3.1] provided
v/[a] is in V(y) for some positive
scalar [a], and setting F(y, v) = 0
otherwise.... In applications, it is
sometimes useful to employ this ex-
tended definition of the distance
function. [McFadden, 1978, page 28]

When y contains more than one
element, efficient production of y can
be described in terms of the distance

function

[FJ(Y. v) = maxtlal>0 l 1/[al‘v € V(y)]

for (v, y) Y and v strictly positive;
the frontier satisfies [F](y, v) = 1.
[Fuss, McFadden, Mundlak, 1978, page
227]

following

From this one could easily, but mistakenly, infer a basis

for frontier production functions. Unfortunately, some

authors make the connection directly:

Notice

In addition, admissable frontier
functions must be continuous, quasi-
concave, and exhibit strong free dis-
posability of inputs. [Kopp and
Diewert, 1982, page 322]

that the conditions for the "frontier functions,"

 

46

continuity, quasi-concavity, and free disposal, are ex—

actly those conditions that apply to distance functions

(see Sections A2.1.2.4. and A2.2 in Appendix Two).

3.1.2. FRONTIER PRODUCTION FUNCTIONS ARE NOT DISTANCE
FUNCTIONS:

Frontier production functions are not distance
functions because the two types of functions have dif-
ferent origins; they are derived from different theoret—
ical premises. A frontier production function is derived
from a production solid. The frontier production function
describes the surface of a production solid that includes
both "technically efficient" and "technically inefficient"
points. "Technically efficient" points are those points
that are on the production surface (or function) and are
supposed to represent those points of production for which
it is true that the firm represented by that point "yields
the greatest output for any set of inputs, given its
particular location and environment (French, 1977, page
94]." What is frequently not made explicit, but can be
easily inferred from this definition of "technical effi-
ciency," is that a firm can be "technically inefficient"
which is represented by a point within the interior of the
surface of the production solid at the same level of
output. That is, "technical inefficiency" means that a
firm 'yields lggg that the greatest output for any set of
inputs, given its particular location and environment.’

(It will be shown in testing the second hypothesis that

 

 

47

this distinction between "technically efficient" and
"technically inefficient" points of production is incon-
sistent with reality, and therefore confusing and mis—
leading.)

A distance function does not originate as a descrip—
tion of a surface to a production solid. A distance
function is a true production function or surface.
McFadden's (1978) fundamental purpose in defining the
distance function is to obtain the 'interiors' to iso-
quants necessary to satisfy the convexity conditions re-
quired to make his "formal" mathematical proof to duality
theory. But, there are no true or observable interior
points of production that represent a solid interior to a
distance function. The 'interior' to a distance function
is represented in Figure 3.13 by point v in V(YO)' This
is not the same 'interior' as the interior to a frontier
production function, point P in Figure 3.1A. Properly
understood, the 'interior' of a distance function is not
that of a solid (points at the same level of production),

but is composed of higher order isoquants (see Appendix

Two). That is, point ve:V(yo) but f(v) ? YO' Rather f(v)
= Y1 where Y1 > YO' In addition, this 'interior' is
'swept out' for p11 practical purposes by invoking free
disposal. Free disposal is used to make f(v') = f(v).
Therefore, any 'interior' points from production solids
that might exist within a distance function are for the

sake of mathematical convenience physically and econom-

 

48

ically superfluous in any description of reality. Free
disposal eliminates fill the differences between v and v'
whatever their origin.

Frontier production functions are different than
distance functions and the first hypothesis is rejected.
Frontier production functions describe the surface of a
production solid. The existence of observable interior
points to an isoquant at the same level of production is a
fundamental condition for frontier production functions.
Only the observable interior points allow one to identify
"technical inefficiency;" that is, being in the interior
of a surface isoquant of a frontier production function.
This means that the isoquants for the complete production
set are planes rather than lines. This is not true for a
distance function which obtains its 'interior' points by
mapping higher order isoquants into a lower one. The
'interior' to the lowest bounding isoquant of an input
requirement set is not made up of additional points of
equal production, but of points of higher or greater
production. That is, an input requirement set is created
conceptually by mapping the production surface of three
dimensional space into a plane. It looks like a conven-
tional isoquant map. It is true that every point within a
given input requirement set, designated by the lowest
bounding isoquant, produces "at least as much" output as
the points on the lowest bounding isoquant, since all the
points within the lowest bounding isoquant are points on

higher isoquants which therefore produce more ("as least

49

as much") output.

3.2 TEST OF THE SECOND HYPOTHESIS

The second hypothesis to be tested is that TE is not
the same as PE. Because a frontier production function
describes the surface to a production solid, frontier
production function theory distinguishes between two sepa—
rate and mutually exclusive types of efficiency, TE and
PE. In frontier production function theory, TE is getting
the most output from a given set of inputs, given a pro-

cess, being on an isoquant rather than in the interior of

an isoquant. PE is using inputs in their least cost
combination ratio. The condition that TE i PE has two
implications. The first is that one can have two firms

that are both price efficient when only one of them is
technically efficient while the second is that one can
have two firms that are both technically efficient when
only one of them is price efficient.

The second hypothesis will be tested in two steps.
The first step will test whether or not the first implica-
tion, that one can have two firms that are both price
efficient when only one of them is technically efficient,
conforms to the principles of duality theory and free
disposal. The second step will test whether or not the
second implication, that one can have two firms that are
both technically efficient when only one of them is price

efficient, conforms to the principles of duality theory.

50

It will be shown that both implications violate duality.
Conversely, it will be shown that TE 5 PE. Additionally,
it will be shown that the term "technical efficiency
(TE)," as it is used in the frontier production function

theory, is ambiguous, confusing, and misleading.

3.2.1. THE CONDITIONS FOR THE TESTS:

Whether measuring distance functions or frontier
production functions, one assumes that any two firms, "A"
and "B", are using the ppm; process and the gppg cost
function, and that the principles of duality apply to the
distance (production) or frontier production function, and
the cost function, which is to say, that for both firms
their production or cost functions are identical. There-
fore, any technically efficient point on a frontier pro—
duction function is implicitly associated with correspon—
ding prices in cost space.

Duality between product space and cost space means
that if one holds output constant and changes prices,
e.g., by a scalar, then one maps from the same isoquant in
product space to different isocosts in cost space. Simi-
liarly, if one holds total cost constant and varies input
quantities, e.g., by a scalar, then one moves from the
same isocost line in cost space to different isoquants in
product space. Finally, if one holds both output and
total cost constant then duality means one isoquant maps
to one isocost line (this is a special case). If one

holds neither output nor total cost constant, then duality

51

does not apply since points in product space can map to
more than one point in cost space, and vice versa.

Polar reciprocal sets represent the set of points for
which duality exists between points in input (product)
space (excluding Stage III) and cost space. In order to
define the full set of mappings one must assume that both
price and input ratios vary in both spaces simultaneously.
This means that a duality mapping is always between a
point that is technically efficient in input space and a
point that is price efficient in cost space. All points
in cost space represent least cost bundles by definition
of a cost function.

The theory of frontier production functions assumes
that firms are purchasing their inputs in perfectly compe-
titive markets so that input prices are fixgg and the same
for all firms. This means that only one price ratio
defines PE, the ratio of the given fixed prices.

Consider the polar reciprocal sets of a distance
function and what frontier production functions might
suggest is true that would assure that frontier production
functions conform to the same principles of duality that
distance functions do (see Figure 3.2). Figure 3.2A,
V(y), is the input-conventional input requirement set from
a distance function and 3.28, R(y) is the input conven-
tional factor-price requirement set from a cost function.
Together they represent the distance function and the

total cost function respectively for both firms "A" and

52

 

 

 

 

 

 

 

 

A
VIN V(yo)
5
'1‘0
"”‘ ' :7.
2
I
A. . . :1—0
1 a”. I
v: o
5
’2
1- c
I a a .
2 o y. '{1
y “r " T
\
O 'rqdlllc (annotfl r1,
0 ‘0 1 ’. O
L '2 o '3 C .
‘_W - —. w - -
'1‘. '9;
I
IN
' My.)
1
to: y. it up" in (no ). at. c:2
P1 1
'“"-F
‘ I: I.
:
my: “Inning from.) at:
.0 2
f1 1
Ila!- 0- '—.
I
'2 1
I
' O
1 I
I
'1 o
10'- - ..
I: o
I' ,
‘ I ‘0
. J—
‘0
:2 I, 0
use .0
N
to:
I; y f- 7. '.
'l e
‘5 I; o
v >
or y. A' ’3
9‘ o
“r " T
I: o

FIGURES 3.2

AN INPUT REQUIREMENT SET V(y) FROM A DISTANCE FUNCTION (A)
AND A FACTOR-PRICE REQUIREMENT SET R(y)
FROM A COST FUNCTION (B)

53

"3°" Let points A0 and BO represent two firms "A" and "B"

respectively. Either all the firms on the isoquant yo
produce at the same total cost (a special case), or one

firm is the least cost producer. Assume that for output

Y firm "A" (point A ) is the least cost producer. The
O 0

prices for WhiCh A0 has a mapping in cost space are PlA'O

and P2A'O; i.e., A0 maps to A'0 in Figure 3.28 at output
YO and at a total cost of co. Similiarly, 80 maps to 8'0
at a total cost C1 > CO. In Figure 3.2A, the ray OAo

represents a path along which the ratio of the prices for
the inputs (PZA'O/PIA'O) remains constant. At point A0 in
Figure 3.2A the line bb' represents the budget constraint
With slope ‘ PZA'O/PIA'O. Point Ao maps to point A'o in
Figure 3.28 when the input prices are PIA'O and pzA'o and
total COSt equals CO' Point 80 in Figure 3.2A maps to
point 8'0 in Figure 3.28 when the input prices are PlB'o
and PZB'O, and total cost c1, and to point 8'1 when input
prices are PIB'l = kPlB'O and PZB'l = kP28'2, and total
cost equals c2 = kc1 (where k equals some coefficient of

proportionality greater than 1.0). Therefore, the ray
OB'1 in Figure 3.28 represents the locus of points in cost
space that map from B0 in product space as prices and
total cost increase in proportion. Notice that at any
P0111t on the ray 03'1, the ratio of input quantities
(VIBO/szO), remains constant since all points on the ray
map from 80. This is analogous to all the points on the
ray OAo having a constant price ratio. Point Ao maps to

A'o, and point 80 maps to 8'0, and vice versa, by duality.

54

The price level is the set of prices that will keep
the total cost relationship between firms constant when
output is held constant but the ratio of input quantities
used varies. Notice again that if the general price level
faced by "A" and "8" were to increase by |1-k|, Ao would
map to A'1 and 80 would map to 8'1. That is, relative
prices and relative total cost would remain the same in
comparing "A" and "B," but the ceteris paribus conditions
would not be violated only if the prices for 293p "A" and
"8" increased by the same proportion |1-k|; if the prices
paid by "A" and "8" remain of the same magnitude. The
magnitude, or price level, paid by the firms is important
since "A" and "8" would both continue to produce yo at the
higher price level only if the price for y were also in-
creased by the proportion |1-k|. Otherwise, "A" and "8"
would produce Y1' Y1 < yo, at A2 and 82 respectively since
the income from the sale of the output would only allow
them to purchase a lesser amount of both v1 and v2 at the
higher input price level.

In demonstrating why duality theory refutes frontier
production functions, close attention to three aspects of
duality theory is especially important: (1) Both the input
(product) space and the cost space for the firms con-
sidered must conform to the assumptions of being input—
conventional. This means the input requirement sets and
the factor-price requirement sets for the firms considered

must display not only regularity conditions, but also

55

display (1) monotonicity. or free disposal, and (ii)

strict convexity from below [McFadden, 1978].

It should be noted that the one-
to—one link between the input—conven—
tional classes described above does not
hold between input—conventional cost
structures and the input-regular pro—
duction possibility sets. Distinct
input-regular production possibility
sets may yield the same input-conven—
tional cost function. However, while
going from the production possibility
set to the cost function can entail a
real loss of technological information
in this case, the information lost is
precisely that which is superfluous to
the determination of observed
competitive cost minimizing behavior.
[McFadden, 1978, 22]

... However, input-conventional cost
structures and distance functions are
defined to have identical mathematical
properties with respect to their second

arguments, input prices or inputs
respectively. [McFadden, 1978, page
26]

(2) The dual prices represented in the cost space of the
duality mapping are the prices which would make the pur—
chased input bundle the least cost bundle. The prices
associated with a particular input bundle may or may not
be market prices. Market and dual prices are not neces—
sarily equal, and may necessarily be different. One must
keep very close track of whether or not prices are market
prices, dual prices, or both. Between firms for which
duality is assumed to hold, only the dual prices are
relevant for the purposes of making comparisons between
the firms whether or not they are market prices. (3)

Finally, and most importantly, the level of prices must

56

remain constant across firms being compared. This means
that the difference between the dual prices of two firms
may be different with respect to their ratios, but one
firm cannot have dual prices of a greater absolute magni-
tude than another. To do so would violate the usual
ceteris paribus conditions. For example, firm "A" may
have dual prices (P150, PZAO) and firm "8" dual prices

(P B0, 9230) where PIAO/PZAO # PlBO/PZBO. But firm "A"

1
cannot have dual prices (Ple, pon) and firm "8" dual
prices (kPIBO, kPZBO) where |1—k|, and k # 0 represents
some percentage difference in the prevailing price level
faced by "A".

3.2.2. WHY FRONTIER PRODUCTION FUNCTIONS ARE NOT COMPATIBLE
WITH DUALITY THEORY - FIRST PART:

This section will test the implication that two firms
can be price efficient when only one of them is technical—
ly efficient. This means that both firms are on the price
efficient ray, e.g., line OAo in Figure 3.2, but one of
them is pp; on an isoquant (it is a true interior point),
while the other one is on an isoquant. That is, one firm
is technically efficient, and the other one is "technical—
ly inefficient," meaning it could produce more output with
the same resources than it is 1p £353 producing without
changing the amounts of the inputs being consumed in

production.

57

In this section a proof will be offered to demon—
strate that frontier production functions are incompatible
with duality theory and free disposal given the charac—
teristic of frontier production functions that there is an
interior to an isoquant at the same level of production.
Apart from the fact that it has already been shown that
production sets cannot be solid due to the laws of thermo-
dynamics (see Chapter Two and Appendix One), it will be
shown that solid production sets are contrary to duality

and free disposal.
3.2.2.1. THE INTUITIVE ARGUMENT:

Consider Figure 3.3 which represents the presumed
input requirement set and factor—price requirement set for
a frontier production function. If one can produce the
same output as one produces at 80 using less of both
inputs and the same process, i.e., produce at A0 with
absolutely no other changes, then why cannot one produce
the same output as one produces at A0 using less of both
inputs? Differences in TE, as the term is used in the
frontier production function theory, must involve some—
thing else in addition to merely a change in the level of
V1 and v2. Since that something else is never defined or
explicitly included, its cost is implicitly ignored, and
therefore comparisons of "technical efficiency" are incom—
plete and invalid.

Consider the polar reciprocal sets in Figure 3.3 for

two firms "A" and "8" which have the same technology and

58

A
[IN V(yol
'1
can. el to: both I. and I1
. a ‘2
”1
A0
'1
slop. - -»

'2

I. if y. or II it 71

 

 

 

 

 

A. 0: I1 yo or 11
’0 again c° to: A9
< .1090 .— '_2
’1
\
0 V: I
1‘ '
P: Ityol
equal: 7. if upping (m A. “I, a: :1
”1
'W'“fi'
or 2
equal- ’1 :2 min from I: at :1
”1 '
I10" " F
2
’1
. new - --
'9 f ’2
o C
A. 1
.1 c. «all: y. u A.
'0 )N .. ti
’1
o . P2 ’
FIGURE 3.3

AN INPUT REQUIREMENT SET V(y) FROM A DISTANCE FUNCTION (A)
AND A FACTOR-PRICE REQUIREMENT SET R(y)
FROM A COST FUNCTION (B)

59

the same cost function conforming to the assumptions and
principles of duality.

Assume for an initial situation that firm "A" is
represented by point A0 and firm "8" is represented by
p01nt BO in Figure 3.3A, that both firms produce exactly
the same product yo, and that both firms purchase their
inputs in a perfectly competitive market at fixed prices
(P1, P2). According to the theory of frontier production
functions, this means that firm "A" is both technically
efficient and price efficient, while firm "8" is price
efficient but not technically efficient, as 80 is a "true"
interior point to the isoquant yo, That is, firm "8" uses
more Of both inputs, V1 and v2, to produce the same amount
of product as firm "A."

Therefore in Figure 3.3, using frontier production
function theory it appears that AC and BO map to points
A'O and 8'o respectively. Indeed, if one calculates the
total cost for firms "A" and "B" at the given market
prices (P1, P2), then C(80) = c1 is greater than C(Ao) =
Co, and those values would be located in cost space at 8'0

and A'o respectively.

3.2.2.2. PROOF USING DUALITY:

Notice that at 8'0, the dual prices for firm "8" are

at a higher price level than the market prices and dual

prices for "A." (P1, P2), but in the same ratio, i.e. they

are higher by a constant, |1-k|, where k > 1.0. Now,

 

(I1

‘1

’1

"I

[(4)

(n

60

using duality, find the points in product space that map
from A'o and 8'0; the points in product space for which
A'o and 8'0 represent the least cost bundle as required by
the definition of a cost function. Notice 8'o will map to
31 since B'o's dual prices are P1 = kP1 and P2 = kP2: that
"8" has dual prices that are |1-k| times greater than "A."
But at 81, f(Bl) = y1 = y0 only if 81 is at A0, since
there cannot be two different least cost bundles at the
same level of output. Otherwise, production at 81 must
equal yl, where y1 > yo; that is, there are two least cost
bundles, but at two different levels of output. Notice
production at 31 is not equal to production at 80 so that

81 is not lacated at 8

0’

In neither case does 8'o map to 80. Therefore, our
initial assumption is inconsistent with duality since firm
"8" cannot be at both points 80 and A0, or points BO and
81, at the same time. Thus, there cannot be any frontier
production function "technically inefficient" points that

conform to duality. Two firms cannot be price efficient

when only one is technically efficient.
3.2.3.3. PROOF USING FREE DISPOSAL:

One might reason that since point 80 is a "true"
interior point, it does not map to a point 8'o in cost
space, but to 3'1. For instance, McFadden (1978) suggests
that the 'interiors' to the isoquant of a distance
functions map to exteriors of factor-price requirement

sets, and vice versa. In so doing one keeps production at

 

61

a constant level in product space and total cost at a con—
stant level in cost space therefore maintaining duality.
In order to map BO to 8'1, one must have dual prices for
the greater quantities of the inputs used at 80 that are
lower than the market prices (P1, 92), lower by a factor
|1—k|, where k < 1.0, which is a violation of the ceteris
paribus conditions since this implies a change in the
price level paid by 80' Therefore, a correction or ad-
justment must be made to keep all comparisons between "A"
and "8" on an equal footing. This means that given fixed
market prices (91' P2), one must freely dispose of the
difference in total cost between "8" and "A," c1 — CO' in
order to have points in cost space for which total costs
are equal (as they are for 8'1 and A'o),

When one uses free disposal in product space one
freely disposes of both input app output (see Appendix
Two). In cost space, total cost corresponds to the output
of product space, while prices correspond to the inputs of
product space. Therefore, when one freely disposes of the

difference in total cost between "8" and "A," this free

disposal is the same as disposing of the difference be-

tween the dual prices for 80 and the market (also dual)
prices for A0. It is the same as moving from 8'1 to A'o
in cost space by free disposal. By the same argument,
free disposing of the total cost difference between "8"
and "A" is the same as disposing of the difference between

the dual prices for 8'0 and the dual prices for A'o; that

62

is, it is the same as moving from 8'0 to A'O in cost space
by free disposal. But by duality, if one freely disposes
in cost space, one must freely dispose in product space
and vice versa. Therefore, moving from 8'1 to A'o, or
from 3'0 to A'o, in cost space by free disposal means one
is necessarily moving from 81 to A0 (where 81 # A0), or 80
to A0, in product space respectively at the same time,
which eliminates any difference in "technical efficiency,"
as the term is used in the frontier production function
theory, between firms "A" and "8."

