.4 '-..

0-169

Date

This is to certify that the

thesis entitled

presented by

iobort Francis Allen

has been accepted towards fulfillment
of the requirements for

i‘ If? . Ll . degree in :C ill-I'Q'Hif’ffi

 

 

January 6, 1969

 

 

 

 

AN EVALUATION OF THE "CAMBRIDGE
CRITICISM" OF NEOCLASSICAL THEORY

By

Robert Francis Allen

AN ABSTRACT OF A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Economics

1969

 

 

ABSTRACT

AN EVALUATION OF THE "CAMBRIDGE CRITICISM"
OF NEOCLASSICAL THEORY

BY

Robert Francis Allen

The chief characteristic of the neoclassical,
real-capital model is the assumption that there exists a
function Q = F(K,L) that is homogeneous of degree one in
the single homogeneous capital good (K) and a single
homogeneous labor force (L). This implies (a) that the
real wage varies inversely with the interest rate, and
(b) that the real capital—labor ratio varies directly with
the wage-interest ratio. Hence, an increase in the real
wage leads to an increase in the capital intensity of tech—
niques, and permanently sustainable net national product
increases as the rate of interest declines.

Recently, Pasinetti, Morishima, Bruno-Burmeister-
Sheshinski and others proved it is possible for a tech-
nique that is the most profitable of all feasible techniques
at some relatively high wage rate to also be the most
profitable at some relatively low wage rate. The technique
"comes back" as the wage falls monotonically toward zero,
with the consequence that an unambiguous ordering of
techniques is impossible. This phenomenon (labeled the
reswitching of techniques) is said to establish the simple
technological fact that there may exist a "set of blue—

prints" for which there is no one—to—one correspondence

 

Robert Francis Allen

between the real capital—labor ratio and the wage-interest
ratio. If so, it is not generally true that a fall in the
interest rate entails the adoption of more capital-intensive
techniques with the consequence that the permanently sus-
tainable consumption stream is greater.

With minor exceptions, recent writers have employed
two-sector, fixed-proportions, heterogeneous capital models
in which the labor—output ratio of one of the sectors is
used as an independent index of capital intensity (i.e., as
the aggregate capital-labor ratio). This study analyzes
several real-capital models and concludes that the use of
the labor-output ratio as an independent index of capital
intensity is definitely unwarranted.

The basic error implicit in the use of this index is
the assumption that a technique of production represents a
unique aggregate capital-labor ratio. This will not
generally be the case. Changes in the wage-interest ratio
imply changes in relative commodity price. This in turn
implies changes in the composition of demand between
sectors. When factor intensities differ between sectors,
the assumption of full employment (common to all these
discussions) implies that the economy's endowment of
capital and/or labor is a variable within the context of a
given technique.

To allow for changes in the composition of demand,
this study introduces a simple demand relation into the

reswitching model. More specifically, in a two good,

 

Robert Francis Allen

heterogeneous capital model, the demand relation is %;= f(p)
Where M is a capital good, 0 a consumption good, and p the
relative price of the capital good in terms of the consump-
tion good. The comparative static properties of the model
are investigated and the implications of changing relative
commodity price for the reswitching phenomenon set out.

Two fundamental points emerge from this analysis:
first, the real capital embodied in a technique is a
decreasing function of the rate of interest, and second, a
switch of techniques exerts an independent influence on
relative commodity price.

The first point implies that for any two techniques
there will generally be some critical rate of interest for
which the aggregate capital-labor ratio is the same for
both techniques. Thus whether reswitching means that the
aggregate capital—labor ratio has reversed itself or not
depends first of all on whether the economy has passed the
critical interest rate before reswitching occurs.

The second point means that, at any specified wage-
interest ratio, relative commodity price may fall, remain
unchanged, or rise g_,g result 2£_entrepreneurs' switchigg
techniques. When the switch itself causes relative com-
modity price to fall, it is virtually impossible for
reswitching to destroy the neoclassical link between the
aggregate capital-labor ratio and the wage-interest ratio.

This study concludes that when the interrelations

between factor prices, factor endowments, and relative

 

3 Robert Francis Allen

commodity price are fully set out, the Cambridge criticism
does not appear as serious as recent discussions would have

one believe.

 

AN EVALUATION OF THE "CAMBRIDGE

CRITICISM" OF NEOCLASSICAL THEORY

By
Robert Francis Allen

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Economics

1969

 

ACKNOWLEDGMENTS

Upon completion of a project such as this, one's
thoughts naturally turn to those individuals who, by virtue
of their encouragement and guidance, are largely responsible
for whatever merit others may find in it. The individuals
in question in this instance are Professors Charles E.
Ferguson, Victor E. Smith, Byron W. Brown, John P. Hender-
son, and my wife Merna.

My greatest debt is to my thesis supervisor
Charles E. Ferguson. It was Professor Ferguson who sug—
gested that this study be undertaken, and it was his
encouragement and guidance that shaped my determination to
pursue what at times seemed to me a very illusive idea.

For his brief but pregnant comments at critical points
during the course of this study I am especially grateful.

I wish to thank Professors Smith and Brown of my guidance
committee for their detailed comments on a late draft of the
manuscript. The end product has benefited materially from
their efforts. Final responsibility for the content of this
study is my own.