The possibility that one might define or locate a
point 3'1 in cost space, where cost is equal to co, raises
two other issues. First, this may be the reason that some
frontier production function authors maintain that im-
provements in "technical efficiency" are costless [Bagi
and Huang, 1983], while others maintain that there is a
cost to "technical inefficiency [Kopp, 1981a, and Schmidt
and Lin, 1983]." Second, and more importantly, the exis-
tence of a point 3'1 at a total cost co would seem to
violate the assumption of strict convexity from below that
is implicit in a cost function. Therefore, if cost
functions cannot have exterigrs at a constant total cost,
in order to maintain strict convexity from below, distance
functions cannot have interiors at a constant production.
Thus, the implication that two firms can be price effi-
cient when only one firm is technically efficient is
refuted. Any firm that is price efficient is always

technically efficient.

63

3.2.3. WHY FRONTIER PRODUCTION FUNCTIONS ARE NOT COMPATIBLE
WITH DUALITY THEORY - SECOND PART:

Section 3.2.2. refuted the implication of frontier
production functions that TE and PE are mutually exclusive
and that two firms can be price efficient when only one of
them is technically efficient. This section will consider
the other implication that two firms can be technically
efficient when only one of them is price efficient since
the fact that PE implies TE does not necessarily mean that
TE implies PE.

Recall the frontier production function definition of
TE, that one produces the greatest amount of output given
a bundle of inputs and a specific process. This defini—
tion leaves open the possibility that one can produce
something less than the greatest amount of output and
still consume all the inputs. This would mean that pro-
duction functions have true interiors at the same level of
production. Section 3.2.2. eliminated using this possi-
bility to distinguish between PE and TE. This is not
surprising since the theory of thermodynamics clearly
demonstrates the same thing, where thermal efficiency is
considered to be an example of technical efficiency (see
Appendix One). In thermodynamics, technical efficiency,
or thermal efficiency, is determined only within the con—
text of comparing value of input (heat) to value of output
(work), a point made in economics by Knight (1933) and

Boulding (1981).

64

Therefore, the term "technical efficiency (TE)," as
it is used in the frontier production function theory is
ambiguous, confusing, and misleading. The term efficiency
implies a comparison. In order to make a comparison, one
must first have a basis for making the comparison; one
must in effect have a numeraire. The frontier production
function theory definition of TE, that one produces the
greatest amount of output given a bundle of inputs and a
specific process, is ambiguous, confusing, and misleading
since for any point on a production function there is only
ppg level of output possible. That is, any point on a
production function not only represents the greatest
amount of output given that bundle of inputs, it also
represents the smallest amount of output given that same
bundle of inputs, if all the inputs are consumed (which is
true by definition). Therefore, to define TE as being on
any point on an isoquant is ambiguous, confusing, and
misleading since by itself there is no basis for compari-
son; there is no counter point that represents any other
degree of "technical efficiency" given the same set of
inputs and the same process. Production means creating
utility by changing the time, form, or location of inputs.
Therefore, efficieny involves implicit comparisons of
value (Knight, 1933; Boulding, 1981]. Value must be
measurable for there to be a basis for comparison between
two points of production: one needs a numeraire.

Physical measures alone divorced from their specific

value (i.e., from a numeraire) are an inadequate for

65

comparing efficiency. Marginal physical products of
inputs, which are determinate once a production function
is defined, have value, as does the output they represent.
They are technical coefficients of production. But as
physical quantities their specific values are not indi-
cated and, as such, they are an inadequate basis for
making efficieny comparisons.

Consider the case where one is faced with a set of
measurements of marginal physical products for various
sets of the same types of inputs used in the same produc-
tion process. If TE was determined by the set of coeffi-
cients of the largest magnitudes (hence the largest
value), then TE would imply selecting those input bundles
where average physical products are their largest, which
represents the unit elasticity point on the production
function, or the boundary between Stages I and II (for a
further discussion of this case see Chapter Four). If
price efficient points are always technically efficient,
as Section 3.2.2. showed, then the set of techniclaly
efficient points must include points that are at points of
unit elasticity on the production function. Even if one
assumes that all the unit elasticity points will lie on
one isoquant, which is not generally true, one might still
ask if any one of the unit elasticity points is more
"technically efficient" than the others. This is the same
as asking if any one point on an isoquant can be more

"technically efficient" than the others.

66
Consider two sets of inputs, v10 and v2O that produce

a given level 0f output, Y0. The marginal physical pro-

ducts are determinate with MPP(v10) = X01 and MPP(V20) =
X02. If one changes the amounts of the inputs used, e.g.

to V11 and v21, then the marginal physical products will

no longer be X01 and x0 1

1
2!

but MPP(v11) = x
o

1 and MPP(v21)

> X11 and

2!

= x where x01 ¢ x1 and x 2 ¢ x12. If x0

O O
X 2 > x12, then f(vlo, v2 ) = y1 where y1 > yo. There-

1, then X02 < X12, for f(vlo,

1 1

fore, in general, if X0 > x1

1

v20) = f(vll, v21) = yo. In this case, notice that the

inherent values of v11 g v10, and v21 ¢ v2O

so that one
might ask if one set of inputs (v10, v20) or (v11, v21) is
"technically more efficient" in producing yo. To answer
this question, one needs to have a basis of comparison, a
basis for valuation; i.e., one needs a numeraire or a set
of relative prices.

In economics, the utility function provides a basis
for measuring value, by allowing one to derive the demand
for all commodities based on the value (utility) they each
provide. When one selects a numeraire, one commodity, the
utility function serves as a basis for evaluating the
relative value of all other commodities (including money)
by comparing them to the numeraire; specifically by estab-
lishing how much of a specific quantity of a specific
commodity will be equal in value (utility) to a specific
quantity of the numeraire. A change in the underlying

utility function will change the relative measures of

value.

67

In economics, relative prices represent specific
measures of value because they are determined by normali-
zing commodities on a numeraire. Clearly, associated with
any specific production function are an infinite number of
sets of relative prices, each set determining a unique
total cost function. The relationship between the rela-
tive prices for any two points on the production function
remains the same, in direction if not in magnitude, re-
gardless of which set of relative prices is selected to
measure the value of the physical quantities if they are
based on the same preselected utility function. In gen-
eral, changes in relative prices from one set of relative
prices to another is accomplished either by vector multi-
plication (changing the utility function) of all prices
and/or a change in the choice of numeraire which deter-
mines the measurable degree of difference between relative
prices. A change in the utility function causing a change
in relative prices is accomplished by vector multiplica-
tion of prices, where the elements of the vector are not
equal. Any comparisons using the resulting two sets of
prices is fundamentally a comparison of the change in
utility. A change in numeraire, holding utility con—
stant, will change relative prices, but not the direction
of change between those relative prices, only the magni-
tude; i.e., the ranking of preference for the individual

commodities will be unchanged.

 

68

Therefore, while the prices for the physical quan—
tities represented by a point on a production function can
change, any set of prices is as valid a representation of
the instrinsic value of those physical quantities as any
other set of prices, if they are derived from the same
utility function; i.e., one holds the utility function
constant. If one holds the utility function constant,
then any change in prices will be the result of either
scalar multiplication, multiplying all prices by a con—
stant, which leaves relative prices unchanged, or a change
in the numeraire, which leaves the fundamental values of
the commodities unchanged. This is why economists prefer
to make comparisons based on sets of "real" prices, sets
of prices using the same numeraire that are not different
by a scalar, and why in a duality mapping a valid compari-
son of points must hold relative prices at the same level
of magnitude, or price level, as was pointed out in Sec-
tion 3.2.1.

In this thesis, all the prices are designated as
algebraic unknowns or variables, which means they can be
any set of pea; prices from the infinite number of sets of
prices with no change in the arguments presented, assuming
there is only one underlying utility function upon which
all comparisons are based. This is a necessary ceteris
paribus condition for comparisons to be on an equal
footing. If two points on a production function are
designated as equally technically efficient at the same

level of output, as the term is used in the frontier

69

production function theory, and the two points represent
two different levels of utility, then since production is
by definition a change in the utility of inputs through a
change in time, form, or space, the two points are in fact
on two different value productivity functions since they
represent two different levels of utility (or output).
Specifically, this means that any two points on an iso-
quant that are designated as equally technically efficient
must reflect the same level of utility on the same utility
function; i.e., two or more points on an isoquant can be
equally technically efficient only if everything is held
to be the same. That is, once one has normalized all
prices by selecting a utility function and a numeraire,
there is only one set of relative prices associated with
the value of, or physical quantity of, any given bundle of
inputs and output.

Therefore, "TE," as the term is used in frontier
production function theory, is either associated with spe-
cific relative prices by virtue of being associated with a
given utility function and a numeraire, or it is mean-
ingless since there is no basis for the comparison of
efficiency of "TE" points. Specifically, if "TE" is not
meaningless, then any point that is techniclly efficient
is also price efficient. For the purposes of testing the
implication that two points can be technically efficient
when only one of them is price efficient, it will be

assumed that the term TE is meaningfully distinct from PE.

 

 

 

70

3.2.3.1. THE INTUITIVE ARGUMENT:

Consider the implication of frontier production
functions that TE and PE are mutually exclusive and that
two firms can be equally technically efficient at the same
level of output (utility) when only one of them is price
efficient. If two firms are equally technically effi-
cient, it means that the two firms are on the same iso-
quant and can produce pp mpg; than that output with the
(value of, or quantity of) the resources at their disposal
when input (and output) prices 3;; fixed. If two firms
"A" and "8" are producing the same output Y0' and they are
purchasing their inputs in perfectly competitive markets,
so that only one of the two firms, assume it is "A", is
price efficient, then the firm that is not price effi-
cient, "8," is not producing as much output (utility) as
it could given the (value of, or quantity of) the resour-
ces at its disposal. Therefore, "8" is not technically
efficient. If "8" is not price efficient, then its cost
is not as low as it could be if it were a least cost
producer; its total cost is higher than it would be if it
were a least cost producer. Therefore, in the perfectly
competitive markets, "8" can costlessly rearrange its
input bundle so that the inputs are used in the same ratio
as they would be were "8" a least cost producer, i.e., in
the same ratio as the input bundle used by "A," and at its

same (original) cost produce more output that "A."

71

3.2.3.3. PROOF BY DUALITY:

Consider Figure 3.4, which is essentially like Figure
3.2. Frontier production function theory suggests that
firms "A" and "8" are initially at points A0 and BO res—
pectively, while prices are fixed at (PIA'O, pzA'o) for
both firms. That is, both "A" and "8" are equally techni-
cally efficient for output yo, but only "A" is price
efficient. Recall that TE means producing the greatest
amount of output (utility) possible from a given set of
inputs in a given process. It was pointed out above that
this definition is meaningless unless one recognizes that
with every set of physical quantities there is associated
a specific set of relative prices (subject to change only
by vector multiplication or a change in numeraire). For
the frontier production function definition of TE to be
rhetorically and internally consistent, it must not change
whether prices are considered or not; the definition of TE
must be invariant with respect to prices, or a lack of
prices, if there is to be a meaningful separation of TE
from PE. But this is not true. Including the PE condi-
tions, concurrent with the TE conditions, for the puposes
of making comparisons between firms means relative dual
prices are established. Frontier production functions
cannot be used to consider TE without concurrently con—
sidering prices as a measurement of the value of the

technically efficient bundle.

 

72

 

 

 

 

 

 

 

 

A
VA\ “1.)
1
b
Vx‘o
IlopI - --
'2 O
A. '1'.
I1 I10” - -—.-
v: o
5
’2
II c
I I 2
2 I ’ 7:;
° “9' ‘- 1T
’1 I11
\
° " 1. c 1 is c u I T: ’
(“I ’ A' A 3' °
L I, o I: o
1* fi —. w ~ —.
'1‘. '1'.
I
’A My.)
1
to: y. 1.! I391” tro- n. It :2
'1 1
I '1
(z...— 2
r
to: y: it upping from I I: c
.. 1. 2
'1 l
“09- '- '——.'
I
P I
. 2
v o
I r
'1 o
10'- '- .
I ’0
2
I' .
1 212°
“”'—F
I o
‘2 2
'3‘. '0
lot
.i 4 l r ’. I'
I! o
I: o
5
,>
v
or y 2
o ”A;
“r " "T"
I: o

FIGURES 3.4

AN INPUT REQUIREMENT SET V(y) FROM A DISTANCE FUNCTION (A)
AND A FACTOR-PRICE REQUIREMENT SET R(y)
FROM A COST FUNCTION (B)

73
But fixing prices (which fixes the efficient price
ratio) by assuming that both "A" and "8" buy their inputs
in the same competitive market, and that the prices they
face are the prices that make AO price efficient, (PIA'Ov
PZA'O), means PE is determined, but that 80 does not map
to 3'0. For 80 to map to 8'0, firm "8" would have to have

dual prices equal to (PIB'O, P BlO), at a total cost of c

2 1
when it is paying market prices equal to (PIA'O, PZA'O).
Therefore, calculate C(80), the cost at "8," at

A

prices (PIA'O, P2 '0) using the cost function shared by

"A" and "8," and find the location for the resulting value

in the factor-price requirement set.

Notice that C(80) # c1 at prices (PIA'O, pzA'O). If

C(80) = Cl, then 8'o represents a least cost producer at
both market prices (PIA'O, P2A'O) and at market prices

(PIB'O, PZB'O) at a total cost c1, using inputs V130 and

szO to produce yo. But this would violate the condition

that if one holds output constant and changes prices, then
one moves from the same isoquant in product space to
different isocosts in cost space since PIA'O ¢ pIB'o and

A' Bl
P2 0 # P1 0.

Notice, too, that C(80) { c1 at ,prices (P1A O.

8' 8'
then P O and P O are not the

1' 1 2
Al
, rather (P1 0, P

Al
P2 0). If C(80) < c

least cost prices at 8 A'0) are.

0 2

The location of C(80) must be on a ray from the

origin in the factor—price requirement set so that the
ratio of the inputs in the input bundle purchased by firm

"8," (VIBO/VZBO), is not changed. Therefore, C(80) is 10-

74

cated not at 3'0, but at 8'1 and at a total cost c2 > c1.

I
Given 80 at fixed prices (P1A O, PZA'O) using the cost
function it shares with "A," firm "8" is located at 8'1 in
the shared factor-price requirement set. That is, if

firms "A" and "8" share the same production and cost
function, then at fixed market prices the locations for
those firms in cost space are at A'O and 8'1 respectively.

The prices (PlB'l, PzB'l) represent not the market
prices which were used to calculate the value C(80), but
the derived prices that make the input bundle represented
by 8'1 a least cost bundle, and the price ratio that would
make the firm "8" PE, since all points on the cost
function are least cost bundles by definition of a cost
function. The bundle purchased by "B" is in fact not
price efficient, since the ratio of the dual prices,
(PlB'I/PZB'l), is not equal to the ratio of the dual

prices, (PIA'O/PzA'O) for the price efficient producer,

"A." Notice that at 8'1 the dual prices (PIB'l, PZB'I)
are absolutely greater than the prices at 8'0, (PIB'O,
923'0). That is, 913'1 = kPlB'O and 923'1 = kPZB'o, where

k > 1.0. At the fixed prices P A'0 and PZA'O, "8" enters

1
the duality mapping at 8'1. This implies that "8" is not

only generating dual prices that are inefficient,
Bl Bl Al Al

P1 1/P2 1 # P1 O/P2 0, but also generating dual prices

that are higher than the level of prices paid by "8."

That is, the magnitude of the dual prices paid by "8" are

greater than the magnitude of the dual prices paid by "A,"

75

by |1-k|.

Because of this change in the magnitude of the dual
prices generated by "B" at 3'1, one must make an ad—
justment to eliminate this implied difference in the mag-
nitude of the dual prices generated by “A" and "B" in
order to maintain ceteris paribus conditions in the com-
parison. Now, having located 8'1 in cost space, under the
assumption that both firms face the same fixed market
prices, where does 3'1 map back to in its corresponding
shared technical (product) space by duality? That is,
what point in the technical (product) space would map to
8'1 in cost space where the dual prices P13'1 and sz'i
would be at the same price level as the dual prices gen—
erated by "A." The point is either 81 or 32, not to 30'
Therefore, the bundle purchased by firm "8" no longer
represents the same technically efficient level of output
(utility) "A" does. 8y duality, points in product (cost)
space have a unique one—to—one mapping with points in cost
(product) space.

If one wishes to maintain "8"'s greater expenditure
capacity, "8"'s larger budget, or total cost, c2, then "8"
must produce a greater output y2 > yo, to provide "8" with
the greater compensating income to cover its greater cost
c2. This means "8" would be represented by 81 in the
duality mapping.

If one wishes to maintain the same budget constraints
for both "A" and "8" at CD and c1, thereby maintaining the

original total cost relationship between the two firms,

76

then "8" must purchase fewer inputs and produce a lower
output Yl' where y1 < yo at c1, and therefore be repre—
sented by 82 in the duality mapping.

In either case, within the duality mapping "8" cannot
be technically efficient at a production level y0 while

paying market prices (PlA'O, P A'O). Therefore, there ap—

2
pears to be a contradiction between the initial situation,
two firms "A" (represented by A0) and "3" (represented by
80) both technically efficient at the same level of ouput
(on the same isoquant), and the final situation, two firms
"A" (represented by A0) and "8" (represented by 81, or

82), on two different isoquants. The firm "8" cannot be

at 80 and 81, or 82, (on two isoquants) at the same time.
The conclusion is that being off the isocost line, not
being price efficient, means by duality that the firm is
also off the isoquant Y0 and not technically efficient at
the same level 0f output Yo. Therefore, the only firm
that is technically efficient on the yo isoquant is "A,"
because it is the only firm that is price efficient. Firm
"8" could produce more output with the budget (resources)
that allows it to produce yo (when its not price effi—
cient), or it could produce less output if it can spend no
more than c1. Therefore, TE 5 PE always, by duality.
This should not be a surprise to persons familiar with
Knight's 1933 and Boulding's 1981 arguments.

Buying 80 inputs at market prices (PIA'O, P A'O) is

2
identical to buying 81' or 82, inputs at dual prices

77

(PIB'l, PzB'i). The duality exists only between 8'1 and

81' or 82, for firm "8" at the fixed market prices (PIA'O,

P A'0). That is, there is no duality mapping on the cost

2

function for 80 at prices (PIA'O, PZA'O) and an output

level yo, since 80 is not a least cost bundle for yo and

prices (PIA'o, PZA'O). Therefore, in measuring efficiency
for firms for which the duality principle is assumed, the
relevant points for firm "8" are 81, or 82, and 8'1, which

means that at the assumed market prices "8" is neither

technically efficient nor price efficient at output level

Yo ; only "A" is either, and it is both, technically effi—

cient and price efficient at output level yo,
3.4. SUMMARY

Distance functions are based on production functions

"hi ch are input-conventional, meaning they conform to
regularity conditions, free disposal (monotonicity) , and
stI‘ict convexity from below. Using duality, one can find
corresponding cost functions which must also display free
d1 eposal and strict convexity from below (quasi-concavi-
ty ) . Distance functions are valid representations of
be ality (excluding Stage III of production). It might
eaGily be mistakenly inferred that frontier production
fulctions are also valid representations of reality be-
c§‘use they look the same as distance functions and bear a

g“:I-perfical appearance to distance functions when graphs of

the two are compared (see Figure 3.1) .

..lnfi
UIIS“

9"?

\n I

3....
.‘v s

..
an .
3V0:

bani

' r!

78

Section 3.1 tested, and rejected, the hypothesis that
distance functions and frontier production functions are
the same, since they arise from different theoretical
origins. Distance functions arise from a production set

that has no true interior points of production to its
isoquants at the same level of production. Frontier pro—
duction functions arise from a production set that is a
solid set with interior points to its isoquants that
represent the same level of output as the isoquant.
Therefore, distance functions can have no "technically
ine fficient" points of production, as the term is used in
fzrc>11tier production function theory, within its corres—
Ponding production set.