I wish to thank Professor John P. Henderson, Director
of Graduate Programs in Economics, who on numerous occasions
has helped me with a difficult decision or problem, not only
while this thesis was in preparation, but throughout my

ii

 

 

three year residence at Michigan State University. And
last, but not least, I wish to thank my wife Merna who gave
generously of her time and patience to type several drafts
of the manuscript and check for accuracy and consistency in
matters of style. Most important has been her understanding
and encouragement without which this study could not have

been completed.

iii

 

 

l— , Jami, ,

 

TABLE OF CONTENTS

CHAPTER PAGE
I. NEOCLASSICAL THEORY AND THE "CAMBRIDGE
CRITICISM" . . . . . . . . . . . . . . . . . . 1
Introduction . . . . . . . . . . . . . . . . . 1
Simple Neoclassical Theory . . . . . . . . . . 5
The Cambridge Criticism . . . . . . . . . . . 12
II. SWITCHES OF TECHNIQUES AND STEADY—STATE

CONSUMPTION LEVELS . . . . . . . . . . . . . . 27
Introduction . . . . . . . . . . . . . . . . . 27
Levhari's Nonswitching Theorem . . . . . . . 28

Capital Intensity in a Heterogeneous
Real Capital Model . . . . . . . . . . . . 3O
Switches of Techniques in a Two-Sector
Heterogeneous Real Capital Model . . . . . . 3h
The Factor-Price Frontier . . . . . . . . . 35
Reswitching on the Grand Factor—Price
Frontier . . . . . . . . . . . . . . . . . hl
Steady-State Consumption Levels . . . . . . . R9
The Traditional Ordering of Consumption and
the Interest Rate . . . . . . . . . . . . so
The Consumption Plateau in a Hetero-

geneous Real Capital Model . . . . . . . . 56

iv

 

CHAPTER H _ U PAGE
. Unorthodox Behavior of Steady-State

Consumption . . . . . . . . . . . . . . . 60

Conclusion . . . . . . . . . . . . . . . . . . 66

III. CERTAINTIES AND AMBIGUITIES IN THE

RESWITCHING DISCUSSIONS . . . . . . . . . . . 69
Introduction . . . . . . . . . . . . . . . . . 69
Samuelson's "Parable" . . . . . . . . . . . . 71
Hicks' Two Rules . . . . . . . . . . . . . . . 78

The Implications of a Nonlinear Process

 

Frontier . . . . . . . . . . . . . . . . . . 86

IV. INTERRELATIONS BETWEEN FACTOR PRICES, FACTOR
ENDOWMENTS, AND RELATIVE COMMODITY PRICE . . . 96
Introduction . . . . . . . . . . . . . . . . . 96
The Model: Variable Coefficients . . . . . . . 98
The Model: Fixed Coefficients . . . . . . . . 109

Reswitching and the Aggregate

Capital~Labor Ratio . . . . . . . . . . . . 120
V. SUMMARY AND CONCLUSIONS . . . . . . . . . . . . 139
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . 150

TABLE

I.

II.

III.

IV.

LIST OF TABLES

Aggregate Capital-Labor Ratios for Two
Techniques . . . . . . . . . . . .
Computed Values of Relative Commodity Price
for Technique Alphal (pd ) and Technique
Betal (p9) . . . . . . . . . . . . . . . .
Computed Values of Relative Commodity Price
for Technique Alphaz (p“) and Technique
Beta2 (p9) . . . . . . . . . . . . . . . .
Computed Values of Relative Commodity Price
for Technique Alpha3 (p“) and Technique
Beta3 (p!) . . . . . . . . . . . . . . . . .

vi

PAGE
93

. 117
. 118
119

 

 

LIST OF FIGURES

FIGURE PAGE
1. The Surrogate Factor-Price Frontier . . . . . . . 9
2. Joan Robinson's Productivity Curve . . . . . . . 19
3. Effect of an Increase in the Real Wage on the

Capital Intensity of Techniques . . . . . . . . 21

 

h. The Factor-Price Frontier . . . . . . . . . . . . 37
5. The Grand Factor-Price Frontier . . . . . . . . . 39
6. Reswitching of Techniques . . . . . . . . . . . . AZ
7. Reswitching and Descartes' Rule of Signs . . . . R6
8. Steady-State Consumption and the Rate of
Interest . . . . . . . . . . . . . . . . . . . 53
9. The Surrogate Frontier and Steady-State
Consumption . . . . . . . . . . . . . . . . . . 55
10. The Factoerrice Frontier and Steady-State
Consumption . . . . . . . . . . . . . . . . . . 58
ll. Reswitching and Steady~State Consumption . . . . 61
12. "Perverse" Ordering of Steady-State Consumption . 62
130 Conventional Ordering of Steady-State

Consumption . . . . . . . . . . . . . . . . . . 65
1h. Samuelson's Special Capital Model . . . . . . . . 73
15. A Switch of Techniques in Samuelson's Model . . . 75

16. An Equal-Proportions Production Capital Model . . 80

vii

FIGURE - PAGE
17. Switches of Techniques in an Equal-Proportions

Production Model . . . . . . . . . . . . . . . 8h
18. A Nonlinear Capital Model . . . . . . . . . . . . 89
19. A Switch of Techniques in a Nonlinear Model . . . 92
20. The Nonlinear Factor-Price Frontier . . . . . . . 112
21. Effect of Switches of Techniques on Relative

Commodity Price . . . . . . . . . . . . . . . . 121
22. Reswitching of Techniques . . . . . . . . . . . . 123
23. Effect of Reswitching on Capital Intensity:

 

Relative Commodity Price Constant . . . . . . . 127
2h. Effect of Reswitching on Capital Intensity:

Relative Commodity Price Falling . . . . . . . 130
25. Effect of Reswitching on Capital Intensity:

Relative Commodity Price Rising . . . . . . . . 132

viii

 

 

CHAPTER I
NEOCLASSICAL THEORY AND THE "CAMBRIDGE CRITICISM"
I. Introduction

Capital theory has an ancient, if not entirely
honorable, heritage. I think it is fair to say, however,
that serious theorizing began with Bohm-Bawerk,l who formu-
lated the period of production model and grappled with the
concept of an "average period of production". He was
followed in this endeavor by Wicksell who, it should be
noted, finally abandoned the notion of an average period of
production.2 Nonetheless, the "average period" was crucial
to their theory.

Bohm-Bawerk regarded capital as a "subsistence fund"
or what in classical economic literature was called a wages
fund. Its function was to support labor during the interval
of time that must elapse between the first application of
inputs and the later emergence of output. With a given

amount of labor, the longer the production period the more

 

lEugen von Bohm-Bawerk, The Positive Theory 2;
Ca ital, trans. William.Smart (London: MacMillan and
Company, 1891).

2Knut Wicksell, Lectures 9;; Political Econom ,
volfi)I, trans. E. Classen (New York: The Macmillan Company,
193 .

 

 

2
capital per man that would be required. Conversely, the
more capital that is made available the longer could be the
production period for any given amount of labor. If we
think of the "subsistence fund" as allowing the "roundabout"
use of labor, we may say that a longer period of production
represents a more "roundabout" or capital-intensive method
of production.

The basic idea to be conveyed by the "average period"
is that for a given labor force any increase in the period
of production means that the ratio of real capital to real
labor has risen. In other words, the "average period" is
to be taken as an independent index of the capital intensi-
ty of production, and this is What makes the concept of the
"average period" so important.3

Thus the average period of production is really a

measure of the quantity of real capital available in the

 

3On the interpretation of the "average period" we

rely on Kaldor, though the question is an open one. See
for example Nicholas Kaldor, "On The Theory Of Capital: A
Rejoinder To Professor Knight, " Econometrica, VI (April,
1938), pp. 163 176. ‘ Kaldor has argued that Whatever
validity attaches to the "average period" is due to its
ability to serve as such an index. p. 169. He further
takes as a premise of Austrian theory the idea that

. . . with the aid of the concept of the 'investment
period', the heterogeneous mass of capital goods can be
reduced to homogeneity, and thus 'capital' can be treated
as a quantity per se." p. 163. Hicks has taken essentially
the same position as Kaldor but attempts to adjust the
concept for Knight's objections about the consistency of
the index. John R. Hicks, Value And Ca ital (2d ed.;
OXford: Clarendon Press, 19W 2 , pp. “213- -226. On the other
hand see Joseph A. Schumpeter, History of Economic Analysis,
ed. Elizabeth B. Schumpeter (New York: Oxford University
Press, 1963),p pp 903 909. but especially p. 901, ft. 26,
where Mr. Kaldor is chastised.

 

3

economy. It is the means by Which we are able to reduce the
heterogenous "subsistence fund" to a homogeneous quantity
per se. The "average period" can then be combined with a
homogeneous labor factor and a theory of production formu~
lated. The most important relations that can be shown to
hold in this framework are those between the rate of inter-
est, the real wage rate, and the length of the period of
production. In particular it can be shown that every fall
in the rate of interest leads to more capital intensive
production." This is a result of great importance for it
is upon it that the most basic propositions of modern neo-
classical capital theory are made to stand.5

Yet, as everyone knows, the Austrian view of the
production process is not the one that has gained favor for
purposes of enunciating a theory of production and distribu-
tion. This is presumably due to the difficulties inherent
in the measurement of the "average period".6 Consequently
our understanding of much of the real economic world has

come to be based on what Professor Samuelson calls the

 

"Robert Dorfman, "A Graphical Exposition of Bohm-
Bawerk's Interest Theory," Review 2: Economic Studies,
XXVI (February, 1959). pp. 153-38.

5The truth of this statement is very carefully and
completely brought out in Chapter II of the present paper.

6As Kaldor has pointed out " . . . the question of
whether the concept has meaning should be kept rigidly
separate from the question whether it is relevant." Kaldor,
pp, 233,, p. 16h, the italics in original. He has shown
its relevance.

 

 

 

h
"J. B. Clark Neoclassical fairy tale".7 This fairy tale

consists of a set of propositions that may be rigorously
derived from the assumption that there exists a function
(1) Q = F(K,L)

that is homogeneous of degree one in a single homogeneous
capital good (K) and a single homogeneous real labor force
(L).