Frontier production functions have the fundamental
pI'OEJerty that TE and PE are mutually exclusive conditions
of reality; that there are in reality these two different
types of efficiency, rather than only one type of effi-
ciency, simply efficiency. If these two kinds of efficien-
CY validly reflects reality, they should conform to the
principles of free disposal and duality between product
and cost space. The hypothesis that TE and PE are not
the same was tested in Section 3.2 in two parts by testing
8&9 Erately the two implications of the hypothesis. The
rib Qt implication is that two firms can be price efficient
wh§m only one of them is technically efficient. It was
shQ'Vm in Section 3.2.2. that this first implication is
curEttradicted by free disposal and duality and therefore

t
list a firm that is price efficient is necessarily techni—

 

79

cally efficient at the same time. The second implication
is that two firms can be equally technically efficient at
the same level of output when only one of them is price
eefficient. Section 3.2.3. demonstrated that this implica-
tLion is contradicted by duality, and therefore that any
firm that is technically efficient is necessarily price
efficient at the same time. Therefore, TE and PE are
always identically equal and the two adjectives "techni-
cal" and "price" add nothing to the discussion.

The consequences of testing these two hypotheses are
twofold. Frontier production functions are not the same
2:5; distance functions since they arise from a different
theoretical origin than distance functions, that of a
solid production set including both "technically effi—
c“1<311t" and "technically inefficient" points of production.
firrleslrefore, in production theory, only distance functions
sl'1<>1..1ld be used as the basis for theoretical explanations
of reality (excluding Stage III of production), or for any
amp drical analysis that excludes Stage III of production,
since distance functions alone reflect the physical reali-
t3? that production sets are hyperplanes (surfaces) in
irllgivut space rather than solids (see Appendices One and
TNQ ) .

The consequences of testing the second hypothesis,
tkjt‘sht TE and PE are not the same, are that the terms "TE"
aljt‘ii "PE" as they are used in the frontier production

E
uhction theory are ambiguous, misleading, and confusing,

since they
There is 0
ES“ def
the ml”
1‘] “835
ietemi

r. v'JLAv

parin

80

since they imply that there is two kinds of efficiency.
There is only one -- simply efficiency. Since production

is by definition a change in the time, form, or space of
the utility of the inputs, a production function implicit—
ly measures utility or value. Therefore, in order to
determine the degree of efficiency for any point on a
production function, one must first have a basis of com—
paring the intrinsic value represented by a point; i.e.,
one must first choose a numeraire. Once a numeraire is
designated, all points on a production function are asso—
ciated with a set of relative prices that serve as a
measure of their value or utility. A change in those
relative prices can be accomplished only by vector multi-
Plication, which means that one has changed production
(utility) functions, or by a change in the numeraire,
which will not change the direction of differences between
‘tllee original relative prices, only the magnitudes of those
dj~152ferences. Therefore, "TE" as it is used in the fron-
tier production function theory must be implicitly asso-
Ciated with specific relative values, or prices, or it
'inm]E>>lies a comparison of points without a basis for com—
pat‘ison. In this case, the adjective "technical" loses
'1‘:WEE: meaning as the distinction between "T8" and "PE" is
‘vjl ”t:hout foundation. If technically efficient points in
ft‘SDntier production function theory are associated with
sgQcific relative values, or prices, which would be impli-
ctdl Utly logical since they stands as counter points to

9% :lnts that are price efficient, points which are expli-

113' 35

c‘.
are tec
vice v1

6 P)"j

5 vs.“

versa

vs.
»9\ )

g)

81

citly associated with values, or prices, then points that

are technically efficient must also be price efficient and

vice versa. TE always means PE by duality; a firm that is
'technically efficient is always price efficient and vice

tiersa. The converse is also always true; a firm that is

IlOt price efficient is not technically efficient and vice

versa. Consequently, in discussions of efficiency, the

aicijectives "technical" and "price" are ambiguous, mis-

leading, and confusing, and might best be dropped from the

discussions. Efficient production is always determined by

treaJlues as Knight (1933) pointed out over fifty years ago.

 

I.
.-
N.

b0:

['9pr
(Brent

SubjeC.

functiolz
ductjm’

his wee.

5
ad,”

the boll:

CHAPTER FOUR

A NEW CASE
There is one other case worth considering. It arises
as a result of calculating a unit isoquant. This chapter

ijull show how a point on the production surface becomes an
';lxnterior' point to the unit isoquant for another point of
Llcauner production on the production surface due to the
at\r¢eraging process itself. This is a new case since it has
not previously been dealt with in the frontier production
function literature. However, unlike a frontier produc-
‘tlicazi function, the definition of technical efficiency is
fundamentally different, there is no distinction between
{rig and PE, and the unit isoquant so described may have
both 'interior' and 'exterior' points.

Consider points Q and P in Figure 4.1. Both points
represent production of the same output, y, using dif-
ferent quantities of the same variable inputs, v1 and v2,
s"I‘laject to the same quantity of an identical fixed input,
:ZEB - Therefore both points are on the same sub-production
futtinction (surface). P represents a higher level of pro-
d.":""'§tion, but is a member of Q's input requirement set.
{Pljl43ls means that both P and Q can be projected into the V1
3116 V2 plane. In the (V1,v2) plane the isoquant for Q is
the boundary to Q's input requirement set, within which P

82

83

 

 

 

FIGURE 4.1

«A' PRODUCTION FUNCTION (SURFACE) SHOWING
TOTAL PHYSICAL PRODUCT (TPP),
AVERAGE PHYSICAL PRODUCT (APP),

AND MARGINAL PHYSICAL PRODUCT (MPP)

0F V1 AND vg IN LEAST COST COMBINATION
GIVEN 3 EQUAL TO A CONSTANT

/\
Y TPP
I
P I
3‘2 I I
I I
I I
. I I
Y1 ----------- I I
I I I
I I I
I l I
| I I
I I I
I l I
I | I
| I I I
l I I I
| I. I I
I I |
I I I
| |
. . I
I
I I I
I I I I APP
I l I \
0 MPP & x?

84

will lie (see Figure 3.1A in Chapter Three). Notice that
Q represents the point of production for which APP (av—
erage physical product) is at a maximum, and therefore
represents the boundary between Stages I and II of produc—
tion.

Now construct the unit isoquant through point Q for

the set of production points from which these two observa-

tions are drawn (see Figure 4.2). Point Q maps to Q' and
point P maps to P'. Point P' is an 'interior' point to
the unit isoquant through Q'. Assuming that both Q and P

use inputs in their least cost combination, both points Q'
and P' are "price efficient," but point P' appears to be
"technically inefficient" as compared to Q'. Notice that
at Q one obtains maximum output per unit input, i.e.,
maximum APP. At P one obtains less output per unit input,
i.e. lower APP. Therefore, Q is on the unit isoquant
where one is using minimum input per unit output, and P is
in the 'interior' where one uses more input per unit
output. By definition of the unit isoquant, P' represents
less output per unit input than point Q' because point P
represents a lower APP than point Q. This means that the
discrepancies between Q' and P' in Figure 4.2 come about
as a direct consequence of the averaging process itself.
The difference between Q' and P' are not due to any
intrinsic "technical" or physical inefficiency originating
at a true interior point to Q as would be construed from
frontier production function theory. In constructing the

unit isoquant one in effect shifts one's attention from

85

 

 

"‘I'fv

 

FIGURE 4.2

A UNIT ISOQUANT

86

points on the TPP (total physical product) curve to cor—
responding points on the APP curve (see Figure 4.1). That
is, clearly there is less output per unit input, i.e.,
more input per unit output, at P (P') than at Q (Q') due
to the law of diminishing returns. At P (P‘) the MPP

(marginal physical product) is lower than it is at Q (Q').

4.1. IS THIS A NEW DEFINITION OF TE 7

This new definition of TE does not depend on interior
points to a production function. It does not mean one can
get more output with the same inputs at P', as would be
concluded from frontier production function theory.
Therefore, to conclude that Q (Q') is "technically, effi—
cient" would be misleading. The point P' (P) is only less
"technically efficient" than Q (Q') in the sense that one
can achieve a higher APP using less input than is used at
point P (P') by moving back to point Q (Q'). If one were
to define this difference in average physical output
potential between points P (P') and Q (Q') as a difference
in "technical efficiency" between the two points, one
would be creating a definition for TE that is fundamental-
ly different than the "TE" that is defined in the frontier
production function literature. Recall that in the fron—
tier production literature a point is TE if, and only if,
one can not get more output (TPP) from the same inputs.
Clearly, at point P (P') one cannot get more output from

the same inputs; one can only get a higher APP from the

87
same variable inputs by leaving some variable inputs
unused. This difference in potential APP is clearly un-
derstood in traditional microeconomic theory. Equally
clear from traditional microeconomic theory is the inap-
propriateness of recommending that one always maximize
APP; that one maximize TE where TE means maximum average
physical product. In traditional microeconomic theory the
relevant issue is whether or not a higher APP is wggth
producing. This is a case of properly treating economic

adjustment with respect to variable inputs.

4.2. TE NOT SEPARATE PROM PE

This case still does not separate TE and PE. A point
P (P') would be off the unit isoquant for Q (Q') due to:

(1) either P (P') not properly equating its MRP (marginal

revenue product) to its MFC (marginal factor cost), or
(2) to P (P') paying lower prices for inputs v1 and v2
(having a higher budget constraint). P' cannot be alloca-

tively efficient, a least cost producer, if Q' is, and
vice versa, when both firms have the same budget con-
straint and pay the same prices and opportunity costs for
all their inputs, both variable and fixed. This is clear
due to the same arguments that were advanced in Chapter
Three regarding distance functions and frontier production
functions.

If P' (P) and Q (Q') are considering different within

firm opportunity costs for the identical quantity of fixed

1DPUt 23, then both might be least cost producers and be

88

producing different levels of output. Recall from tradi—
tional microeconomic theory that inputs become fixed in
production when their within firm opportunity cost (shadow
price) is between their out of firm opportunity costs
(acquisition and salvage price) [Johnson and Quance,
1972]. The different firms represented by P' (P) and Q
(Q') might maximize the within firm opportunity cost for
the fiXEd input 23, the economic returns to 23, at dif-
ferent values between the acquisition price and salvage
price that both firms face. If P' (P) allocates less of
its budget to compensate for the use of 23, pays less rent
to 23 than Q (Q') does, then P' (P) can "purchase" more of
the two variable inputs v1 and v2 than Q (Q') can. This
problem can be avoided if both firms endogenize the quan-
tity allocation 0f 23 using the method outlined by Edwards
(1958); i.e., both firms will be evaluating 23 at the same

within firm opportunity cost.
4.3. 'INTERIOR' AND 'EXTERIOR' POINTS

Unless the unit isoquant drawn through Q (Q') is the
boundary between Stage I and Stage II due to the firm
being in long run equilibrium within a perfectly competi-
tive industry, there are both 'interior' and 'exterior'
points. Notice that if point P in Figure 4.1 were to
represent the least cost producer, and one were therefore
to draw a unit isoquant through P (P'), then Q (Q') would

become an 'exterior' point to the unit isoquant through P

89
(P'). That is, one could still increase "TE", increase
physical output per unit input, by operating at point Q
(Q'). In this case 23 for Q (Q') would be earning eco—

nomic rent; 23 for Q (Q') would have a higher within firm

opportunity cost (shadow price) than for P (P').
4.4. SUMMARY

This chapter has considered a valid case where one
can construct unit isoquants that have 'interior' points.
It is clear that the interpretation of these unit iso—
quants is not the same as the interpretation of a frontier
production function. The 'interior' points to the unit
isoquant represent higher levels of total production than
points on the unit isoquant, rather than equal levels of
production. An 'interior' point becomes an 'interior'
point due to the fact that at higher levels of production
the law of diminishing physical returns results in lower
total output per unit input which means one uses more
input per unit output. In addition, unless the unit
isoquant represents the boundary between Stages I and II
of production, where average physical production is at a
maximum, a production set for firms that are in long run
equilibrium in a perfectly competitive industry, the unit
isoquant will have both 'interior' and 'exterior' points.
The term "technical efficiency" is really a misnomer in
this case, since it has nothing to do with getting maximum
output from a given set of inputs. In this case, techni-

cal efficiency implies only that one is producing where

90

average physical product is at a maximum. Clearly, one
chooses to be technically efficient or maximize APP only

if it is allocatively efficient.

CHAPTER FIVE

TEE FRONTIER PRODUCTION PUNCTION LITERATURE

While the popularity of frontier production functions
is relatively recent, the arguments for their existence
have been around for some time. The erroneous distinction
between TE and PE has flourished largely unchallenged.
The distinction between TE and PE seems to have been
simply postulated and accepted without any serious atten-
tion to its logic or its conformity with the laws of
nature. How did such misguided logic establish itself in
economic theory? What are the original sources of error?
If the logic of frontier production functions is inconsis-
tent with legitimate economic theory, the frontier produc—
tion function literature should have discredited itself.

This chapter will examine some of the more salient
literature relevant to frontier production functions. An
exhaustive review of all the literature is unnecessary
since it is highly repetitious. The chapter will attempt
to explain briefly how the errors established themselves
and evolved despite the internal inconsistences revealed
in the foregoing chapters. It will be shown that these
inconsistences substantiate the arguments offered in Chap-
ter Two; that frontier production functions arise due to

specification error and/or aggregation error.

91

92

5.1. LINEAR PROGRAMMING AND TREE DISPOSAL
5.1.1. EDOPHANS:

In tracing the development of technical and price
efficiency, free disposal, and frontier production
functions, there does not appear to be any literature
prior to the development of linear programming, or activi-
ty analyisis.

One of the earliest references to technical efficien—
cy may be found in Koopmans (1951). The article is a
discussion of how one finds the efficient set amongst the
feasible set. Koopmans creates ambiguity by saying that
he will be discussing the "efficient point set in the
commodity set." It is clear that he means isoquant in
using the term feasible set rather than the efficient set
(a maximum profit point, or an expansion path), since he
is referring to a production function. His postulate that
the isoquant is ”the efficient set" is never substan-
tiated; indeed what an "efficient set" means is not dis—
cussed. If an isoquant is itself efficient, then the
tangency of the isoquant and the budget constraint is a
different type of efficiency and one can infer that there
is technical and price efficiency. If the isoquant is a
technically efficient set, then one can infer that a
technically inefficient set must also exist, i.e., the
interior to the isoquant. None of this is explored in the

article. Koopmans' use of "efficient" is an unfortunate

 

93

word choice that is never explained. The same is true of
his use of the term free disposal.

The concept of free disposal is not developed, but
alluded to within the context of developing the conditions
and characteristics of activity analysis. It is evident
from the introductory remarks that free disposal depends
on the perfect complementarity assumed for each activity

being analyzed.

.. and situations where some factors
can only be combined, within the
technological principle involved, in
fixed ratios to each other (limita-
tional factors). The second type of
situation can only be reconciled with
the notion of a production function
defined in the whole factor space by
allowing the production manager to
discard parts of the factor quantities
specified as being available. The cor-
responding production functions have
kinks at the points where the ratios of
available factor quantities coincide
with the technical ratios specific to
the process in question. [Koopmans,
1951, page 33-34; underlining added]

In linear programming this type of free disposal is repre-
sented by slack and surplus activities, which serve as a

place "freely" to "dispose," or "discard," "available" but

unused amounts of resources in order to establish the

equality constraints of the algorithm. Seldom are the
quantities of inputs in the slack and surplus activities
interpreted as having been consumed in the production
process: they are not inputs to production. Instead sur-
plus (slack) activities can be thought of as warehouses in

which inputs can be stored at zero cost, i.e. disposed of

94

freely -- at no cost. Having inputs available does not
mean they are consumed. In addition, it is worth noting
that in a linear program any input always has a shadow
price associated with it. Consequently, in linear pro-
grams the physical quantities of inputs are not separated
from their values.

Free disposal probably originated in the slack acti-
vities of linear programming. This in itself casts some
suspicion on its theoretical rigor since linear program-
ming is not a general theory of production but is, in-
stead, a operational method for optimizing returns under a
set of extremely restrictive and often unrealistic as-

sumptions which simplify calculations.

5.1.2. BOLES:

Boles' (1966, undated) contribution is worth noting
since he develops an algorithm closely related to a linear
programming algorithm. Boles restricts his discussion to
the mechanical aspects of constructing an algorithm that
will compute technical and price efficiency indices.
While he provides no new insight into the theory, the
algorithm he develops could have a very practical applica-
tion in applied work, in determining the degree of speci-
fication or aggregation error one has in a sample and the
need to carry out additional detailed investment/disin-
vestment analysis of the causes of apparent interior

points -- that is, by recognizing that Boles measures of

95

technical and price efficiency are essentially measures of
specification or aggregation error, his algorithm serves a

potentially legitimate and useful diagnostic value.

5.2. FRONTIER PRODUCTION FUNCTION THEORY

5.2.1. FARRELL:

Farrell (1957) is credited with formally introducing
the theory of frontier production functions. It is clear
that Farrell developed his ideas from association with
linear programming.

the treatment of the efficient

production function is largely in-

spired by activity analysis....

(Farrell, 1957, page 11]
Except for extending his theory to include situations
where constant returns to scale need not be assumed
[Farrell and Fieldhouse, 1962], and some elaboration and
interpretation [Bressler, 1966], Farrell's theory has been
largely accepted without critcal challenge. Nerlove
(1965) and Yotopoulous (1974) point out some critical
ambiguities, but still accept the fundamental premise.

It is the virtue of the present method

that it separates price from technical

efficiency. [Farrell, 1957, page 264]
Farrell himself states that the concept of technical effi—
ciency is a consequence of aggregating inputs, management,
and measurement error. Recognizing this destroys the

distinction between TE and PE.

96

Price efficiency deals with choosing the optimum set
of inputs along an isoquant. Price efficiency (allocative
efficiency) is the point where an isoquant is tangent to a
budget constraint. This determines the quantities of
variable inputs V(J) purchasable with a given budget that
will maximize production. Similarly, technical efficiency
maximizes physical output from a given bundle of inputs.
If technical efficiency and price efficiency are dif-
ferent, then it implies that V(J) can produce different
levels of output, each level having a different value.
There are two possible explanations.

Technical efficiency implies that given amounts of
the V(J) can produce different amounts of an output.
Clearly, two different outputs can not be on the same
isoquant since one output is smaller than the other. If
the sub-production function is the same for both sets of
inputs, then the two isoquants are in the same input
requirement set, and the "extra" input and output must be
freely disposed in order to move from the higher to the
lower isoquant. But because of polar reciprocal sets, the
TE and PE of the two points on the two isoquants are the
same and there can be no difference between TE and PE.
This flatly contradicts the statement that Farrell's
method "separates price from technical efficiency."

The second case suggests that the quantity of output
forthcoming from V(J) is not uniquely determined by the

production processes. Nowhere is there an explanation of

97

how identical inputs can produce different output; that
is, there is never an explanation of what accounts for the
differences in production. Therefore, price efficiency
does not insure maximum value, only a proportionate rela-
tionship; between the v's in V(J). If the bundle is used
technically efficiently one gets one output but if its
used technically inefficiently then one gets a different
output. Therefore, the value of V(J) is not uniquely
determinable.

If one looks closely, what Farrell treats as tech-
nical inefficiency results from comparisons across sub-
production functions (specification error) or from aggre-
gating heterogeneous inputs (aggregation error). Indeed,
one can infer that mis-aggregation 2: inputs creates an
apparent "technical inefficiency" from his suggestion that
dis—aggregating inputs improves the technical efficiency
of the firm.

It will be seen that, in accordance
with the theory developed in Section 3,
the introduction of a new factor of
production in the analysis cannot
lower, and in general, raises the
technical efficiency of any particular
[observation]. Thus, the more factors
that are considered, the more are ap-
parent differences in efficiency ex—
plained as being due to differing in-
puts of the factors. [Farrell, 1957,
page 269]
Since inputs are defined to be quality (form) specific,

one cannot have differences of quality between units of

the same input, without committing an aggregation error.

98
Management is better regarded as choosing production

processes, input levels, and hence, output levels than as
a factor of production. Consequently, it does not affect
the "technical efficiency" of a given set of inputs.
Farrell is ambiguous when he says,

... the technical efficiency of a firm

or plant indicates the undisputed gain

that can be achieved by simply 'gin—

gering up' the management. [Farrell,

1957, page 260]
Management chooses the input requirement set. Choosing
one set over another set is not maximizing output from a
given set of inputs, or 'gingering up' technical efficien-
cy, but an economic adjustment among the alternative sets
[Edwards, 1958]. For different input sets, the physical
output will be different, but not for identical sets. This
is a physical (technical) phenomenon, but implicit is a
comparison of two input requirement sets with different
fixed inputs. If a one chooses one set rather that an-
other, one is implicitly changing sub-production functions

in the process.