The assumption that such a function exists is the
chief characteristic of the neoclassical, real—capital
model. The relations between the rate of interest, the real
wage rate, and the capital intensity of production that can
be shown to hold in this framework are identical to those of
the Austrian model. It may therefore seem to be a matter of
indifference that the production function with its J. B.
Clark capital has been generally preferred to the Austrian
model with its Bohm-Bawerkian "average period" for purposes
of passing on this body of theory. Yet the difference
between them does appear substantial. The Austrian model
views the process of production as a process that takes
place over Eigg, It is the element 3: £igthhat is the

single most important characteristic of the Austrian model.8

 

7Paul A. Samuelson, "Parable and Realism in Capital
Theory: The Surrogate Production Function," Review 9;
Economic Studies, XXIX (June, 1962), pp. 193‘207.

8"From the technical point of view, the essentially
new contribution of Bohm-Bawerk's system to economic theory
is the mastery of the element of time." Joseph A. Schum-
peter, "Schumpeter on Bohm-Bawerk," Th2 Develo ment 2;.
Economic Thought, editor, Henry W. Spiegel (New'York: John
Wiley and Sons, 196A), p. 376.

 

 

 

S

The real capital model pretends that time does not matter.
.This difference in the-models is actually quite critical and
has in fact been responsible for the emergence of what is
now called the "Cambridge Criticism" of neoclassical
theory.9

Recently there has appeared a symposium10 devoted to
the Cambridge criticism. The upshot of the symposium is
this: what the critics have been trying to say is both
correct and important. It is important because it shows
that the parables that collectively make up simple neo-
classical theory are £23 generally valid.

This may be a very difficult pill to swallow. _Yet as
Professor Samuelson says "we must respect, and appraise, the
facts of life."11 As a first step in this direction let us
set out completely the theory in question and the essential
nature of the Cambridge criticism. Doing so will bring out
the truth of the above statement concerning the element of

time in production.
II. Simple Neoclassical Theory

One of the more famous analogies to be found in the

history of economic analysis is contained in the writings

 

9Section two below demonstrates that it is Mrs.
Robinson's attempt to resurrect BohmmBawerk's view of pro~
duction that leads to the criticism.

lo"Paradoxes in Capital Theory: A Symposium," The
puarterl Journal of Economics, LXXX (November, 1966),
.303 $3

11Paul A. Samuelson, "A Summing Up," The uarterl
Journal of Economics, LXXX (November, 1966),p $83.

 

 

6
of J. B. Clark. For Clark capital is like a waterfall and

12 The

capital goods like its individual drops of water.
individual drops of water may come and go but the waterfall
will always be the waterfall. The interpretation that has
been placed on this by the weight of authority is that there
is a £231 substance called capital Whose depletion is con-
tinuously replaced so that the substance itself remains
homogeneous.l3

In a somewhat similar fashion Clark drew a distinc-
tion between Pure Labor and individual laborers.l(-L With
the aid of this real homogeneous capital and labor, Clark
developed his central thesis that in a static state there
is a single, natural, and universal law that governs the

distribution of the product between wages and interest.
This is, of course, the law of marginal productivity.15

 

12John B. Clark, _T_h__e Distribution o_f_ Wealth (New York:
The MacMillan Company, 1927), pp. 116~HO. -"The articles that
embody the fund are, like particles of water in a river,
vanishing things; while the fund itself, like the river, is
the abiding thing." p. 157. The waterfall is Schumpeter's,
Histor , pp. 312., p. 902.

13For example, Schumpeter, ibid. The question of
just what Clark did mean by capital is, like the "average
period", an qpen question. Clark wavers between a value
concept and the physical concept. For example, in refer-
ence to capital, he says "It is value embodied in goods
the identity of which is perpetually changing, .
p. 120, ft. 1, _p.. 213. But this is a question we cannot
pursue here.

 

lLLClar , g3. cit., p. 157. "Men are as perishable as
are capital~goods, but labor is as permanent as is capital."

15Ibid., p. 200. "One law governs wages and inter-
est--the law of final productivity."

 

7

, But our interest in Clark is not due to his law of
distribution, though this was an important contribution to

the development of economicanalysis. Rather our interest

 

stems from his notion of capital, which has proven to be a
very fruitful simplification. When capital-~defined as a
single kind of physically homogeneous capital good-~13 com-
bined with labor in a simple neoclassical production func-
tion, a truly imposing list of propositions or parables can

be made to appear. In particular it can be ahown that,

 

given competitive markets, (a) the wage rate varies inverse-
ly with the interest rate, and (b) the real capital-labor
ratio varies directly with the wage-interest ratio, so

(0) distributive shares are well defined; and (d) net
national product per worker, or the permanently sustainable
consumption stream, varies inversely with the rate of
interest.

That is quite a bit to prove; but having proved it,
neoclassical theory establishes an orderly relation between
the physical realm of production and the commodity and
factor markets. Let us now set out the basic, rigorous
model in its most demanding detail.16 To this end, assume
that there exists a production function
(2) Q = F(K.L)

that is homogeneous of degree one in the single homogeneous

 

16For an expanded version see C. E. Ferguson, The
Neoclassical gheorypg£_Production and Distribution, forth-
coming from Cambridge University Press.

 

8

capital good (K) and labor (L). By its homogeneity property

(3) _ tQ, = tF(K,L) = F(tK,tL)

and letting t =‘l

L

_ K _ K
U4) %- ME) - F(-.1).

Thus we may write the production function as:

(5) Q = 1.51%).

The operation of diminishing returns ensures that

(6) sniff.) < 0,

whereas the economic operation of enterprise requires
(7) F'(%) > 0.

We further require competitive imputation, so

(a) w =93 = Fug) - (swig) > o,

 

0L
and
= 23 :_-_ I .12
(9) r OK F (L) > 00
Now differentiate (8) and (9) to obtain
dw
.___.. __ _ _I_{_ "(K > 09
(10) d(-I-{-) "' (L)F \I)
L
and
£13: .. K
(11) d(-—) = F ('13-) < 0.
Then dividing (10) by (11) we obtain
dw
d(K)
I :: g2! : ”K
(12) dr dr .1: <0.
MIX)
Finally, the Marshallian elasticity of (12) is given by:
(13) - E.§_.==.E§ 2 ratio of aggregate relative shares.
w dr wL

NOw we have established the essential results of neo—
hclassical theory. _Let us look at it graphically. Panel a,
Figure 1, is a graph of inequality (10). It shows that the

 

 

an»:
4?

 

 

SB
0
m
s
a

a

o

w

m

p

o

m

ones

a

nHl

a

H

o

o

m

A

o

a

p

fl

0

A

m
H
Ham
0
.H
comm
H
o

 

m
m
n
o

H

m

3

m

w

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o

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lO
wage rate varies directly with the capitalwlabor ratio.
Similarly, Panel b illustrates inequality (11): the rate
of interest varies inversely with the capital-labor ratio.
The two relations are brought together in Panel c, which
depicts inequality (12) and emphasizes that the wage rate
must vary inversely with the rate of interest. Thus proposi-
tion (a) is established.

The curve in Panel c is what Samuelson calls the Sur-
rogate Factor—Price Frontier.l7 By equation (13) the
Marshallian elasticity of this curve is the ratio of aggre—
gate relative shares. Thus we have proposition (c).

The Surrogate Frontier also verifies our propositions
(b) and (d). we shall set these out in terms of the capital
intensity of the economy's technique of production. The
reason for so doing is that these concepts (capital intensi—
ty and the technique of production) have played a very
prominent role in the recent criticism of neoclassical
theory. It will be useful to begin with a clear picture of
What these concepts mean.18

‘With a given technology and a large number of firms
each producing a single product, we may view each firm as
having a production constraint of the form:

(In) y = f(1.k).

 

17Paul A. Samuelson, "Parable and Realism in Capital
Theory: The Surrogate Production Function," loc. cit.

18The following discussion is based on R. G. D. Allen,
Mathematical Economics (London: MacMillan and Company, 1957),
pp. 332-El.

 

11

where y stands for output and l,k represent the services of
the firm's variable production inputs. This function sum-
marizes the efficient combinations of input services open
to the firm, and it is often assumed that the function is
characterized by constant returns to scale. The feature of
the production function of immediate interest is the implied
continuous substitutability of inputs. Each combination of
inputs represents one efficient method of producing the
given output; hence we may refer to any particular combina-
tion as a method of production. If we call 1 labor and k
capital, a method of production is simply an efficient
capital-labor ratio, and we have a large number of produc—
tion methods that are easily substitutable for one another.

Since each firm is a singlemproduct firm, we may
group them by product and speak unambiguously about m dis-
tinct industries (one for each product). If an arbitrary
set of prices for labor and capital is specified, each
industry would be observed to adopt a best method of pro-
duction (i.e., capitalwlabor ratio), and the aggregate of
these methods would constitute what is called the "tech—
nique of production". Hence a technique of production is
the aggregate capitalmlabor ratio that contains one method
of production for each of the m industries in the economy.

It is now clear from equation (12) that the slope
at any point on the frontier of Panel c, Figure l, corres-
ponds to one technique of production open to society.

Furthermore, the quantity of capital embodied in any

 

l2
technique is simply the slope of the frontier at any point
multiplied by the amount of labor utilized at that point.
So we have the proposition that an increase in the equili-
brium wage (or decrease in the rate of interest) will
bring about an increase in the capital intensity of the
technique "employed" by the economy. This is the critical
point contained in proposition (b).

Finally equation (A) together with equations (6)
and (7) yields proposition (d): a fall in the rate of
interest will lead to an increase in the productivity of
labor though at a decreasing rate.

On such simple relations as these do we base much
of our understanding of factor pricing, product distribu-
tion, and the very size of the product that is to be
distributed.

III. The Cambridge Criticism

What we now call the Cambridge criticism of simple
neoclassical theory first appeared explicitly in Mrs.
Robinson's article on the Production Function.19 At the
time, it appeared as though she did not realize how really

fundamental and important her discovery was.20 Her primary

 

19Joan Robinson, "The Production Function and the
Theory of Capital," Review Q§_Economic Studies, XXI (1953),
pp 0 81-1060

2OPerhaps it would be more accurate to say that she
did not then, and does not now, feel that the criticism is
as fundamental as others would have it appear. In the
recent edition of her Accumulation she still maintains that

 

13

concern, at least ostensibly, was with J. B. Clark's
capital21 and the practice of thinking about an equilibrium
position as though it were a position towards which the
economy is moving.22 Mrs. Robinson was irritated because
the neoclassical economist insists on pretending that he
can compare different steady-state, equilibrium ratios of
capital to labor and thereby come to an understanding of
the actual changes that take place between factor ratios 2g.
3392,23 She is interested in the problems of accumulation
and her suspicion is that the neoclassical economist has
been throwing mud in the water.