The two problems mentioned above can arise due to

measurement error. In discussing the practical problems
of measuring inputs, i.e., measurement error, Farrell
aknowledges,

... that a firm's technical efficiency
will reflect the quality of its inputs
... [Farrell, 1957, page 260]

Farrell goes on to explain that discrepancies in the

99

measurement of inputs also affects price efficiency, "but
in rather a complex way, so that problems are best discus-
sed ad hoc." As an example he states,

If labour input were measured in man-

hours, as is conceptually correct, this

too would affect price efficiency, but

if in numbers of men employed, it would

affect the firm's technical efficiency.

[Farrell, 1957, page 261]
This suggests that Farrell regards men as fixed inputs but
man hours as variable inputs. Farrell seems to be sugges-
ting that price efficiency is associated with variable
inputs and technical efficiency with fixed inputs and that
variable inputs cannot be used technically inefficiently
while "fixed" inputs cannot be used in uneconomic propor-
tions. When Farrell attempts fully to illustrate the dif-
ference between price and technical efficency he attri-
butes the differences to specification error (comparisons
across sub-production functions), or error in aggregating
inputs.

Farrell's paper contains repeated instances of ambi-
guity about what one is to assume initially about the
production function that gives rise to the "efficient"
production function. He is repeatedly ambiguous about how
price and technical efficiency can be separated and iso-
lated without interactions. Finally, he repeatedly
neglects the implications noting that what he calls
"technical efficiency" depends on input mis-specification

and/or aggregation errors.

100

5.2.2. BRESSLER:

Bressler's (1966) summary of the frontier production
function theory has been presented in Chapter Two. In
addition, Bressler offers three explanations for ineffi-
ciency in a firm: (1) operating with excess capacity, (2)
inefficient use of technology, and (3) using an outmoded
or inefficient technology [Bressler, 1966].

Operating with "excess capacity" clearly focuses
attention on the role of fixed inputs. If having excess
capacity means that to be profitable one must expand use
of variable inputs, then the firm is simply inefficient.
A second interpretation is more conventional: excess
capacity suggests the need to make length of run ad-
justments (changing sub—production functions), which is
treated below.

Since the production possibilities set defines the
technology, Bressler appears to be using a different defi-
nition of technology in cases two and three. Technology
is not chosen, but given by a state of knowledge. Bres-
sler's technology appears to be the same as the input
requirement set, or technique, of Appendix Two. Bres-
sler's cases two and three, efficient use of technology
and choice of technology, imply efficiency in general
rather than just "technical efficiency." In light of the
theory of production functions presented in Appendix Two,

one can conclude that the choice of technology and the

101

efficient use of technology are not merely physical
(technical) decisions but are inherently economic deci—
sions based on evaluating the costs and returns involved.
Choosing a technology raises the issue of technical change
and makes the selection of fixed and variable inputs
carrying old and new technologies endogeneous to the sys-
tem (see Edwards, 1958). Changing technology implies
changing sub-production function. Using a technology
efficiently means first, finding the tangencies between
the budget constraints and isoquants and then the high
profit level of production. "Using an outmoded or ineffi—
cient technology" suggests that one has not properly con-
sidered the economies of investing in or using inputs
carrying new technologies and disinvesting in or ceasing
to use inputs carrying old technologies.

Bressler's comparisons of efficiency are directly
attributable to making comparisons across sub-production
functions. As mentioned in Chapter Two, Bressler points
out that economic efficiency has a direct relationship to
average costs. Specifically, if one maps the inverse of a
production function's longer-run average total cost curve,
one finds a curve representing maximum efficiency for the
production function at each level of production, e.g.
point A maps to point D in Figure 5.1. Each level of
production represents an isoquant. If one observes a
point within this "efficient envelope," point E, to what
may it be attributed? There are only two possible expla-

nations.

102

 

 

 

 

/\
Index of LIULC
Average
Cost SRAC
A
I
I B
l
l
I I c
um I L
I I
I I
I
l l
I d f I I
n ex 0 I —$
. , E
Efficiency ”
D
88
\
Output 7

FIGURE 5.1

A SHORTER-RUN AVERAGE COST CURVE (SRAC), A
LONGER-RUN AVERAGE COST CURVE (LRAC), AND AN
"EFFICIENT' ENVELOPE" (EE)

103

First, if the efficiency envelope represents maximum
efficiency at each level of output (for each isoquant),
then the efficient envelope is an expansion path. One
would be inefficient if one used inputs in proportions
different from the proportions determined by the tangency
of the isoquant and the budget constraint. One would be
inefficient, but still be on the isoquant. Point E must
be on the same isoquant as D, but not on the budget con—
straint. D and E produce the same amount of output.
Thus, E is inefficient from an allocative point of view
and does not illustrate technical inefficiency. Technical
efficiency cannot be determined from this presentation.
Technical inefficiency, as Bressler uses the term, is
ambiguous.

The second possiblity is that the interior of the
efficient envelope maps back to the interior of the
longer-run average cost curve, e.g., point E to point B.
But what is in the interior of a longer-run average cost
curve? The longer-run average cost curve is an envelope
of shorter-rgg average Egg; curves. Consequently, being
off an efficient envelope means being on a different
short-run average cost curve which, by duality, means
making a comparison across different sub-production
functions without considering the economies of shifting
between them; i.e., of making a comparison between firms
which are using different levels of fixed inputs without

due attention to investment and disinvestment theory.

104

Consequently, being at point B (off the efficient
envelope) is due to the law of diminishing returns when
certain inputs are fixed at different levels for A and B.
This is a curious treatment of efficiency because the
values of the difference in the amount of fixed inputs are
not taken into account.

If only the variable inputs are taken into considera-
tion, a firm operating on the SRAC at B in Figure 5.1
appears inefficient because it is not operating on its
longer-run average cost curve at A. This is inappro-
priate. The efficient firm would still be experiencing
decreasing costs on its sub-production function at A, and
would continue to expand output thereby reducing average
total cost at least to 8. Furthermore, A is only per-
tinent in the longer-run, and with respect to the longer-
run, A is not efficient, C is. Therefore, A is not an
efficient point on either the SRAC or the LRAC. It is
simply inappropriate to compare A and 8 without con-
sidering the economies of investing and disinvesting in
the inputs which are treated as fixed.

In his conclusion, Bressler aknowledges the aggrega-
tion problem, but without aknowledging any of the critical
problems it raises as to the legitimacy of the frontier
production function.

First, all these methods are subject to
essentially similar problems of aggre-
gation; for example, if we use some
aggregate measure of capital inputs in

any of these approaches, we are ig—
noring the fact that capital is a non-

105

homogeneous input and $1.00 of capital

applied marginally in a small or a

large business may represent quite

different real inputs. Stated another

way, it makes a lot of difference if

our marginal capital input is in the

form of power shovels or hand shovels.

[Bressler, 1966, page 136]
In effect Bressler recognizes that aggregation hides
specification error and that when one compares firms, one
frequently compares two different sub-production
functions. Further, he implicitly recognizes that dif—
ferences between firms can be attributed to the influence
of differences in fixed input on output.

In summary, Bressler, too, fails to make valid dis-
tinctions between price efficieny and technical effi-
ciency. The difference which appears to him to exist
between the two kinds of efficiency are due to different

sub-production functions (specification error) or aggrega-

tion error.

5.2.3. NERLOVE:

As was mentioned earlier, Nerlove (1965) discusses
some of the ambiguities in Farrell (1957). Despite recog-
nizing the short comings of Farrell's approach, Nerlove
attempts to retain the technical/price efficiency dichoto-
my and "attempt[s] to generalize Farrell's work."

Nerlove summarizes the conventional assumptions about
production and concludes that there will be no difference
in output for profit maximizing firms using the same

inputs in the same environment (subject to the same

106

uncontrollable random inputs of the same magnitudes).

Those differences which do exist must

therefore be due to differences among

firms with respect to their technical

knowledge and fixed factors, their

ability to maximize profits, and their

economic environments. [Nerlove, 1965,

page 87]
Different technical knowledge implies different production
possibilities sets and/or technical change. Different
technical knowledge and different fixed factors suggests
different sub-production functions which alone are not
grounds for evaluating differences in efficiency between
firms using a physical criteria alone. The ability to
maximize profits involves, among other things, finding the
tangency between the budget constraint and an isoquant
which is not determined solely by technology. Different
environments result in different sub-production functions,
as defined in Appendix Two. None of these situations
allows for "technical inefficiency."

Nerlove quickly pinpoints the weakness of Farrell's

argument.

His measure may be divided into two

components. The first, technical effi-

ciency, relates to an improper choice

of production function among all those

actually in use by firms in the indus-

try. The second, price efficiency,

refers to the proper (or improper)

choice of input combinations.
[Nerlove, 1965, page 88]

The choice of "production function," which corresponds to

107

the choice of sub—production function of Appendix Two,
means choosing the amount of the fixed inputs. Nerlove
does not define what "improper" means. Had he done so he
would have had to consider the economies of investing in
variable inputs and of disinvesting in what might be fixed
inputs. This may be similiar to Bressler's (1966) refer—
ence to excess capacity. But his comment makes it clear
that differences in "technical" efficiency are due to
being on different sub-production functions, i.e., dif-
ferent levels of fixed input. He does not consider the
question of the opportunity costs associated with choosing
one sub-production function over another, i.e., whether
the total net value of one bundle of resources, both
variable and fixed, is greater than or lesser than another
bundle of resources.

Nerlove elaborates on this same essential issue in
his discussion of Farrell's "quasi-factors" [Farrell,
1957]. Quasi—factors are those inputs which are defined
in Appendix Two as uncontrolled random inputs, the collec-
tion of which Nerlove calls "the environment." Nerlove
implies that firms with different amounts of quasi-factors
will produce different levels of output, which is to be
expected inasmuch as the quasi-factors act as fixed inputs
and influence the onset of diminishing returns. Regarding
firms purely with different amounts of quasi-factors as
the "same" constitutes a specification error -— e.g., dif—

ferent sub-production functions are regarded as the same.

108

The last point is clearly crucial to

any definition of relative economic

efficiency; in general, some attempt

must be made to standardize environment

in the construction of the measure, or

else the measure will reflect not mere-

ly differences in efficiency but also

the degree to which the environment of

a particular firm is favorable or un-

favorable. [Nerlove, 1965, page 90]
He continues by pointing out that distinguishing between
price efficiency and technical efficiency is tantamount to
mixing short-run considerations (what he calls price
efficiency). and long-run considerations (what he calls
technical efficiency).

If price efficiency applies only in the short-run,
then it deals with the variable inputs. To make efficien—
cy comparisons across firms one must hold the levels of
fixed input identically constant among firms. Conse-
quently, differences in efficiency will be due only to
differences in the success of each firm in accurately
finding the tangency between its budget contraint and an
isoquant and its high profit points.

If technical efficiency applies only in the long—run,
then clearly it deals with economic adjustment or changing
the amount of fixed input -— with investment and disin-
vestment. This means that changing the amount of fixed
input will change technical efficiency and technical effi—
ciency must result from inappropriate comparisons across
different sub—production functions. This creates two

problems: (1) what are the criteria for making changes in

fixed inputs in order to change what is called technical

109

efficiency; and (2), a firm cannot be technically effi-
cient and price efficient simultaneously except at point C
in Figure 5.1. A firm cannot be both shorter-run and
longer-run efficient except at point C since in general
there will be the inherent contradiction noted above that
B is efficient but A is not.

Unfortunately, after clearly identifying the short
comings of the frontier production function approach,
Nerlove equivocates by attempting to "generalize
Farrell's" method. In so doing he implicitly assumes that
a firm with less fixed input than another firm is "tech-
nically" less efficient. In assuming that firms are 2353
to choose whatever level of fixed input they want in order
to minimize long-run average costs Nerlove implicitly
assumes there is no cost in changing sub-production
functions. If this were true, all firms would be opera-
ting at the minimum of their long-run average total cost
curve and there would be no such thing as inefficiency,
"technical" or otherwise.

In Appendix Two it is noted that Edwards (1958)
offers a way out of this dilemma by showing that the
fixity or variability of inputs can be endogenized to the
system for the purpose of finding the most efficient
point, without changing any of the basic conclusions of

the theory presented.

110

5.2.4. YOTOPOULOUS:

Yotopoulous (1974) comes closest to dispelling the
technical efficiency fallacy. He repeatedly identifies

the real sources of "technical inefficiency."

The difference in output between the
"average" firm and the extreme positive
outlier is used to measure the techni-
cal inefficiency of the average firm.
Another interpretation, of course,
could have the "average" firm represen-
ting the norm and positive outlier
representing an unusual endowment of
some fixed factor of production, such
as entrepreneurship, or luck. It may
represent the classical source of error
in measurement or of noise in the uni-
verse, and as such it can imply nothing
systematic about efficency. [Yotopou-
lous, 1974, page 264]

This clearly suggests that given a comparison between two
firms, the "technically efficient" firm is so due to some
additional amount of some fixed factor though entrepre-
neurship is a poor candidate for reasons discussed earlier
in considering Farrell's contribution. When firms have
the same amount of fixed input, and face the same prices,
than differences in efficiency are due either to error in
maximizing profits which is simply inefficiency; or to
differences in the uncontrollable random fixed inputs,
which cannot be appropriately called "technical efficien-

CY. u

... the remaining differences in obser-
vable input mixes can be attributed to
two factors. First, they can be traced
to differences in nonmeasured fixed
inputs of production. These can be

111

readily captured through the analysis
of variance as used to measure manage-

ment bias. They constitute the com-
ponent of technical efficiency. Se-

 

cond, the results can be attributed to

residual differences that are due to

imperfect equalization of marginal pro-

ducts to opportunity costs. These

constitute the component of price effi-

ciency. [Yotopoulous, 1974, page 269]
Like Nerlove (1965), Yotopoulous failed to exploit his
insights; instead he attributed "technical" efficency to
the "environment" and to uncontrolled random inputs. This
makes differences in "technical" efficiency among firms
due to chance or to being on different sub-production
functions. Yotopoulous constructs, hypothetically, the
set of situations where one might observe differences in
technical efficiency between two firms. He specifically
attributes the differences in "technical" efficiency to
being on different sub-production functions: The differ—
ences in firms which he treats as differences in "tech-
nical" efficiency are due to differences in the amount of
fixed input being used.

In Panel II comparison of technical

efficiency becomes possible since the

isoquants belongs to production

functions that differ only by the con—

stant. This term represents differen-

ces in endowments of fixed factors as

well as the impact of nonmeasurable

inputs, such as entrepreneurship.

Technical efficiency is the shorthand

notation for such differences.

[Yotopoulous, 1974, page 266]

In his discussion of these various cases, only the "effi—

ciency" of the variable inputs is actually compared. He

112

does not introduce the economies of investing or disin-
vesting to change sub-production functions. In effect,
technical efficiency becomes a comparison of the produc—
tivities of the variable inputs, ignoring his own critical
point that these productivities will differ with dif—
ferences in the quantites of fixed inputs. This is the
same thing as ignoring the contribution to production and
the costs of the fixed input(s).

Despite repeated statements demonstrating that
technical efficiency cannot exist if one maintains the
usual set of assumptions about production, as stated in
Appendix Two, Yotopoulous retains a belief that something
called "technical efficiency," and its logical corollary
"technical inefficiency," exist. Firms that have the same
production function, but different amounts of fixed input,
are said to have "neutral differences in technical effi-
ciency." That is, firms on different sub-production
functions display no differences in technical efficiency,
which contradicts everything he has previously developed.
He states:

Technical inefficiency, on the other
hand, is related to the fixed resources
of the firm. It is an engineering
datum and as such, at least in the
short run, it is exogeneous and part of
the environment that is taken as given.
[Yotopoulous, 1974, page 271]
What this suggests is that the amount of fixed input used

is either not a choice -- an example of an uncontrollable

113

input -- or that the choice is made without any reference
to costs. 7

Like Nerlove (1965), Yotopoulous fails to exploit the
opportunities revealed by his own observations and in-
sights. Consequently, he presents a "computational form
that combines the three elements, technical, price, and
economic efficiency." Prehaps because his "form" excludes
consideration of fixed inputs, and fixed costs, he does
not detect the inconsistency of his "form" with his own

arguments.
5.3. FORMAL MICROECONOMIC THEORY

Unfortunately, some of the leading microeconomic
texts of the last twenty years have institutionalized the
error of distinguishing between "technical" and "price"
efficiency. As in the frontier production function
literature itself, the distinction is postulated with
little, if any, attention to the logic of such a dis—
tinction, or its contradictions with the usual assumptions

made regarding production processes.
5.3.1. HENDERSON AND QUANDT:

In the "Basic Concepts" of production theory, Hender-
son and Quandt (1971) state that the "production function
states the quantity of his [the manager's or entrepre-
neur's] output as a function of the quantities of his
variable inputs." One or more additional inputs are con-

sidered fixed within the production function. Conse-

114

quently, for two firms to have the same production
function, they would have to be constrained by the same
amounts of the same fixed inputs. This corresponds to the
firms having the same sub-production function as defined
in Appendix Two.

In defining technology, and explaining how it is
different than a production function, Henderson and
Quandt contradict the laws of thermodynamics. They begin

by stating that,

The entreprenueur's technology is all
the technical information about the
combination of inputs necessary for the
production of his output. it includes
all physical possibilities. [Henderson
and Quandt, 1971, page 54]

It would seem they have defined the production possibili-
ties set. Appendix One and Two make it clear that the
laws of thermodynamics substantiate that the production
function is a surface and not a solid; no isoquant has an
interior due to identical quantities of variable inputs
producing more than one quantity of output given the same

fixed inputs. However, Henderson and Quandt contradict

this in their next sentences.

The technology can state that a single
combination of [V1] and [V2] can be
utilized in a number of different ways
and therefore can yield a number of
different output levels. The produc-
tion function differs from the tech-
nology in that it presupposes technical
efficiency and states the maximum out-
put obtainable from every possible
input combination. The best utiliza—

115

tion of any particular input combina-

tion is a technical, not an economic,

problem. The selection of the best

input combination for the production of

a particular output level depends upon

input and output prices and is the

subject of economic analysis. [Hender-

son and Quandt, 1971 page 54]
This suggests that within the production possibilities
set, two identical sets of inputs can produce different
sets of output. The crucial phrase is "utilized in a
number of different ways." The inputs can be used in
different ways only if they are used in different rela—
tionships to each other with respect to their time, form,
or location. In any of these cases the inputs are not
identical. As is pointed out in Appendix Two, the time,
form, and space (location) of an input is held constant in
its definition; two apparently identical inputs that dif-
fer in either time, form, or space are in fact different
inputs. Consequently, "a single combination" of two iden-
tical variable inputs cannot be "utilized in a number of
different ways." The "way" the inputs are "utilized" is
captured within each input vector in the input requirement
set. The choice among the different ways of using inputs
is as economic as any other choice: it depends on whether
or not "it pays" to make the change in time, form, or
location, and is therefore economic and not merely
"technical." If one observes different quantities of
output as a result of using two identical bundles of

variable inputs, then the difference must be due to dif—

ferences in the amount of the fixed input[s] used. Then,

116

by Henderson and Quandt's own definition of a production
function, the different outputs are produced by different
production functions, not by a difference in technical

efficiency.

5.3.2. MCFADDEN:

McFadden's (1978) development of microeconomic theory
is ironic because he maintains a distinction between
technical efficiency and price efficiency, and at the same
time develops the duality theory of polar reciprocal sets
that eliminates any possible distinction between the two.
In fact it is done within the same context, distance
functions (see Appendix Two).

His treatment of distance functions creates ambiguity
and misunderstanding as was suggested in Chapter Three.
The similiarity of the unit isoquant of a frontier produc-
tion function, and a distance function is more than coin-
cidence (see Figure 3.1). It was suggested that a fron-
tier production function is mistakenly identified as a
distance function, where "a" is a measure of "technical
efficiency." This suggests that the scaling process im-
plicit in distance functions changes the quantities of v,
but not the original marginal physical products associated
with them, which is not generally observed if there are
fixed inputs.

A positive input bundle (v, z) is

efficient for an output bundle y and
distance function F if F(y,v, z) = 1

117

and any distinct postive input bundle

(v', z) with (v', 2) <= (v, 2) has F(y,

v', z) < 1. Alternately, define an

input bundle (v, z) to be efficient for

an input requirement set V(y) if any

distinct input bundle (v', z) with (v',

2) <= (v. 2) has (V'. 2) ¢ V(YI-

[McFadden, 1978, page 30; z's added]
What gets lost in this scaling is the corresponding expec-
ted changes in marginal physical products. However, this
frontier function, or "efficient set" is slightly dif-
ferent than the frontier in Chapter Two. In including a
reference to a graph of an isoquant (see McFadden, 1978,
page 17), McFadden indicates that "inefficient" points are
points 23 the isoquant. Since the distance function moves
all "interior" points to the isoquant by free disposal,
the inefficient points must be points that are not on the
expansion path for a given level of output.

McFadden developes distance functions chiefly as one
way to prove duality between production functions (exclu-
ding Stage III) and cost functions. The one to one cor-
respondence between prices and inputs, between physical
quantites and their values, is summarized in the defini-
tion of the polar reciprocal sets that were discussed in
Chapter Three. Polar reciprocal sets prove there is no

possible distinction between technical and price efficien-

cy.
5.4. OTHER LITERATURE

The foregoing has dwelt on the literature important

in establishing the erroneous theory of frontier produc-

118

tion functions. There is a growing volume of other lit-
erature that may be grouped into three categories: (1)
methods for estimating frontier production functions, (2)

applications of frontier production functions, and (3)
miscellaneous or related concepts. This other literature
will be dealt with only briefly since it does not focus on
the development of the theory of frontier production
functions per se, but rather accepts the premise that
frontier production functions exist.

The first group concerns itself with how one might
estimate frontier production functions, or measure tech-
nical and price efficiency between firms. This literature
would be of more value if it focused instead on issues of
specification error or aggregation error which together
constitute the discrepanies between firms that are mis-
takenly attributed to differences in technical efficiency.
The contributions of Boles (1966, undated) within the con-
text of linear programming, or activity analysis, has been
alluded to above. Timmer (1971) attempted to measure
"technical efficiency" using a specific functional form
and a mathematical programming algorithm. It marks one of
the first attempts to measure "technical efficiency" para-
metrically. The more recent literature attempts to de-
velop a method of estimating frontier production functions
parametrically using an econometric approach [Forsund,
Lovell, and Schmidt, 1979, Forsund and Hjmalmarsson, 1974,

Forsund and Hjmalmarsson, 1979, Schmidt and Lovell, 1977,

119

Schmidt and Lovell, 1978]. The focus of this literature
is on the specification of the error term, which has
either a one sided distribution (all the errors are of one
sign), or is divided into two components, one representing
the usual error term while the other is one sided repre-
senting differences due to "technical inefficiency."

Despite the attention to the development of the
theory of, and methods for estimating, frontier production
functions, there have been few attempts to apply frontier
production functions [Bravo-Ureta, 1983, Hall and LeVeen,
1978, Lesser and Greene, 1980, Seitz, 1966]. Most appli-
cations have used a non—parametric method, while applica-
tions using a parametric method have served chiefly as
examples of a new or improved method.

The last group contains literature that does not
always refer explicitly to frontier production functions
but clearly offer theories of management or decision-
making that are closely akin to the concept of a frontier
production function [Charnes, Cooper, and Rhodes, 1978,
Charnes, Cooper, and Rhodes, 1981]. The best known, and
prehaps best example of this literature, is Leibenstein's
theory of "X-Efficiency" [ Leibenstein, 1966]. The theory
of X-Efficiency suggests that different firms using the
same inputs to produce the same output will have different
degrees of efficiency depending on how the production
process is organized in practice. While the concept may
be useful to a decision-maker in a very applied sense, in

the strictest sense, it violates the definition of homo-

120

geneous inputs given in Appendix Two as the inputs are not
time, form, and location specific and ignores the costs

and returns involved in investing and disinvesting.

5.5. SUMMARY

Perhaps the oddest aspect of the frontier production
literature is that it appears to be a theory developed as
a consequence of a technique (linear programming), rather
than the other way around. In the earliest literature,
dealing with activity analysis, what is called technical
efficiency is assumed or postulated without a clear des—
cription of what technical efficiency means, or how it
differs from the traditional concept of economic efficien—
cy. Since there is no explicit discussion that would
suggest that an isoquant can be represented as a plane,
one would expect more suspicion of an "unit isoquant"
which creates a plane by dealing in average production,
rather than the more traditional focus of 3933; produc-
tion.

Implicit aggregation and specification error is evi-
dent from a careful reading of Farrell [Farrell, 1957,
Farrell and Fieldhouse, 1962], and several successors,
including Bressler (1966), Nerlove (1965), and Yotopoulos
(1974). Despite this, each author eventually (uncritical—

ly) accepts the existence of a frontier production

function without attempting to specify it in a proper

manner. Thus, the concepts of frontier production

121

functions have been uncritically incorporated into state—
ments of microeconomic theory [Henderson and Quandt, 1971,
Malinvaud, 1972, McFadden, 1978, Quirk and Saposnick,
1968, Varian, 1978]. Ironically, McFadden's effort to
prove duality mathematically provides the theoretical
refutation of the existence of frontier production
functions by describing the characteristics of distance
functions and polar reciprocal sets [McFadden, 1978].
None—the-less, he simultaneously maintains the logically
inconsistent notion of "technical efficiency." This in-
consistency appears to be the result of his discounting
the importance of those aspects of traditional theory that
are commonly considered to be 'economically irrelevant;'
i.e., Stages III, because normally all the points in
Stages III are inefficient [McFadden, 1978]. Unfortun-
ately, this oversight by McFadden (1978) creates ambiguity
in those aspects of the theory that he preserves.

Finally, there is the other literature described in
5.4 above, which adds little to the debate of whether or
not frontier production functions exist, but if properly
directed, might provide methods for dealing with the prob-
lems of specification and aggregation error and lead to

incorporation of investment/disinvestment theory.

CHAPTER SIX

CONCLUSION

Frontier production functions are supposed to be
different from, and better than, traditional production
functions. They are supposed to represent the "best per-
formance" obtainable by a firm given some set of endow-
ments and some given technology. They do this by distin—
guishing between the "technical" and "price" efficiency of
the 'best performer' and the other firms included in the
comparison. This dissertation maintains that rather than
measuring "best performance," frontier production
functions measure specification and/or aggregation error.
This is because the theoretical basis upon which frontier
production functions identify "best performance" is in-

valid.

6.1. VARIATIONS IN DATA IN APPLIED WORK

In any applied work there will be variation in the
data one collects. The real world does not conform to
theoretical conditions of homogeneity, perfectly cOmpeti-
tive markets, etc. This lack of conformity does not mean
that microeconomic theory is failing to explain reality.

Microeconomic theory, like all theory, is a guide to
analysis after one has decided what is important and what

122

123

is unimportant to the issue being investigated. One must
take a preliminary step of using one's judgement as to
what constitutes "sameness" and what constitutes "close
enough" when specifying a sub-production function and
aggregating data across observations for the purposes of
the investigation. Once the degree of "sameness" and
"closeness" has been determined, the remaining variation
in the data is implicitly "unimportant" and irrelevant to
the inferences to be drawn from the analysis. The statis-
tical properties assumed to be exhibited by the data as
specified and aggregated serves as a means of filtering
out the remaining variation. The remaining "noise" is
assume to be captured in an "error term" and estimation is
done at the mean, or average value, of the data.

The variation that remains means that if one plots
the data there is a scattering of points, a distribution
of the observations. The question examined in this dis-
seration is essentially which observations in this distri-
bution should serve as the bench mark from which compari-
sons of the observations might be made. As such, it
examines the nature of the remaining variation; it ex-
plains what causes the variation one observes between
observations after one has specified a common functional
representation for the technical relationship among the
specified and aggregated variables of production. If one
chooses the traditional bench mark of the data's mean,
then one is implicitly assuming that the remaining varia-

tion is due to random unexplainable and inconsequential

124

phenomena, that the variation that remains is simply due
to the "unimportance" of measuring and explaining the
remaining imperfections of the real world.

If one chooses the theory of frontier production
functions, then one is assuming a bench mark of one
extreme of the observations, which is an implicit as-
sumption that the remaining variation is systematic and
due to consequential and measureable causes, "technical
efficiency" and "price efficiency." Therefore, the "less
efficient" observations all lie within the interiors of

the isoquants of the extreme observations.

6.2. ISOQUANTS DO NOT HAVE INTERIORS

Chapter Two, Three, Four, and Five discuss the fron-
tier production function and traditional theory explana-
tion of these interiors to isoquants.

Chapter Two is a summary of the characteristics of
frontier production functions that make them different
from traditional production functions. Frontier produc-
tion functions originated from the concept of the "unit
isoquant." Certainly plotting real production data, with
its inherent "remaining" variation, on a per unit of input
to per unit of output basis will reveal a scattering of
the observations, as was suggested above. Two cases for
inferring that this scatter of observations represent
solid production sets was discussed. It was shown that

both cases of these frontier production functions were a

12S

violation of traditional microeconomic theory due to
specification and/or aggregation error in identifying the
frontier production function.

Chapter Three explored the critical case of whether
or not frontier production functions are accurate repre-
sentations of distance functions. Superficially, they
appear to be the same. They have the same basic as-
sumptions of concavity and monotonicity, and they both
have 'interiors.‘ The issue is whether or not the unit
isoquant is the same as an input requirement set. Using
the principles of the duality theory that distance
functions serve to prove [McFadden, 1978] and the free
disposal that is simultaneously assumed for both produc-
tion and cost space, it was proved by contradiction that
frontier production functions are a violation of the
theory of distance functions. This was accomplished by a
careful accounting of what is in the 'interior' of a dis-
tance function (higher isoquants) compared to the interior
points conceived to be within the surface of a frontier
production function.

Chapter Four presented a new case of an unit iso-
quant. It explained what the unit isoquant demonstrates
in reality. It returned to the original formulation of
the frontier production function and correctly inteprets
the fact that observations on production on a per unit of
input to per unit of output basis provides an unit iso-
quant with an 'interior.‘ It pointed out that "technical

efficiency" would identify the point of maximum average

126

physical output, the boundary between Stages I and II on a
production function with both stages in the production of
firms in long-run perfectly competitive equilibrium. As
such, "technical efficiency" is identical to traditional
economic efficiency.

Chapter Five reviewed the salient frontier production
function theory revealing that the proponents of this new
theory repeatedly and consistently reveal the apparent
interior points of frontier production functions to be due
to specification and/or aggregation error yet do not
explore the theoretical consequences of such revelations.
6.3. THE INTERIORS IN FRONTIER PRODUCTION FUNCTION THEORY

AND ”TECHNICAL EFFICIENCY"

Frontier production functions represent not just a
change in the "bench mark" observations in applied work
from the mean points of the data to the extreme points of
the data. Frontier production function theory is ques-
tionable as the apparent 'interior' points result from
specification and/or aggregation errors.

The traditional view of the world is that identical
circumstances result in identical outcomes (see Appendices
One and Two). It is this stability of the real world that
makes events predictable. By contrast, the fundamental
postulate of frontier production function theory is that
identical circumstances may result in different outcomes
and that the differences are not attributable to chance

variations in uncontrolled variables. This is what

127

explains an interior to frontier production functions that
is non—existent in physical reality.

Frontier production function theory suggests that two
production events can differ in their degree of "technical
efficiency" and that a "technically less efficient" pro-
ducer can become more "techncially efficient." The new
theory does not explain how this can be accomplished. If
one is to improve one's "technical efficiency," then one
presumably must "improve" or change one's initial circum-
stance in some way. But frontier production functions
assume there is no difference in the initial circumstances
of the "technically efficient" and "technically ineffi-
cient" producers; they have the same inputs; the same
output, and the same technology. If the change involves
changing the time, form, or location of some aspect of
production (other than the differences in time, form, or
location that are assumed to be "unimportant"), then a
specification and/or aggregation error has been committed
in identifying the differences in "technical efficiency."
If one is to change one's "technique" then one must change
the sub—production function one is using in production,
and "techncial efficiency" becomes an incomplete compari-
son across sub-productions.

The error involved in conceiving that frontier pro-
duction functions have interiors which can be technically
corrected is that of ignoring the contribution and oppor—

tunity cost of changing sub-production functions by inves—

128
ting or disinvesting in "fixed" inputs.
6.4. INTERIORS TO FRONTIER PRODUCTION FUNCTIONS
WOULD MEAN ”TE" t "PE"

If gig points of equi-production are on an isoquant,
then a producer cannot be "price efficient" without simul—
taneously being "technically efficient." Being "price
efficient" means being on the expansion path of produc-
tion. Moving along the expansion path of a full produc-
tion function means changing sub-produciton functions by
investing or disinvesting in fixed inputs. This means
that the fixed inputs are temporarily variable inputs. In
order to completely account for "price efficiency" one
must account for the prices (opportunity costs) of the
fixed inputs (see Edwards, 1958).

Can a firm be "technically efficient," but not "price
efficient" by being on the same isoquant as the "price
efficient" firm, but off the expansion path? If the
prices paid for all inputs are different for the two
firms, then in a world in which duality is assumed to
exist, both firms will be "price efficient" if they are
both "technically efficient," as was demonstrated in Chap-
ter Three. If prices are the same for both firms then the
firm that is not "price efficient" is not "technically
efficient," since it can achieve more output with its
resources by selling and/or buying its inputs without
changing its total investment or expenditures. That is.

the "price inefficient" firm will pay a greater cost for

129

its bundle of inputs, so that its bundle of inputs has a
greater value. That is, the firm implicitly has a greater
budget constraint. By trading some of its inputs for more
of the other inputs in the market place it can "cost-
lessly" rearrange its bundle to be a "price efficient"
bundle still of greater value than the bundle of the
originally "price efficient" firm. Thus, the adjusting
firm will be implicitly able to produce a greater output,
since it will have more of all inputs due to its larger
constraint.

The opportunity cost principle means that physical
quantities of commodities are inseparable from their value
(prices). The measure of thermal efficiency (an example
of "technical" efficiency in engineering) is defined as
useful output to costly input. In this treatment both
inputs and outputs are measured in a common physical
denominator. When this ratio is unity, efficieny is 100
percent. Similarly, economic efficiency is marginal reve-
nue product to marginal factor cost. When this ratio
equals unity for all inputs, including fixgg inputs, effi-
ciency is 100 percent. In this definition both commdities
are measured in a common denominator of prices. When one
finds the profit maximizing point by equating ratios of
marginal physical products to the corresponding price
ratios, one is implicitly comparing useful output to
costly input where a common denominator measurment that is

either physical or value is lacking.

130

6.5. THERE IS ONLY ONE TYPE OF EFFICIENCY

Production efficiency is measurable only within some
clearly defined set of circumstances or constraints. If
the preconditions for comparing the efficiency of two or
more firms are identified, the decisions made as to what
constitues "sameness" and "close enough," then efficiency
can be measured. This efficiency can not be separated
into "technical efficiency" and "price efficiency" since
the two are identically equal. If they were not equal, as
suggested by frontier production function theory, then
some aspect of the established preconditions has been
violated, either in theory, or in practice. Efficiency is
efficiency which is a maximizing of profit, when inputs of

given value are used to produce the greatest value.

6.6. FINAL CONSIDERATIONS

There are three issues that might deserve further
attention.

(1) For expediency, the economics of making input
fixity and variability endogeous to the input requirement
set has been avoided in this work. That one can endo-
genize decisions to invest and disinvest fixed inputs has
been established by Edwards (1958). That so doing does
not alter the conclusions about efficiency has been postu-
lated, rather than proved. The fixed inputs were not made
endogeneous lest they become confused with variable

inputs. Even when they are endogeneous, they behave as

131
fixed inputs, not as variable inputs.

(2) The methods for separately measuring "technical
efficiency" and "price efficiency" on frontier production
functions clearly do not do so. What is actually measured
is specification error and/or aggregation error when pro-
duction function analysts measure technical inefficieny.
Both of these errors create significant problems in
applied work since both bias estimates of a system's
parameters. Proper identification and measurement of
these errors might allow one to aggregate or disaggregate,
specify and respecify, production relationships in applied
work in order to minimize, or at least explicitly account
for, the degree of specification or aggregation error
present in the analysis.

(3) Finally, more attention should be paid to whether
or not the proof of duality theory requires free dispo-
sal, input requirement sets, and distance functions when
nonstochastic interior points are known to be absent for a
production function. These three concepts assist proving
duality theory by excluding Stage III of the production
function and by eliminating possible 'interior' points.
Aside from this they add little if any insight into pro-
duction theory. They are often misunderstood, and conse-
quently, researchers attempt to introduce the concepts
into the analysis of applied problems resulting in ques-
tionable inferences, and misleading prescriptions.

Free disposal, which has no counterpart in reality,

is especially misleading. Without free disposal there can

132

be no input requirement sets and distance functions. Free
disposal is accomplished mechanically by introducing a
scalar into the analysis. The scalar, is the factor by
which all additional inputs have to be reduced to sink to
the bounding lowest isoquant, within a given input re-
quirement set, the 'scale' by which 21; the output and
input levels must be scaled back to shrink production to
the level of the lowest isoquant. It is a measure of the
"distance" between the two isoquants. Thus, by using free
disposal, within the context of a distance function, one
can make any two sub-production functions identical simply
by reducing one to the other by scalar multiplication.
The only means for determining the scalar is as a function
of what needs to be disposed in order to eliminate any
difference between two sub-production functions, or two
isoquants. The scalar has no counterpart in physical
reality, and suggests consequences that obscure the true
differences between two sub-production functions or iso—

quants, and what accounts for those differences.

APPENDICES

APPENDIX ONE

EFFICIENCY AND THE LAWS OF THERMODYNAMICS

Economic theory is largely a theory about the be-
havior of people. It focuses on how people make choices,
and offers guidance in selecting the best among competing
choices. Each choice represents an opportunity; an oppor-
tunity to experience the utility embodied in that choice.
To say that something has utility means that it has value
for the consumer. That is, if a commodity can provide
utility it has value. In economics the value of an oppor-
tunity (in some sense a measure of utility) is captured by
the concept of opportunity cost. In physics the concept
of utility is captured in the concept of work. Because
work can provide utility it has value.

In production economics one speaks of transforming

"inputs" into "outputs." Production is a cycle, wherein
energy is transformed. It is assumed that one is able to
derive utility from the output. Since the transformation

is usually physical in nature, and often involves changes
in the gpgpg of the production system, technical efficien-
cy is relevant. Thermal efficiency, hereafter treated as
synonomous with technical efficiency, is one of several
types of technical efficiency. By examining what thermal
efficiency is, one can appreciate what technical efficien-

133

134

cy means in general.

This chapter will first define the terms and rela-
tionships of thermodynamics that are pertinent to economic
production theory as it relates to efficiency. The second
section presents the fundamental definitions. The third
section will focus particularly on the first and second
laws of thermodynamics and demonstrate that the "technical
efficiency" of frontier production functions cannot exist.
The fourth section will define and explain technical effi-
ciency, or thermal efficiency, as it is used in thermo-
dynamics. It will be shown that technical efficiency in
thermodynamics is the same as economic efficiency. The
last section will explain specifically how and why TE and

PE are inconsistent with thermodynamic theory.

A1.1. DEFINITIONS FROM THERMODYNAMICS

The laws of thermodynamics are observations on the
physical relationships in the transformation of energy
from heat to work or vice versa. Energy is the ability to
do work. There are two types of energy; energy that is
stored and energy that is in transition. Work and heat
are energy in transition.

The laws of thermodynamics explain the work (heat)
that can be obtained from resources, and some of the
restraints for doing it. The first law is the well known
principle of the conservation of energy. The second law
deals with the the amount of thermal energy that will

become "useful" work by means of a given cycle. It leads

135

to a definition of thermal efficiency which is a type of

technical efficiency.