Like Bohm—Bawerk before her, Mrs. Robinson is struck

by the following observationgzu

 

the “perverse case is " . . . not of great importance."
p. 109, ft. 1, Joan Robinson, The Accumulation of Capital
(2d ed., New'York: St. Martin's Press, 196

21Robinson, 92. 9313., p. 81. "The student of
economic theory is taught to write 0 = f(L, C) Where L is a
quantity of labour, C a quantity of capital and O a rate of
output. . . ." but he is never told "in What units C is
measured."

22Ibid., p. 85, "The neoclassical economist thinks
of equilibrium as a position towards which the economy is
tending to move as time goes by. But it is impossible for
a system to g2§_into a position of equilibrium, for the
very nature of equilibrium is that the system is already in
it, and has been in it for a certain length of past time. "
Italics in original.

23This is a " . . . profound methodological error,
which makes the major part of nee-classical doctrine
spurious." Ibid. , p. 8h.

2L"Ibid., p. 82. Italics added. This no doubt is
what Bohm- Bawerk had in mind when he said "That roundabout"
methods lead to greater results than direct methods. . . .
but "It must be emphatically stated that the only basis for

 

1Q

To- -day, in country Alpha, a length of roadway is being
cleared by a few men with bulldozers; in Beta a road
(of near enough the same quality) is being made by
some hundreds of men with picks and ox-carts. In
Gamma thousands of men are working with wooden shovels
and little baskets to remove the soil. When all
possible allowances have been made for differences in
national character and Climate, and for differences

in the state of knowledge, i§_seems pretty clear that
the main reason for this state of affairs is that
capital in some sense is more plentiful inA lpha _than
in Gamma. Looked at from the point of view of an_
individual capitalist, it would not pay to use Alpha
methods in Gamma (even if unlimited finance were
available) at the rate of interest which is ruling,
and looked at from the point of view of society, it
would need a prodigious effort of accumulation to
raise all the labor available in Gamma even to the
Beta level of technique.

Thus there is a real problem with which neoclassical
analysis pretends to cope. Indeed as Mrs. Robinson says
there is even an "element of common sense" entangled in the
production function that has been brought to bear on the
problem.25 In an effort to rescue it (the element of common
sense) Mrs. Robinson is willing to play the neoclassical
game though "with due regard to its limitations." Thus she
proceeds to re-examine the steadywstate equilibrium rela-
tions between the quantity of capital, the labor force, and
the state of technical knowledge.

It is during this re-examination that Mrs. Robinson
stumbles upon what appears now to be the Achilles heel of

simple neoclassical theory. What she discovered is very

 

this proposition is the experience of‘practical life."
Bohm-Bawerk, 22, cit., p. 20.

25Ibid., p. 83. "We cannot abandon the production
function without an effort to rescue the element of common
sense that has been entangled in it."

 

 

15
simply stated as follows: under the postulate of a given
"state of knowledge," if we observe the changes that take
place in the technique of production as the real wage rate
rises (and interest rate falls), we may find that the tech-
nique being adopted is less capital intensive rather than
gggg capital intensive.

Such a possibility is labeled "perverse" by Mrs.
Robinson and she devotes very little attention to it.26
But as we now know, it is a very important seed that has
been planted here. For--and this is the Cambridge
criticism--onoe this is admitted our propositions (b) and
(d) of the preceding section have to go.

It will now be useful to look at Mrs. Robinson's
argument in greater detail. Doing so will help to clarify
the above statement. In addition it will bring out a
point which is of some interest in itself: recall that
the basic characteristic of the Austrian model is the
element 2;.timg whereas the basic characteristic of the
Clark model is a timeless production function. Mrs.
Robinson's chief contribution, judging from all that has

transpired since the appearance of her article, stems from

 

26It is interesting to note how one of the most
persistent of the critics is unable to admit to the basic
flaw in the theory in question. Perhaps the explanation is
that the flaw turns out to be the very "element of common
sense" that we Should like to preserve. Or perhaps it is
merely that "A good deal of exploration . . . is needed
before we can say whether the above is a mere theoretical
rigamarole, or whether there is likely to be anything in
reality corresponding to it." Robinson, ibid., p. 106.

 

16

the fact that she had put time back into the analysis of
production. For although the result was a very "incon-
venient" production function, it threw open for all to
see the very doubts that had previously plagued Wicksell
and were destined to lead to what we now call the Cambridge
criticism.27

To re-examine the neoclassical argument about the
movement of the capital-labor ratio, Mrs. Robinson found
that it was first necessary to put definite meaning to the
concept of a "quantity of capital". This is an index
number problem28 and it presupposes that there is something
meaningful that is called capital. After some preliminary
discussion lamenting the very notion of capital-in-general,
she decides that "to treat capital as a quantity of labour
time expended in the past is congenial to the production-
function point of view, for it corresponds to the essential

nature of capital regarded as a factor of production."29

But, and here is the critical point, "the past labour time

 

27In the Lectures he wrote that " . . . it appears
inconceivable g priori that an increase of capital could,
ceteris paribus, coincide with.a decrease of both wages and
rent--though.the question should he further_investigated."
22, 213,, p. 18?; Italics added.

 

28This is the message of those who were first to
react to Mrs. Robinson's article. See David G. Champernowne,
"The Production Function and the Theory of Capital: A Com-
ment," Review g£_Economic Studies, XXI (1953-195u), pp. 112-
135. Also Robert M. Solow, "The Production Function and the
Theory ofBCapital," Review 3; Economic Studies, XXIII (1955),
pp. 101-0 .