A1.1.1. SYSTEMS:

In thermodynamics, one investigates the nature and
behavior of physical systems. Systems interact with each
other, or their environment, producing work or heat in the
process; energy is transformed. In order to study the
interaction between two systems, they must be insulated
from the environment so that no thermal energy escapes
into the environment and is thereby unaccounted for in the
interaction between the two systems. This is called an
adiabatic system. In order to simplify the investigation
of the principles of thermodynamics, it will be assumed
that one has a system and its environment (which is in
effect another system) and that together they represent an
adiabatic system.

A system is either a particular collection of
matter, a gloseg:system, or a particular region of space,
an open system. Briefly, in a closed system no matter can
cross the system boundary, while in open system matter can
cross the boundary. Interactions will occur when the
gpppg of the system is out of equilibrium with the state
of the environment, and the system is not insulated from
the environment; when there is no barrier to the exchange
of energy between the system and the environment. When

there is an exchange of energy between a system and its

136

environment, the energy is transformed such that the sys-
tem may change state, and/or some of the energy is
transformed into heat and/or work. This appendix will be
confined to exploring the characteristics of closed sys-
tems for the sake of brevity. All of the principles
governing the characteristics and behavior of closed sys-

tems can be easily assigned to open systems.

A1.1.1.1. STATES:

The state of a system is evaluated by its eguation-
of state. In a very simple case, for an ideal gas, this

relationship might take the form,

(A1.1) pv = RT
where
p = PRESSURE

v = VOLUME

:0
II

CONSTANT; USUALLY A CONSTANT ASSO-
CIATED WITH A PARTICULAR GAS

T = ABSOLUTE TEMPERATURE

When a system exchanges energy with its environment, it
often results in a change in the values of the variables
in the equation of state. For example, for a change in
state of a system (A1.1), P, v, or T would change in

value.

137

A1.1.1.2. PROCESS:

The path which describes the exchange of energy
between a system and its environment is called a process.

A process is any transformation of

a system from one equilibrium state to

another. A complete description of a

process typically involves specifica-

tion of the initial and final equilib-

rium states, the path (if identifi-

able), and the interactions which take

place across the boundaries of the

system during the process. Path in

thermodynamics refers to the specifica—

tion of a series of states through

which the sytem passes. [Wark, 1983,

page 10]

In thermodynamics, a process transforms a system from one
state to another state. A process is a path function
since it encompasses the system's changing states from its
inital to its final state. The values of the initial and
final states of the system in thermodynamics may, or may
not, be equal. That is, the system may return to its
initial state. If the initial and final state of the
system are equal then the inital state does not equal the
final state for the system's environment. That is, either
the state of the system changes, or the state of the
environment changes, or both. The change occurs because
energy has been exchanged between the two. It is the work

and the final states of both the system and its environ-

ment that correspond to output in economics.

138

A1.1.1.3. PROPERTY:

A property of a system is a characteristic, or para-
meter, of the system. It is a ppint fppctigp, since it is
the value of a property at a point. Examples of proper—
ties are pressure, volume, temperature, energy, mass, and
entropy, but not heat nor work (the difference between
temperature and heat is explained below). In Figure A1.1,
a system may change state, from $1 to 82, by either Pro-
cess A or Process B. In either case, the properties of Si
and 82 respectively are the same; they are not tuniquely
determined by A nor B. The properties p,v, and T, become
the means whereby the change in the state of the system
can be measured.

When the properties of the system are different than
those of its environment, i.e. it is out of equilibrium
with its environment, the system can

...interact with the environment and

produce work until the system reaches a

state where such potential differences

do not exist. For any system, this

state is called the dead state because

the system can do nothing more. [Dixon,

1975, page 231]
Whenever a system is out of equilibrium with its environ—
ment, the two states differ, there is the potential for an
exchange or transfer of energy between the two which will

result in work becoming heat or vice] versa, until the

system reaches its dead state.

139

A1.1.1.4._§OMOGENEITY:

Properties of a system are classified as either ex—
tensive or intensive. The distinction is important since
it has a bearing on the definition of homogeneity in
economics. The distinction is as follows:

Imagine a whole system divided into a

number of parts. If the value of a

property for the whole system is equal

to the sum of its values for the

various parts of the system, then it is

called extensive. ...intensive proper-

ties have meaning at a point or local-

ly. That is, we can talk about the

local pressure or temperature but not

about the local mass or volume because

the latter have no meaning. [Dixon,

1975, page 59-60]
Examples of extensive properties are mass (M) and volume
(V), while examples of intensive properties are pressure
(p) and temperature (T). Extensive properties may be
effectively converted into intensive properties by divi—
ding them by mass, which then equals an
average specific property, and finding the limiting value
of this quotient at a point, which is called a local
specific property. For example, the local specific volume
of a system is:

(A1.2) v :- lim (A V/AM)

£5V-+ 0
where
v = LOCAL SPECIFIC VOLUME OF THE SYSTEM
V 2 VOLUME OF THE SYSTEM

M = MASS OF THE SYSTEM

140

Extensive properties are such that they cannot vary within
the system: they are the same everywhere. If the same is
true for the local specific properties, the system is said
to have uniform properties throughout. A system whose
properties are uniform throughout is homogeneous. It will
be assumed throughout that all systems are homogeneous.
This is the same as assuming that all inputs, both vari—

able and fixed, into a production system are homogeneous.

A1.;$l.5. REVERSIBILITY:

Thus far there has been no discussion of the direc—
tion of change when a system changes state. If one starts
with state S1 in Figure A1.1, and arrives at state 82 by
means of process A, and then 'backs up' from S2 to Si by
means of reversing process A, then the process is called

reversible.

A process executed by a system is
called reversible if the system 33g ipg
envirgpment can be restored to their
initial states and leave no other ef—
fects anywhere. Another term for re-
versible might be completely restor-
able. The definition requires that
work and heat exchanged between a sys-
tem and its environment in a reversible
process can be restored to each in
exactly the same form so that both [the
system and its environment] are re-
turned completely to their their ini-
tial states. [Dixon, 1975, page 174]

A process is reversible if its effects on a system and the

system's environment can be completely returned without

141

52
Process A 2

2 Process B

51

FIGURE A1.1

THE CHANGE IN THE STATE OF A SYSTEM (81 TO 82)
BY MEANS OF EITHER PROCESS A OR B

142
additional inputs. In reality, no 32;; process is rever—
sible, as a consequence of the second law. As an example,
consider a block sliding down an incline. As it does, it
produces heat, i.e., the 'environment becomes hotter.' If
one were then to slide the block back up the incline, the
environment does not become cooler; that is, the process

is not reversible.
A1L131.6. CYCLE:

A gyglg is a sequence of processes operating on a
system such that the final state is identical to the
initial state. Figure A1.2 is a cycle, since it takes the
system from $1 to 82 where 51 = $2. Recall that although
there is no change in the initial and final states of the
system, there is necessarily a change in the state of its
environment, since real cycles are not reversible. As
will be seen later, the development of the second law
requires that one can "imagine" a reversible cycle. In
particular, the definitions of technical efficiency
require using the hypothetical reversiblity of a cycle in

order to measure thermal, or technical, efficiency.
A1.1.2. WORK AND HEAT:
A1.1.2.1. WORK:

As was noted above, work and heat are not properties
of a system. Work is defined at the boundary of the

system. Work is a form of energy, commonly measured as

143

51, $2

Cycle

 

FIGURE A1.2

THE CHANGE IN THE STATE OF A SYSTEM (51 TO 82)
BY MEANS OF A CYCLE

144

force times distance:

(A1.3) w =fr dx

where

8
II

WORK IN FOOT POUNDS

m
II

FORCE IN POUNDS

X = DISTANCE IN FEET

In order that the relationship between a system, its
environment, and work are explicitly clear, the following

definition of work will be used.

Work is done by a system (on another)
when the sole effect external to the
system could be the rise of a weight.
The amount of work done is the product
of the weight (force) times the
distance lifted. By convention, work
done py a system (which could lift
weights in the environment) is taken as
positive for that system; work done pp
a system (the environment lifts weights
within the system boundaries) is taken
as negative. [Dixon, 1975, page 106]

Clearly, there are a number of different types of systems
that can do work. The type of work as measured by (A1.3)
is linear mechanical work. Table A1.1 is a partial list
of other types of systems and the types of work they do
and how that work is measured. Notice that none of the
WORK equations are expressed as inequalities; they are all
equalities. That is, a given system cannot be "technical-
1y inefficient," as the terms are used in the frontier

production literature, in performing work given the same

processes. The amount of work produced by the system is

145

TABLE A1.1

SUMMARY OF VARIOUS WORK EQUATIONS

SYSTEM FORCE DISPLACEMENT WORK
LINEAR MECHANICAL FORCE(F) DISTANCEIdX) dW = F dX
ROTATIONAL

MECHANICAL TORQUE(T) ANGLE(d«) dW = T’du
ELECTRICAL CHARGE VOLTAGE(e) COULOMBS(dQ) dW = —e dQ
ELECTRICAL FIELD VOLT/METER(E) POWER(dP) dW = -VE dP

MAGNETIC FIELD AMPERE/METER(H) WEBER/METER2(dM) dW -VH dM

SOURCE: DIXON

exactly determinant for a given cycle, and does not vary.
One system can be technically inefficient in performing
work as compared to a different system given the same
cycle, or one cycle can be technically inefficient in
performing work as compared to a different cycle given the
same system. Given two identical systems, and identical
cycles, the same force and the same displacement (the same
inputsand the same technology), one will not get different
amounts of work from each; the energy of one system will

not be less efficient than the other.

A1.1.2.2. HEAT:

Heat is another form of energy. It may provide

 

utility and be obtained by transforming work energy. Like
work, heat is measured at the boundary of the system and

its environment and therefore is not a property of the

146

system or its environment. It is thermal energy in tran-
sition due to a temperature difference. Typically, heat
is defined operationally in Btu's; the energy required to
raise the temperature of one pound of water at atmospheric
pressure from 59.5 to 60.5 degrees Fahrenheit. It should
be noted that the temperature of a system may change
without heat being transferred either to or from the
system, e.g., by a change in pressure. One should also be
careful to distinguish between a difference in temperature
between a system and its environment, and a change in
temperature within a system. These two points emphasize
that heat is an interaction between a system and its

environment and not a property of either.

A1.1.3. SUMMARY OF DEFINITIONS:

Energy in most forms is a property of a system, while
work and heat are not. Energy is exchanged between a
system and its environment in the form of work or heat.
Thus, work and heat are defined at the boundary of the
system. A process is the means by which the transfer of
energy is effected between a system and its environment.
Work may be transformed into heat, and vice versa, by a
process. Except in the case of a cycle, the results of a
process acting on a system are changes in the state of the
system. Since real processes are not reversible there is
either a change in the state of the system or in the state

of its environment, or both, in all cases of a real pro-

147

cess acting on system. In production economics, the
inital states of the system and the environment corres—
ponds to the inputs, the production function corresponds
to cycle, and work or heat and the change in the state of
either the system and/or its environment corresponds to

output.
A1.2. THE LAWS OF THERMODYNAMICS
A1.2.1. THE FIRST LAW:

The first law defines the well known principle that
in the absence of nuclear changes or approaching the
velocity of light, energy is neither created nor
destroyed. Energy can be transferred or exchanged between
a system and its environment in the form of work or heat
by means of a cycle. Within the cycle, work may be
transformed into heat or vice versa. But, neither within
the cycle, nor as a result of the cycle, is there a change
in the sum of the energy of the system and its environ—

ment. Therefore, the first law can be stated as:

(A1.4) f(dQ - dW) = 0
where

:6

INTEGRATION OVER A CYCLE

HEAT IN JOULES

W 8 WORK IN JOULES

This suggests that input = output. Given two identical

cycles acting on two identical systems, there can be no

148

differences between them: there is no difference in
technical efficiency between them. Given the same inputs,
gpg pip; ggmg production function pp; always gets the same
output.

If work is energy that provides utility, i.e., work
has value, then technical (physical) efficiency must
implicitly be a comparison of values. Therefore, there is
no basis for a difference between technical (physical)

efficiency and price (value) efficiency.
A1.2.2. THE SECOND LAW:

The second law accounts for the energy that is "lost"
when energy is exchanged between a system and its environ—
ment. Due to the first law, the energy is not truly lost;
rather it is degraded in quality so that it is no longer
available to become work, and therefore loses its value.
This is called degradation of energy. By observation of

the real world:

(A1.5) fid'Q/T < 0

where
Q = HEAT FLOW IN THE SYSTEM OR ENVIRON-
MENT
T = ABSOLUTE TEMPERATURE AT WHICH THE HEAT
FLOWS
This is the Clausius Ineguality. If a cycle were rever—

sible, then in conformance with the above,

IA1.6) fd'Q/T = o

149

where
Q = HEAT FLOW IN THE SYSTEM
T = ABSOLUTE TEMPERATURE AT WHICH THE HEAT
FLOWS
The Clausius Inequality is statement of the fact that in

reality no process is reversible so that,

(A1.7) éd'Q/T <= 0

where
Q = HEAT IN THE SYSTEM
T = ABSOLUTE TEMPERATURE AT WHICH THE HEAT
FLOWS
and

THE EQUALITY HOLDS FOR HYPOTHETICALLY REVER-
SIBLE CYCLES

THE INEQUALITY HOLDS FOR REAL CYCLES

Equation (A1.7) is a statement of the second law. It
reveals the existence of another property of systems known
as entropy (8). Since entropy is a property it is not a
function of the process, but a function of the end states

of the system. It is defined as:

(A1.8) ds = d'Q/T
where

ENTROPY IN THE SYSTEM OR ENVIRONMENT

HEAT FLOW IN THE SYSTEM OR ENVIRON-
MENT

D
II

V-J
II

ABSOLUTE TEMPERATURE AT WHICH THE HEAT
FLOWS

when the process is reversible. While 8 is always a

150

property, in general d'Q/T is pp; a property, because of
(A1.5), the Clausius Inequality: it is a property only in
the limiting cases of a reversible process. One must
imagine a reversible process in order to measure the
change of entropy in the system and/or environment. Since
entropy is a property, it may increase or decrease within
a system, or within its environment. But since in reality
any process goes only in one direction, due to the
Clausius Inequality, the change in the total entropy of a
system and its environment must be positive. Therefore,

in general, for real systems:

(Al.9) ds > d'Q/T
OI‘

(A1.10) ds > O

in an adiabatic real process since dQ = 0, due to the

definition of adiabatic, and T > 0.
A1.2.3. SUMMARY OF FIRST AND SECOND LAWS:

The first law means that the amount of total energy
in a system and its environment does not change. The
second law means that when a real process acts on a system
the total entropy in the system and its environment in-
creases; that there is some amount of energy that becomes
"bound up" as the increase in entropy, and is unavailable
to be transformed into work. If one can imagine a rever-

sible cycle and one can imagine a zero heat flow, then one

151

can imagine a situation where there is no change in en-
tropy for a closed system. In such a situation, all the
energy exchanged between the system and its environment
would be in the form of work. If there is either heat
flow or a reversible cycle, then one can measure thermal
(technical) efficiency.

Technical efficiency can be defined when one has a
reversible cycle using heat as an input and work as an
output. The measure of work output to heat input is the

measure of technical, or thermal, efficiency.

A1.3. TECHNICAL EFFICIENCY

In thermodynamics the definition of efficiency is a

relationship between values; it is a ratio of value.

Efficiency here means the useful output

divided by the the costly input, both

expressed in energy units. [Dixon,

1975, page 15]
In this definition one can substitute "work" for "useful
output." 1p; definition glppg eliminates gpy possible
distinction between "technigpl efficiency" gpg "price
efficiency" pg gpgy Egg pggg ip pp; frontier production
function literature. Since technical efficiency in ther-
modynamics deals with work energy it implicitly deals with
changes in value since work by definition has value: i.e.
provides utility. "Costly" is used in the opportunity

cost sense, since the input has an opportunity cost.

Consequently, in thermodynamics there is no difference

152

between technical efficiency and price efficiency; they
are identical.

Recognizing that the second law is a reality, that
some energy is "lost" when a system interacts with its
environment, technical efficiency becomes a function of
the work, which is a function of the cycle in the exchange
of energy. In order to be perfectly clear that thermal
efficiency is a function of the cycle, one must avoid
misunderstanding another concept in thermodynamics, that
of potential work, sometimes called optimum work, maximum
work, or reversible work. Potential work is a function of
the properties of the system and therefore is not a
function of a process, nor heat nor work.

Potential work is a measure of energy availability
within a specific system. That is, it is a function of
the magnitudes of the properties which determine the
states of the system, measured between two different
states.

The maximum possible work output
that can be produced by a system from a
given state to its dead state is what
is called, appropriately enough, the
work potential. The term availability
is also used.

It should be noted by students
that, for a given environment, work
potential is a property of systems.
...it should be clear at this point
that the maximum work that can be pro-
duced by a system is not a function of
the process. The actual work, of
course, will be a strong function of

the process but the maximum is the
maximum regardless of how it is ob-

153

tained. Hence work potential is a
property. [Dixon, 1975, page 232]

Two things should be evident from this definition of
work potential. The first is, that two systems with the
same properties have the same work potential. Secondly,
given that the second law and entropy account for a "loss"
of some of the energy in a system when it changes state,
the measure of potential work is less than the pppgl
energy in the system.

The definition of potential work suggests that one
might measure efficiency by taking the ratio of work
potential to the amount of work one actually observes
given the operation of one process. This might appear to
be a means of evaluating the "technical efficiency" of a
system. Unfortunately, in the case of a cycle the measure
of work potential equals zero, since in a cycle the system
returns to its initial state. This eliminates any dif-
ferences between the initial and final properties of the
system with which to calculate a measure of work poten-
tial. Therefore, efficiency is a measure of 211 the
processes which use heat input and produce work output
during the cycle.

During a cycle some processes will have positive net
work in and some will have negative net work in. If the
cycle produces positive net work, it will use positive net
heat in a greater amount due to the second law,. even if
the cycle is reversible. Therefore, the measure of effi-

ciency will be less than 100%. Nevertheless, in the case

154

of a reversible cycle one will have a measure of the
maximum efficiency possible. The technical efficiency of
a non-reversible (real) cycle will be less, but always the
same given the same inputs and the same processes, ceteris
paribus.
A1.4. WHAT IS WRONG WITH FRONTIER PRODUCTION FUNCTIONS
FROM THE PERSPECTIVE OF THERMODYNAMICS:

Within the context of thermodynamics thermal effi-
ciency, or technical efficiency, has been defined. Does a
comparison of firm P to firm Q, in Figure 2.1, reflect

this type of technical efficiency? If it does then one of

two situations must exist.

A1.4.1. THE FIRST SITUATION:

In Figure 2.1, P and Q are clearly supposed to be
within the same input requirement set, or sub-production
function, so they are using the same cycle. Firms P and Q
also use the same homogeneous inputs in the same pro—
portions since they are both on a ray from the origin.
Therefore, one can conclude that P and Q have the same
properties in their initial states with the only dif-
ference being that P is some scalar 5 times greater in
quantity than Q. This corresponds to the situation of
getting the same quantity of useful output with different
amounts of the same inputs in the same proportions. Since
they are not on the same unit isoquant in their end

states, assumed to be a dead state, then their end states

155

differ by some factor g; # 3. That is, if the amount of
useful work is the same, then the amounts of waste in the
two systems must be different. Specifically, firm P's end
state will have a higher value of entropy. This means
that the properties in the end states are not the same, so
that the output from the two systems are not the same.
This situation cannot be due to the processes since the
states are by definition not a function of any of the
processes. Therefore, the difference in the end states
must be due to g: the properties in P's end state that are
different than the corresponding ones in Q must be a
function of p. Thus, g, the factor of proportionality,
would have to be an argument in the equation of state,

which it is not in thermodynamics.

A1.4.2. THE SECOND SITUATION:

If the initial states of the two systems are the same
and the amounts of useful output are different, then one
has the situation of two sets of identical inputs pro—
ducing different output. The difference between P and Q
is the amount of actual work done by them. This is not a
function of the properties of either their initial states
or their and states, as was discussed above, but of the
processes each uses. Specifically, if there is a differ-
ence in the amounts of actual work, then in keeping with
the laws of thermodynamics it must be due to each system

using different processes. Using different processes

156

means opoerating with different fixed inputs, i.e.,
following different cycles or being on different sub-
production functions. Comparing the two firms P and Q
using a unit isoquant under these circumstances consti-
tutes an error of specifying the same fixed inputs or

process when in fact they are different.

A1.5. SUMMARY

Given identical systems and identical processes, the
measures of efficiency will always be identical. One
cycle may be less efficient than another cycle given the
same system since it is a different pgph from state to
state, thereby producing different levels of net heat in
and different levels of net work out as the system's
properties assume different values at each ppipp along the
path. Similiarly, two systems might differ in technical
efficiency using identical processes since the states for
each system will differ initially and consequently at each
ppgpp (state) along the path from initial to final state.
Therefore, if one observes two different production situa-
tions that differ in efficiency, either the systems are
different (non-homogeneous inputs) or the processes differ
(different sub-production functions).

Two things should be clear from considering thermal
efficiency in thermodynamics. The first is that there is
no difference between technical efficiency and price effi-

ciency since efficiency measures the ratio of value of

157

output to the value of input. The second is that "tech—
nical efficiency" as used in the frontier production
function literature cannot exist. If one uses identical
inputs and the identical process, then one must get the
identical output. Therefore, if in Figure 2.1, the pro-
cess used by points Q and P are identical, point P can be
less efficient than point Q only if P is producing a
different level of output. Alternatively, if P and Q are
producing identical output, then they it must be the
result of Q and P using different processes. In either
case, the laws of thermodynamics makes it clear that the
frontier production function distinction between "TE" and
"PE" are not valid without violating the basic as-
sumptions of the theory. Therefore, frontier production
functions are the result of either specification error,

and/or aggregation error.

APPENDIX TWO

EFFICIENCY IN ECONOMIC THEORY

To understand much of the frontier production
function literature one must be familiar with the "set
theory" approach to microeconomic theory. Frontier pro-
duction functions are sometimes conceived in that litera—
ture as "distance functions" displaying "strong dispos-
ability," [Kopp, 1981b, Kopp and Diewert, 1982] .

The first section of this chapter will define of
terms. The next section will present the usual as-
sumptions made about production sets and the implications
of the more important two, concavity, which indirectly
suggests the importance of fixed inputs, and monotonicity,
which implies free disposal. Fixed inputs are frequently
ignored in the frontier production function literature
which may explain why aggregation and/or specification
error have been identified as "technical efficiency" in
frontier production functions. In Chapter Three free
disposal and duality theory were used to show that fron-
tier production functions cannot exist if they are dis—
tance functions that display strong (free) disposability.
The third section correctly defines efficiency in relation
to profit maximizing behavior, to clarify the insepar-

ability of the physical and value aspects of production.

158

159

Finally, as a result of the first three sections, the
explicit inconsistencies of frontier production function

theory will be explained.

A2.1. PRODUCTION THEORY

A production relationship maps inputs to outputs,
where input is the range and output the domain. One
should bear in mind at the outset that production is by
definition the creation of value -- production processes
are implicitly normative. Production pg§p§ creating
utility by changing the time, space, or form of commodi-
ties. Prehaps for this reason alone, one cannot consider

physical (TE) and value (PE) efficiency separately.

A2.1.1. WHAT ARE INPUTS AND OUTPUTS:

Inputs are the commodities and services with which
one starts while outputs are the commodities and services
with which one ends a production process even if there is
no change in time, form (quality), or space. Thus, "left
over" inputs are part of the output. Left over inputs
would result from a system that changes state but does not
reach its dead state. In this simple analysis, inputs and
outputs are assumed to be individually homogeneous (have
the same properties), and completely divisible. Addi-
tionally, homogeneous means that each input or output
identified is defined to be alike with respect to time,
place, and form. Therefore, the individual units of a

quantity of identical input or output are indistinguish-

160

able from each other under any and all circumstances, and
in particular, in the way they behave in production.
Notice that this means that the processes becomes a
function of the system and vice versa. One cannot differ—
entiate between units of the same inputs, except with
respect to order in which they are added to production.
This means each pound in ten pounds of the input V(j) is
exactly like any other one pound of that input V(j) in all
its descriptive characteristics. Divisiblity means that
the functional nature of the input (output) is independent
of the units in which it is measured. Five pounds of the
input V(j) added in one pound units will have the same
affect on the output as five pounds of V(j) added in ten
one half pound units.

The law of diminishing utility indicates that the
utility of any particular input (output), will change at
the margin as it becomes increasingly scarce or plentiful.
Indeed, a good that is not scarce does not have exchange
value. This issue of changing marginal value will be side
stepped for the purposes of this disseratation by assuming
atomistic competition so that inputs and outputs can be
treated as having constant prices. In order that there be
no confusion, the part of output that has net positive
value will be called product, that part that has zero net
value will be called waste, and that part that has nega-
tive net value will be called pollutant. Either utils or

dollars are treated as being adequate common denominators

161

of value.
A2.1.2. PRODUCTION SETS:

Output is produced by transforming inputs. From the
laws of thermodynamics it is clear that there is a single
valued relationship between the quantities of inputs
transformed and the quantity of product generated, given
specific processes. That is, using given processes and
specific quantities of homogeneous input, only one quanti-
ty of product will result, ceteris paribus. Because of
this relationship the terms output and product can be used
interchangeably in most situations. For a particular
producing unit, or firm, there is some finite set of
production possibilities, that is described by the collec-
tion of all possibile input bundle combinations and the
quantities of product that result from their transforma-
tion. A vector of input and output quantities can be
variously called a production plan, activity vector, or
netput.

Suppose the firm has n possible
goods to serve as inputs and/or out-
puts. We can represent a specific
production plan by a vector y in Rn
[the positive quadrant of euclidean
hyperspace] where y(i) is negative if
the i h good serve: as a net input and
positive if the it good serves as a
net output. Such a vector is called a

netput vector. The set of all feasible
production plans — netput vectors - is
called the firm's production possibili-
ties set and will be denoted by Y, a
subset of R“. [Varian, 1978, page 3]

162
A2.1.2.1 PRODUCTION POSSIBILITIES SETS:

A producpgon_possibilities set is the collection of
all feasible input/output vectors for some output vector
Y. As such, it represents the technology for Y. The

production possibilities set is:

(A24) Y = (y(I). 31(2). y(3). y(M). V(l).
VIZ). V(3)..... MM. 2(1).
2(2). 2(3). .... ZIP). 0(1).
(1(2)! ”(3)1000: U(Q))
where
y(I) = PRODUCTS FOR I=1 TO M
V(J) = v(J,I) FOR INPUTS J=1 TO N AND
PRODUCTS I=1 TO M
Z(K) = z(K,I) FOR INPUTS K=1 TO P AND
PRODUCTS I=1 TO M
U(L) = u(L,I) FOR INPUTS L=1 TO Q AND

PRODUCTS I=1 TO M

It is worth reiterating that this set is a feasible set,
or set of those production plans that are physically
possible, e.g., the set of netput vectors that conform to

the laws of thermodynamics.

A2.1.2.2. TECHNICAL:CHANGE:

Usually, technical change means that the original
production possibilities set has been expanded by adding
new input:output vectors, or by adding a dimension to the
existing vectors. This is equivalent to adding a prev—
iously unknown input, or unknown way of combining known

inputs (processes), to create new vectors, or a new dimen-

163

sion, to the production possibilities set. Note that this
new input, or new way of combining old inputs, can be used
in various amounts, which is why the change adds more than
one vector to the set.

If this new input results in the possibility of a
given amount of product y being produced at a lower cost,
then one can conclude that the new production possibil-
ities set is more efficient. To ignore the new activity
bundles (to operate only with the opportunities of the old
production set) would indeed be inefficient. Efficiency
is achieved by making an economic adjustment resulting
from comparing the cost of using the new input bundles as
opposed to any of the original bundles. Strictly
speaking, this means not only their cost in operation, but
also the cost of switching from one set to the other by
investing or disinvesting. If adopting or using the new
input results in a lower net value from production, the
new technology is less efficient than the old technology.
Determining efficiency is a question of evaluating cost;
the ratio of the value of useful work to the cost of the
input.

Exactly what causes technical change and how techni-
cal change is accomplished are two complex issues that are

beyond the scope of this dissertation.
A2.1.2.3. PRODUCTION FUNCTIONS:

A production possibilities set may contain one or

more production functipns or processes. The production

164

function specifies a functional relationship between in—
puts and an output such that for any given vector of
inputs there is one, and only one, vector of output. That
is, one cannot have two netputs in the same production

function where:

(A2 2) Y*(I) <> y(I)
and

(AZ-3) (V*(J). Z‘(K). 0*(LII = (V(J). ZIK). U(L))

In addition there is a distinction made between the groups
of inputs in that the V(J)'s are variable inputs,the
Z(K)'s are fixed:inputs, and the U(L)'s are random var-

iable inputs. These distinctions are critical.

A2.1.2.3.1. SUB-PRODUCTION FUNCTIONS:

A sub3prpdugtion:£pngtigp is a "restricted" subset of
the production possibilities set. It is restricted in the
sense that there is only one product, y(I), and some
subset of the inputs remain fixed or constant over the
entire range of production possibilities. The sub-produc-

tion function defines the technical relationship between

inputs and output. It is one technigpe within the tech-
nology for Y. It defines the physical transformation of
inputs into output. It is the totality of the processes

that act on the inputs, or system, resulting in useful

work, or output.

165

(A24) f(YI = MN”). VIZ). v(3)..... NM.
2(1). 2(2). 2(3). 2(P). 11(1).
11(2). u(3). 11(0)) I (Y. v.
z, u) 6 Y]
where

y = VECTOR FOR A PRODUCT I

V(J) = VARIABLE INPUTS FOR J=1 TO N
FOR PRODUCT I
2(K) = FIXED INPUTS FOR K=1 TO P
FOR PRODUCT I
u(L) = RANDOM VARIABLE INPUTS FOR L=1 TO Q

FOR PRODUCT I

Variable inputs, V(J), are inputs over which the
manager has control, and for which the manager may vary
the quantities of the input in the sub-production pro-
cesses within one production period. Variable inputs will
be varied in the amounts used within the production pro—
cesses as a result of assessing their costs, relative to
the value of production within the firm, and relative to
the value of production outside the firm. Their acquisi-
tion prices and their salvage values are always equal.
Their within firm opportunity cost is the same as their
out of firm opportunity cost. Within firm opportunity
cost changes acquisition price and their salvage values
and all three remain equal. Therefore, when product or
variable input prices change, the amount used will neces-
sarily be adjusted upward or downward in order to maintain
efficiency.

Fixed inputs, z(K), are those inputs over which the

manager may have control, but which are not varied in the

166

amounts used within the production processes. They are
defined for commodities and inputs for which acquisition
costs exceed salvage values. The amount of a fixed re-
source acquired by the firm initially is to be determined
after assessing its value in production. The fixed
resource will be acquired up to the point that its mar-
ginal acquisition cost equals its value in production both
expressed as stocks or services. Once acquired by the
firm, its quantity is fixed in production so long as its
within firm opportunity cost, or shadow price, is bounded
by its acquisition and salvage value. That is, having
been acquired, its acquisition price is greater than its
salvage price. Therefore, a change in the fixed input's
acqusition cost or salvage value, the latter reflecting
out of firm opportunity costs, will not necessarily lead
to an adjustment in the amount used, investment or disin-
vestment, by the firm. That is, the input is fixed in
production.

Some fixed inputs are specialized in the sense that

they have no within firm opportunity cost, i.e., they
cannot be used to produce more than one product. The
number of products they can produce is one, the Ith. If

one considers unspecialized inputs capable of contributing
to the production of more than one product, then such an
input may be fixed to the firm, but not within the sub-
production function for one product, since its amount
might be varied between two, or more, sub-production

functions for the multiple products it can produce in the

167

firm. It should be noted that the fixity of inputs can be
made endogeneous [Edwards, 1958] without changing the
conclusions to be drawn regarding either the contribution
of the fixed input to the production processes or to the
definition of efficiency.

Thus, the optimal amount of an input whose acquisi-
tion price equals its salvage price always changes with
its gpg price. However, the optimal amounts of inputs
whose acqusition costs exceed their salvage values do not
always react to changes either in their acquisition cost
or salvage value. Indeed, if the within firm opportunity
cost is between the input aquisition and salvage prices
expressed as flow prices then the optimal amount to use
will not vary and the input is fixed.

Random variable_inputs, u(L), are inputs over which
the manager has no control, and whose quantities (or
quantities) vary randomly among firms and for individual
firms from some normal, or average specification. This
average quantity (or quantity) is usually the first moment
of the distribution from which the quantities of u(L) are
drawn. This average quantity is fixed. Examples would be
inches of rain fall, number of hours of sun light per day,
quantity or land, labor of capital; or air temperature.
Since the random variable inputs are usually measured as
deviations from their averages in any given production
period the expected value so measured is equal to zero.

Since the random variables are outside the control of the

168

manager, and because their expected value is zero, the
average value from which they deviate is fixed. This
assures "average conformity" with the laws of thermo-
dynamics. Unless otherwise noted, fixed inputs and the
average value of random variable inputs are treated in the
same manner.

The sub-production function does three things to a
subset of the production possibilities set; it (1) holds
the quantites and qualities of a subset of the inputs, the
z(K)'s, at a constant level, absolutely or on the
averages, for all the input vectors in the set, and (2)
fixes the distribution from which the random variable
inputs, the u(L)'s, are drawn, and (3) fixes the func-
tional relationship or processes, the f(....), between the
inputs and the output. Notice that this means that the
sub-production function changes if the quantities or
qualities of the fixed inputs change, or the average
quantities or qualities of the random variable inputs
change. Note that the levels of the fixed inputs are held
constant; this means as one increases variable inputs, not
necessarily in proportion, the constraining influence of a
fixed input or an average random variable input may

change.

A2.1.2.3.2. INPUT REQUIREMENT SETS:

A collection of all input vectors capable of pro-
ducing g; least some given level of y is called the input

reguirement set, V(y) [Varian, 1978].

169

(A2-5) V(y) = [(v. z) I (v. v. z) E Y and Y <= 3"]

That is, if (v, z) is in V(y) and (v', 2) >= (v, 2) then
(v', z) is in V(y), but (v, z) V(y'). Notice that this
means that a sub—production plan (v', 2) that is in V(y)
will produce y', where y' > y, and still be in y's input
requirement set. Notice too, that the 2's are fixed in
identical quantities for all the sub-production plans in
the set V(y); if the quantity of one or more z's changes,
one implicitly changes input requirement sets and sub-

production functions.

A2.1.2.3.3. ISOQUANTS:

The collection of sub-production plans that produce
exactly y are called isoquants, Q(y).
(AZ-6) Q(Y) = (V. 2) I (v. 2) V(Y) but
(V. 2) t V(Y')
where

(A2.7) y' > y

All of the input vectors (v', z) > (v, 2) such that (v',
z) e V(y) are not members of Q(y). This definition of
isoquants should make it clear that input requirement sets
are conceived to be, basically, a collection of isoquants.
That is, that the input requirement set V(y) is the iso-
quant Q(y), and all the isoquants Q(y'), where y' > y.
That means that Q(y) acts like a "frontier" in V(y),

within which all the higher valued isoquants lie. Notice

170

that this means a point can be an 'interior' point to an
isoquant and still be in the same input requirement set as
the isoquant. These 'interior' points are not on the
isoquant, because they produce y' > y, but that they are
members of the same input requirement set, V(y). It is
especially important to appreciate that these 'interior'
points are fundamentally different than those of a unit
isoquant or frontier production function (see Chapter

Two).

A2.1.2.4. DISTANCE FUNCTIONS:

McFadden uses the concept of free disposal (discussed
in a following section) to expand, modify, or generalize,
the concept of a production function to what he terms a

gistance fpnction.

The concept of a distance function
comes from the mathematical theory of
convex sets, and was introduced into
economics by Shephard (1970). While
the reformulation of duality in terms
of distance functions is potentially
useful in applications, its primary
appeal comes from the fact that it
allows us to establish a full, formal
mathematical duality between [produc-
tion] and cost functions, in the sense
that both can be thought of as drawn
from the same class of functions and
having the same properties. [McFadden,
1978, page 24]

Formally, the definition of a distance function is:

(AZ-8) F(Y.V. 2) = Max (a > 0 l (1/a * (V. 2) E V(y))

171

This is essentially an application of the implicit-
function theorem [Chiang, 1974]. One should note that
implicit-functions of mathematics do not correspond with
real world production processes. Implicit-functions are,
in this respect, similiar to reversible processes in
thermodynamics; neither can exist in reality due to the
second law.
Figure A2.1 represents a distance function.

As illustrated in [Figure A2.1], the

value of F(y,v, z) is given by the

ratio of the length of the vector (v,

z) to the length of a vector (v‘, 2)

defined by the intersection of the "y-

isoquant" and the ray through (v, z).

[McFadden, 1978, page 25; 2'8 added]
Note what this does: "a," in (A2.8), is an adjustment or
scaling factor that reduces all input bundles on the
isoquants greater than y* (in the interior of y*) to
values equal to the bundles pp the isoquant y*. Clearly,
"a" needs to be a vector, with 1's corresponding to the
2's so the 2's are unchanged in value. Only that portion
of the input bundle (v, 2) equal to (v‘, 2) remains after
scaling, where (v, z) > (v*, z); i.e., (v, z)/a = (v*, 2).
Note: this does not mean that (v, z) is getting less
output from the same inputs as (v*, 2), since both the
inputs and the corresponding output are adjusted for (v.
2). In effect, the distance function finds a scaling
factor that takes any input bundle within the 'interior'

of the lowest isoquant of an input requirement set and

moves it back to that lowest isoquant; it transforms the

 

 

172

(V.z)

(V'.z)

/
(V22)

FIGURE A2.1

A DISTANCE FUNCTION.

173

'interior' so that the 'interior' lies on its boundary.
The physical quantities of "excess" input and the cor—
responding "excess" output, are entirely and completely
removed and eliminated from consideration. It eliminates
the 'interior' for all practical purposes.

There are three facts which must be noted. (1) Dis-
tance functions are made possible, conceptually, with free
disposal. Consequently, the concept of free disposal is
very important and therefore will be dealt with in a
following section. (2) The distance function eliminates
any 'interior' points within the lowest isoquant of an
input requirement set while retaining the input require-
ment set's mathematical properties of convexity. The
convexity conditions are necessary for McFadden to develop
a concise mathematical proof to duality theory. These
conditions would also be met in a monotonic production
function, or in an analysis confined to Stage II (the
rational area) of a traditional production function, as
contrasted to a production solid. Thus free disposal
appears necessary to deal with the 'interior' points of
production solids. (3) McFadden's (1978) original formu-
lation of distance functions did not include fixed inputs,
the duality theory they are used to prove is valid only

for production sets that exclude Stage III of production.

174

A2.2. ASSUMPTIONS ABOUT PRODUCTION SETS

Theories of production often postulate three funda-
mental assumptions. (1) The first is that production is
regplar or ipput regular. The set of all possible produc-
tion plans for some output or group of outputs is non-
empty (you cannot get positive output from zero inputs),
and closed (there is some bounded and describable set of
feasible production plans). This is the standard economic
assumption that there is "no free lunch." (2) production
sets are concave functions, which means one can obtain
input requirement sets that are convex sets. Input re-
quirement sets are convex sets because the lowest isoquant
within the set is a "boundary" to all the higher iso-
quants. Isoquants are not convex sets; they are convex
functions (See Chiang, 1974). The distinction is critical
since it is within the context of convex sets rather than
convex functions that one may provide a proof of duality
[McFadden, 1978]. In practice, this assumption is always
further restricted so that input requirement sets are
strictly convex sets and isoquants are strict convex
functions. This assures that the isoquants are "well be-
haved" (no flat spots). This means, in effect, that
perfect substitutes and perfect complements are ruled out
or are combined into single inputs. Additionally,
Malinvaud (1972) points out that convexity implies that
the bundles of resources used to produce any given level

of output display conditions of additivity and divisibili-

175

ty. This means changes in input quantites can be con—
sidered in infinitesimal amounts. This in turn suggests
that sub-production plans would necessarily display con-
stant returns to scale except for the presence of some
fixed resource [Malinvaud, 1972]. This is why the con-
sequences of fixed inputs in production is very important
and will be dealt with in the following section. (3)
Production displays ponotonicity among the variable in-
puts; that is, additional variable inputs will yield addi-
tional output(s) within the constraint of the fixed in—
put(s). This assumption assures that isoquants will not
converge. This means points of equi-production are on the
same isoquant. This assumption also implies free disposal
among the variable inputs.