29Robinson, "The Production Function and the Theory
of Capital," pp, cit., p. 82.

 

 

17

which produced to~day‘s capital gpods was itself operating
upon preexisting capital goods or natural resources.
At any moment when work is being done to—day's labour is
being added to the product of past labour, which in its own
day was added to the product of still earlier labour."30

Thus if we are to take labor time embodied in capital
goods to establish a concept of capitaleinwgeneral, we must
recognize that it is compound labor time that is relevant.
So Mrs. Robinson defines capital--the factor of production-—
to be past labor time, compounded; 3 interest, embodied in
the stock of capital goods.31 As it turns out, this is a
surprisingly critical step. For from this definition comes
her shifting production function~~called a productivity
curve by her. Let us take a look at it.

First we must suppose that we can draw up a set of

blueprints containing all the techniques of production that

 

30Robinson, Accumulation, gp. ci , p. 121. This
point is less successfully made in her article on the

Production Function, pp, cit., p. 82.

31Robinson, Accumulation, pp, pip,, p, 123. Here it
is clear that this is what she means by capitalwin-general.
It is confusing, but perfectly legitimate, to multiply our
definitions according to our needs. Thus Mrs. Robinson
oalls " . . . the stock of goods in existence at any moment
physical capital. The value of these goods in terms of a
unit of output we call pppital simplicitero Capital valued
in terms of wage units we call real capital: though it must
be observed that there is a slightly misleading flavour
about this term, since the cost of capital goods, in terms
of wage units, includes interest over the time required to
construct them and to use them in production." Robinson,
"Production Function and the Theory of Capital," pp, 213,,
p. 86. Italics in original. It is her real capital that
interests us.

 

 

18

are known to producers. Each technique is defined as a
specific g}; 9; capital ao_0_£1_s that together with a given
amount of labor will produce a certain output. These are
strictly engineering relations. Next we must know how much
capital is represented by each set of capital goods in order
that we might avoid using a technique that involves more
capital per man but the same or less output than some other
known technique (in her terminology this is the problem of
"costing" the capital goods).

Suppose we have done the above and are left with a
set of techniques that may be represented by a curve such

32

as shown in Figure 2. The vertical axis measures output
per man and the horizontal axis the ratio of real capital
to labor. We have four distinct techniques labeled Alpha,
Beta, Gamma, and Delta arranged in terms of increasing
capital intensity. Thus Alpha represents more capital per
man than Beta, Beta more than Gamma, and Gamma more than
Delta.

Now suppose that when the wage rate is given as OW
the techniques Gamma and Beta are equally profitable and
the rate of profit on either technique (equals the rate of
interest in long run competitive equilibrium) is given by

l
N ‘33 The question is: what will happen to the capital

32The following is based on the discussion in her
Accumulation, pp. cit., pp. hllml8.

33For the proof of these relations see ibid. Our
interest is solely in the fact that the fUnction must
shift.

 

l9

 

 

 

W

HIV:

N 0

Figure 2. Joan Robinson's Productivity Curve

2O
intensity of the technique in use as the real wage rises?
The answer: it depends: Remember that capital is defined
as past labor time, compounded at interest, embodied in the
capital goods. Thus as the wage rate rises and the inter-
est rate falls each and every set of capital goods appears
as less capital)“ The production function shifts to the
left.

It may be "reasonable" to suppose that, as the wage
rate rises, the production fUnction shifts so that each new
equilibrium wage-interest ratio corresponds to a more
capital-intensive technique. This possibility is shown
in Panel a, Figure 3. But it is also possible, as Miss Ruth
Cohen has noticed, that the production function will shift
such.that the technique brought into use will be less
capital intensive.35 This possibility is shown in Panel b,
Figure 3. When the real wage is Wl both Gamma and Beta
techniques are profitable. As the wage rate rises to W2
the Gamma technique becomes gradually replaced by the more
capital intensive Beta technique which alone is the most

profitable at wage rates between W1 and W2. At the wage

 

34That the interest rate must fall follows from the
assumptions of a given fully employed labor force and the
division of all product between laborers and capitalists.
Robinson, "The Production Function and the Theory of
Capital," pp, pip,, pp. 85m86. See below Chapter II where
this relation is fully discussed for a somewhat different
real capital model.

35Ibid., p. 9h. "The relation between one curve and
the next depends upon the reaction of the cost of various
outfits of equipment to differences in the rate of interest,
and this depends, as we have seen, in a complicated way,
upon the gestation period and length of life of items of
equipment."

 

 

moswwnnoofi
no eonmoopoa dosages one so one: Hoom one on omoonooH so no oooeem,

 

 

 

 

.m oaswfim
Q Honmm w Honwm
q H ”w
ML 0 2 m2 m2 3. M .. o
1a \\ 4 \
it
a.» fix .33 m» x
1. 3Q «a A‘.
A4 I ;s III..I.II |.I1
“a
.1
= m 3

22
rate W2 both techniques are again profitable and as the wage
rate rises even higher, toward W3, we find that the Gamma
technique has become the most profitable technique. Thus
we have moved from a more to a less capitaleintensive
technique.