While many set theoretically inclined microeconomic
theorists note that only the assumption that production is
regular is essential for most of the important results of
microeconomic theory, concavity (assumption two) and mono-
tonicity (assumption three) are critical to the formal
proofs of gpglity theory, which receives major emphasis in
the more abstract forms of production theory currently
taught in general economics departments (for example see
Varian 1978).

Duality simply means that every point in production
space (excluding Stage III) is associated, by a one to one
mapping, with a corresponding point in cost space (polar
reciprocal sets). Since value and technical relationships

are inseparable, this one to one correspondence is intui-

176

titively obvious. This means the apparent difference
between physical quantity and price is nothing more than a
mathematical transformation, a difference in the units of
measure. Input requirement sets that conform to these
three assumptions are called input-conventional input re-

guirement sets.

A2.2.1. CONSEQUENCES OF FIXED INPUTS:

Fixed inputs play a very important role since they
act as constraints on the amount of total product one can
achieve from adding more variable inputs. They determine
the sub-production function one is using. Thus, the fixed
inputs determine which technical relationships will exist
among the factors of production and output, i.e., which
sub-production function is relevant and, hence, the mar-
ginal physical products (MPP) for the various variable
inputs.

One cannot ignore fixed inputs, especially in empir-
ical work, because without some input fixed in the produc-
tion processes, constant returns to scale would apply and
one could continuously increase (variable) inputs in fixed
proportions and obtain proportionate increases in physical
output. In particular, the law of diminishing returns (or
law of variable proportions) would not apply and marginal
physical products (if not marginal value products) would
always be positive and constant. The second partial deri-

vatives of the production function are negative in all of

177

Stage II and part of Stage I, which means that marginal
physical products are decreasing, giygp that all the other
inputs become fixed. The absence of the law of dimin-
ishing returns, and marginal physical products that can be
negative, result in an incomplete production function, a
production function without Stage I and III, so that pro—
duction can never reach a maximum. The three topics of
constant returns to scale, the law of diminishing returns
(or law of variable proportions), and the stages of pro-
duction, will be reviewed in turn, demonstrating how they

are important to a definition of efficiency.

A2.2L1.1. CONSTANT RETURNS TO SCALE:

Constant returns to scale means that there will be
proportionate increases in output resulting from propor-
tionate increases in all the inputs. Therefore, in
theory, constant returns to scale will exist when there
are no fixed inputs. The relationship between height,
width, volume and mass may also make strict constant
returns to scale physically impossible in reality. Though
this relationship does not fix any inputs, but it does
prevent expansion of all inputs in constant proportions.

Under constant returns to scale (no fixed inputs),
and with prices constant, production will never reach a
maximum, so all production levels are equally efficient.
One can discuss differences in efficiency only within one
of two sets of circumstances; (1) where output is prede-

termined, or fixed, or (2) where the total amount of

178

resources available, the budget constraint, is prede-
termined, or fixed. In the first case, the level and
combinations of variable inputs that achieve minimum cost
for producing the given level of output are efficient. In
the second case, the set of given resources determines the
input requirement set, and the maximum output possible.
With constant returns to scale there is only one sub—
production function, since all inputs are variable. Gen-
erally, one sub—production function may be less efficient
than another, i.e., unable to produce as many units of
value per unit of value consumed in the production proces-
ses as another sub—production function. When one con-
siders changing sub-production functions, one is asking
whether or not is would pay to vary some hitherto fixed
input. The economics of investment and disinvestment was
developed by Edwards (1958). In order to understand why
one sub-production function might be more efficient than
another, one must understand the law of diminishing

returns.

A2.2.lg2. THE LAW OF DIMINISHING RETURNS:

The law of diminishing returns (or law of variable
proportions) states that as one adds to production succes-
sively more equal units of a given variable input while
holding one or more other inputs fixed, the marginal
physical product of the variable input first increases at

an increasing rate, then increases at a decreasing rate,

179

then decreases absolutely. The law of diminishing returns
presumes the existence of fixed inputs. In the case
above, under constant returns to scale, efficiency was
determinable only after something other than inputs became
fixed, because otherwise marginal physical products
remained constant and equal to average physical product.
The laws of diminishing returns, and/or diminishing utili—
ty, influence efficiency since the former affect the mar-
ginal physical products and the latter the value of the
product.

Because of the laws of thermodynamics (see Appendix
One) if one has identical processes using identical vari-
able inputs and the identical quantity of identical fixed
inputs, the marginal physical products will be identical
for every identical marginal change in the variable in—
puts. Consequently, if one observes two production pro-
cesses using identical quantities of identical variable
inputs to achieve different quantities of identical out—
put(s), then the marginal physical products for some in-
put(s) must be different. This can only be true if the
amount of fixed input(s) is different for the two sets of
processes. That is, the sub-production, or processes,
with the lower level of fixed input(s) constrains the
marginal physical product for some variable input(s) and

thus constrains output.

180

A2.2.1.3. STAGE III OF PRODUCTION:

One last consequence to the inclusion of fixed inputs
in sub-production function is Stage III, a region where
product declines after reaching a maximum. Stage III is
particularly important since many contemporary theorist
maintain:

A production plan y in Y is called

efficient if there is no y' in Y such

that y' >= y; that is, a production

plan is efficient if there is no way to

produce more [product] with the same

inputs or to produce the same output

with less inputs. [Varian, 1978, page
One can produce more useful product with less input if one
is in Stage III, i.e., by moving from point D to point B
in Figure A2.2. but staying on the same isoquant. The
monotonicity assumption means that more variable input
always produces more output, but in Stage III the addi-
tional output is waste or pollutant. For example, water
is a necessary input to crop production but too much water
results in reduced product. What happens in Stage III is
that the additional variable input(s) begins to block the
ability of some other input to contribute to the produc-
tion process thereby reducing useful output. When too
much water is added to crops, the roots are unable to get
as much oxygen as previously and this reduces useful
output. This is anexample of the law of diminishing

returns since one input is blocking the other(s) from

contributing, e.g., due to the lack of space for both to

 

181

 

FIGURE A2.2

ISOQUANT SHOWING STAGE III

7sz z

182

operate effectively. One should note that this is not a
change in sub—production function since the levels of the
fixed inputs remains constant. Ignoring Stage III means
the production possibilities set is incomplete; that it
does not contain all the feasible production plans.

In production theory, Stage III is often precluded or
avoided since it is outside the range of "rational" pro—
duction. In Figure A2.2, ABCDA is the full isoquant for
some level of production. Usually, only ABC is considered
rational since the portion ADC is in Stage III. If one
where to observe a point D, one could conclude that one
could achieve the same level of output using less of the
same variable inputs, and conclude that point D was a
point of "technical inefficiency;" that point D was in the
interior of the isoquant ABC. Clearly, D is not in the
interior of the isoquant but is on the isoquant and is
inefficient for exactly the reason that Stage III is
outside the range of rational production; net value of the

output is not at a maximum.

A2.2.2. CONSEQUENCES OF MONOTONICITY:

A2.2.2.1. FREE DISPOSAL:

Free disposal arises as a consequence of the as-
sumption of monotonicity [Varian, 1978]. Monotonicity
implies that if a set of resources (A) is greater than
another similiar set of resources (B) then the former set

(A) can produce pp least as much output(s) as the latter

183

set(B).

The idea is clear: if we can
produce y with a certain input bundle
v, we should be able to produce y if we
have more of everything. This is some-
times referred to as the hypothesis of
"free disposal." For if we can always
costlessly dispose of anything we don't
want, our technology must certainly
satisfy the monotonicity assumption.
[Varian, 1978, page 6]
The "anything we don't want" is both input and output.
Due to the laws of thermodynamics and the one to one
relationship between input and output, it is especially
important to remember that despite the fact that input
requirement sets are defined in terms of inequalities, in
reality one is dealing with sets of equalities. Free
disposal is the means of reconciling the appearance of
producing exactly y with the input bundle (v', z) in-
cluded in V(y), when (v',z) in fact produces exactly y', y
< y'. The input requirement set for a particular isoquant
includes the input bundles for that isoquant and all the
input bundles for all the isoquants at higher levels of
production. The input requirement sets for higher levels
of production are proper subsets of the input requirement
sets of lower levels of production [McFadden, 1978]. It
means that all of the isoquants "within" a given isoquant
y are within the input requirement set for that given
isoquant y. Free disposal does not require that the pro-

duction function is a solid instead of the surface one

ordinarily associates with a production function. The

184

concept of free disposal does not mean that an isoquant
has interior points that are at the same level of output.
Suggesting this is a misunderstanding of what constitutes
free disposal, i.e. a violation of monotonicity.

The notion of free disposal is often misunderstood
which is to be expected as it has no counterpart in the
real world. The nearest exception to this is approached
as two inputs approach perfect complemenatary. McFadden
(1978) suggests that free disposal is essentially a gim-
mick to provide the conditions for the derivative condi-
tions necessary to provide a rigorous mathematical treat—
ment of the theory.

However, the importance of [free
disposal] in traditional production
analysis lies in [its] analytical con-
venience rather than in [its] economic
realism; [it] provide[s] the groundwork
for application of calculus tools to
the firm's cost minimization problem.
[McFadden, 1978, pages 8 & 9]

Free disposal means one may "throw away' commodities
without using up inputs in the disposal process [Pachico,
1980]." This does not mean using up additional inputs,
instead it means that free disposal removes some amount of
input, and its counterpart in output, from production.
The freely disposed input/output is in no sense part of
the production processes. When the additional output y' -
y is freely disposed, the additional resources (v', z) —

(v, z) are also disposed. One must remember that inputs

cause output by being consumed in the processes. Freely

185

dispOsed inputs are not consumed, which is what Varian
(1978) means by "costlessly disposed." Because an input
is never consumed it is never paid for nor does it have an
opportunity cost. Proper accounting in production will
only record those inputs consumed in production.

Free disposal means that one can move from a higher
isoquant to a lower isoquant githout cost, by the free
disposal of the additional output. It also means that
'interior' points of production functions can be swept
out. The inputs that created the output thrown are also
treated as costless in polar reciprocal sets. In the
definition or a distance function, the scaling factor, Q,
is the mechanical means of performing "free disposal."
This scaling factor is needed in Figure A2.1 to transform
(v, 2) so that (v, 2) will produce exactly y*, and not y,
where y > y*, in keeping with the laws of thermodynamics.
Those excess amounts of the inputs in the input bundle (v,
z) are freely disposed, otherwise they would create output
in excess of y‘ if used in production. The output is
reduced precisely because the quantities of inputs used

are reduced.
A2.2.3. DUALITY THEORY AND POLAR RECIPROCAL SETS:

The duality theory proved by McFadden (1978) excludes
Stage III of production since he assumes that all marginal
physical products must be non—negative. Duality means

that all the points in a input-conventional input require—

186

ment set have a unique correspondence to points in the
associated cost space. Therefore, in duality theory the
input-conventional input requirement sets of distance
functions have an analogous counterpart in cost functions,
called factoreprice reqpirement sets, R(y) [McFadden,
1978]. Factor-price requirement sets are defined as:

(A2.9) R(y) = [r >= 0 | r * (v, 2) >= F(y, v, z)

for all positive (v, 2)]

where

r = VECTOR OF PRICES FOR ALL INPUTS

This means that the prices in the factor-price requirement
set satisfy the condition that when multiplied by the
input bundles in the corresponding input requirement set,
the inner product is at least as large as the value of the
relevant distance function F(y, v, 2), which was defined
earlier. This property means that that not only do
input-conventional input requirement sets and factor-price
requirement sets have a unique one-to-one mapping from one
set into the other, but pppp pisplay the property pf gppp
disposal. The logical consequence of this is that for any
point within the "interior" of an isoquant, there is a
mapping of this point into the "interior" of a isocost;
i.e., if one is on a higher isoquant, one is producing at
a greater cost. Inputs have an opportunity cost. If by
invoking free disposal one ignores the additional output
created by additional inputs in production space, then ppp

must necessarily ignore the additional cost for those

187

inputs in cost space. If one can move from a higher
isoquant to a lower isoquant by free disposal, then one ip
necessarily moving from a higher isocost to a lower iso-

cost in cost space, by the same free disposal.
A2.3. MAXIMIZING BEHAVIOR:

Maximizing behavior defines efficient behavior. It
involves both technical (physical) and price (value)
information. The most efficient production is seldom the
maximum average production. Efficiency deals with the
question of relative costs; inputs are used efficiently
when they are used in least cost combination. This is
obvious from the decision rule equating marginal cost with
marginal revenue. This is identical to equating the
ratios of marginal physical products to the ratios of the
respective prices for all the inputs, where the marginal
physical products represent the technical aspect of
production and the prices represent the opportunity cost
aspect. Some of the prices are internal opportunity costs
for unspecialized inputs or "shadow prices" for spe-
cialized fixed inputs [Edwards, 1958]. Notice that this
is identical to the definition of efficiency used in
thermodynamics, i.e, the ratio of useful output to costly

input [Dixon, 1975].

(AZ-1°) MPPIV1I/P(V1) = MPP(v2)/P(v2) =

= "P? (Vn) /P (vn)

Take the case of two inputs with different marginal phys-

188

ical products. One input is necessarily being used in-
efficiently when compared to the other only if the two
inputs have the same price or opportunity cost. One cannot
allocate resources using marginal physical products alone
or prices alone. If technology could be separated from
value, it would seem reasonable to expect that there would
be a rule for maximizing efficiency by equating marginal
physical products without reference to their values and
vice versa. This would suggest that one could allocate
resources solely on technical or price criteria. What
those criteria might be is unclear.
A2.4. WHAT IS WRONG WITH FRONTIER PRODUCTION FUNCTIONS
FROM THE PERSPECTIVE OF ECONOMIC THEORY

First, the analysis of frontier production functions
deal with averages. Recall Figure 2.1. The unit isoquant
represents average input per unit putput. Bressler (1966)
notes that the price line represents "average cost." Both
technical and price efficiency are found by using average
input and average cost rather than marginal input and mar-
ginal cost. This surely conflicts with conventional pro-
duction theory wherein one equates marginal cost to mar-
ginal revenue to find the most efficient point of produc-
tion. Indeed, only in the context of perfect competition
is it true that average revenue equals marginal revenue.
Only in the case of constant returns to scale and fixed
input prices is it true that average cost equals marginal

cost over the whole range of output. Consequently, the

189

only case imaginable where it might be legitimate to
define efficiency in terms of averages would be for con-
stant returns to scale under perfect competition, a very
restrictive and unrealistic case.

Even in the special case of constant returns to scale
in perfectly competitive equilibrium, a firm P would not
be within the interior of the unit isoquant. Firm P uses
more of the variable inputs given the same amount of fixed
input than a firm on the unit isoquant. The assumption of
monotonicity means there must be more output. Constant
returns to scale means that for additional input there is
proportionate increases in output. So, in terms of the
output per unit input, P must lie on the unit isoquant, or
violate constant returns to scale, or use a different
amount of the fixed input. Recall that with any fixed
input, constant returns to scale only exists only within
an infinitely small neighborhood around the point on the
production function where average product equals marginal
product for all inputs.

Assume for a moment that one observes a firm Q and a
firm P both using the same identical sub-production
function. If P is using more variable inputs then it is
in fact producing more output. If P is getting less
output per unit input than Q, it is because P is on a
higher isoquant where the marginal physical products for
the variable input are smaller due to the law of dimin—

ishing returns. In this case, the higher output of P is

190

obscured by the averaging process in finding the unit
isoquant. It is also true that P may be inefficient. P
might be a dairy farmer who continues to feed his cows
beyond the point where the return for the additional milk
produced is greater than or equal to the cost of the
additional feed. The inefficiency is due to value as well
as physical considerations.

If P is producing more output than Q, can one invoke
free disposal to make a comparison of the two firms, and
the inputs they use to achieve the same output, and con-
clude that P is less efficient? No, because if one
ignores the additional output then by duality one must
ignore the cost of the additional output. By duality,
free disposal in production space must be associated with
free disposal in cost space. Therefore, if output is
freely disposed in production space, the cost for the
inputs that produced that output must be freely disposed
in cost space, which makes those inputs free goods, and
economically irrelevant. If one ignores the cost of the
additional output, then by duality one must ignore the
additional inputs. That is, if one includes inputs in the
production accounting, then in order to make the produc-
tion "ledger" balance one must also take account of the
output produced by those inputs. Conversely, if the out—
put is ignored by free disposal, then the corresponding
input must also be ignored, or freely disposed. The
inputs are irrelevant technically and economically because

one has freely disposed of their output. In order to

191

maintain the 1:1 ratio of energy input to energy output
demanded by the law of thermodynamics, if one erases an
amount from one side of the equation (output), one must
erase an equal amount from the other side of the equation
(input). If one were to maintain that in freely disposing
of the output one were converting it from work output to
waste output, then one is implicitly changing sub-produc—
tion functions, or processes, ceteris paribus.
Finally, the quotation:

But [P] uses more of both inputs than

[Q] to produce the same level of out-

put. [Timmer, 1971, page 777]
exemplifies an error repeatedly asserted in the frontier
production function literature; namely that Q produces the
same amount of output as P with fewer inputs. That P
might actually be producing more output, which is freely
disposed, has been discussed above. Suppose that P and Q
are in fact producing the same amount of the same output.
Figure 2.1 suggests that Q uses fewer inputs. This indi-
cates an inherent indexing problem since in this case the
23 for Q is not the same as the 23 for P. That is, Q uses
fewer variable inputs, but can only get more output at the
margin with them if Q has pppp gippp input, due to the
laws of thermodynamics and the law of diminishing returns
to scale. One can compare the "size" of the two input
bundles by evaluating them with respect to their oppor-

tunity costs. This would indicate that economic efficien-

192

cy is identical to "price" efficiency, and that the fron-
tier production function distinction between "price" effi-

ciency and "technical" efficiency is meaningless.

A2.5. SUMMARY

In economics, as in thermodynamics, it is impossible
to separate the physical aspect of production from the
price aspect of production. Efficiency is defined with
respect to the relationship between the two; i.e., effi-
ciency means equating marginal value products with their
respective opportunity costs, or useful output to costly
input.

In production theory the role that fixed inputs play
in determining the technical relationships between all the
inputs in the production process is critical. Indeed, the
level of fixed input determines the sub-production
function. Because some inputs are fixed, the law of
diminishing returns operates, leading to variable returns
for different levels of input, both variable input and
fixed input. Constant returns to scale is a special case
where no input is fixed, and where, consequently, there is
nothing endogeneous to the production system that affects
efficiency unless prices become functions of quantities.
Stage III may result from fixing input and means that MPP
>=< 0. This implies that efficiency is associated with
the location on the isoquant at which one is producing.
It also raises the question of whether or not isoquants

can have "interiors."

193

Free disposal deals with the issue of the 'interior'
to an isoquant. Free disposal is a necessary in order to
understand that input requirement sets are a collection of
upper level isoquants for "one" level of production:
Clearly a paradox. Free disposal is often not clearly
understood, since it has no observable counterpart in the
real world. It serves only to reconcile the paradox that
an input requirement set implies being on and off the
isoquant at the same time. This reconciliation allows one
to define a distance function and give a rigorous mathema-
tical proof to duality (excluding Stage III). Duality is
demonstrated by polar reciprocal sets. Duality means that
the physical aspects of production, marginal physical
products, are inseparable from the value aspects of pro-
duction, prices, through their one to one mapping from
input space (excluding Stage III) to cost space.

Since the physical aspect of production is insepar-
able from the price aspect of production, the notion of
"technical inefficiency," as used in frontier production
functions, has no logical basis. Since an isoquant does
not have a pppp interior at the same level of ouput there
is no theoretical basis for the definition of "TE" as it
is used in the frontier production function literature.
The definition of a frontier production function is a

violation of the tenets of microeconmic theory.

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