If we search for an explanation of this "perverse"
reaction to a change in the wage-interest ratio, we are led
straight to the element of compound interest. In Mrs.
Robinson's words: "This might occur if the plant required
for less mechanized techniques had a much longer gestation
period or working life, so as to be much more sensitive to
the interest rate than that for mechanized techniques."36
'What this means is that the less capital—intensive tech-
nique has been more heavily weighted by the compound
interest factor.37

To sum up, the foregoing demonstrates that the
criticism of neoclassical theory (a~la~Robinson) really
38

comes down to thisz‘ time plays an important role as an

 

361b1d., p. 106.

 

37In other words, her measure of capital is subject
to the same defect that was present in the Austrian measure
(i.e., the average period). A lower rate of interest would
shorten the average period even if production processes
were unchanged.i§ the average period did not neglect com-
pound interest. This is Wicksell's criticism of Bohm-
Bawerk's average period. Actually Bohm-Bawerk did not
include even simple interest in his definition. On this
see Samuelson, "A Summing Up," 2p, gi£,, p. 570.

38Perhaps I am overwreacting to Mrs. Robinson's
capital concept. Nevertheless, I believe that this is the
key to what really disturbs her. She is anxious, as was
Bohm-Bawerk, to master the element of time. The criticisms

 

23
element in production. In particular it is crucial to the
notion of capital and the productivity of labor. 'Yet neo~
classical theory pretends that time does not matter. This
is likely to lead to error.

Surprisingly the error, if there is one, does not
concern time p§£_§§, Rather it concerns the link that the
theory has erected between the wage—interest ratio and the
capital intensity of production (i.e., the capital-labor
ratio). In particular the rate of interest may ngthe a
single-valued function of the capital~labor ratio.

The immediate reaction to this was to some extent
beside the point, concentrating as it did on the index
number problem.39 But the "perverse" reaction of the
capital-labor ratio did bother those who came upon it.

For example, Mr. Champernowne, who was anxious to replace

Mrs. Robinson's capital with something more "convenient",

is quite careful to point out that the "perverse" case

can only be made to disappear by assumption.“’0

But it is Piero Sraffa who finally announces with
some degree of confidence that the "perverse" case must be

Ml

taken seriouslyz

 

that we have mentioned above are repeated by Mrs. Robinson
in "Accumulation and t?e)ProductionuFunction," The Economic
Journal September, 19 9 , pp. h33- 2. See especially

p. E37 and p. hhl.

39Champernowne, loo. cit., and Solow, loc° cit.

”OChampernowne, ibid., p. 119. By convenience, he
had in mind a function that would yield nice aggregate
relative shares.

LLlPiero Sraffa, Production 9;; Commodities :91 Means

 

 

21+
We have seen that as the rate of profits rises there
may be several intersections between the prices at
which the two methods produce, with as many switchings
backwards and forwards from one method to the other.

. . In view of this possibility we cannot (contrary
to what one might have expected) say in general that,
of two alternative methods of production, the one that
corresponds to a Standard system with a higher ratio
of product to means of production . . . will be the
most profitable when the rate of profits is com-
paratively high, and the least profitable when it is
comparatively low.

The critics can no longer be ignored or passed over
by a redefinition of terms. Henceforth we shall have to
qualify our statements concerning the wagewinterest ratio
as it relates to the capital intensity of production.

The possibility that the same technique may be
observed at two substantially different wage rates
(interest rates) has been labeled the "Ruth Cohen Curio~
sum" (Joan Robinson) or more recently the "reswitching of
techniques" (SraffaePasinetti). It is, to repeat, reswitch-
ing and its implications that constitute the Cambridge
criticism of simple neoclassical theory. This implication
is of great importance for capital theory: no longer can
one say that a fall in the rate of interest generally

entails the adoption of more capitaleintensive techniques

 

32 Commodities (Cambridge: Cambridge University Press,
1963), p. 8E. Italics added. The entire thrust of this
work is to determine the effects of distribution on the
relative prices of commodities. The model employed is a
fixed proportions circulating capital model, and the price
of every commodity is determined.by the production condi-
tions. The cost of commodities is reduced to past labor
compounded at interest. The critical point in the book
comes with the demonstration that, as wages rise and
interest falls, relative prices may reverse themselves.
This is used to argue the above point by way of analogy.

 

25

of production with the consequence that the permanently
sustainable consumption stream is greater. A lower rate of
interest may be associated with a lgflgr_rather than a
higher consumption plateau. And though reswitching reveals
this possibility, even it (reswitching) is not necessary.“(2

The exponents of neoclassical theory were not long
in replying to this criticism. The first response came in
the form of a nonswitching theorem}3 Here neoclassical
theory was willing to concede reswitching in the production
of a single good; but not for the economy as a whole. How-
ever, under closer examination, the proof of the theorem
was itself shown to be erroneous and so the theorem had to
go.uh In its place we find a number of numerical examples
of reswitching~weach designed to prove that this phenomena
is indeed possible for the whole basis of production.
Beyond this everyone has been most anxious to spell out the

above-mentioned implication of reswitching for capital

 

hZSee, for example, Michael Bruno, Edwin Burmeister,
and Dytan Sheshinski, "The Nature and Implications of the
Reswitching of Techniques," The Quarterly Journal 23,
Economics, LXXX (November, 1966}, pp. ShB-h9. Perhaps it
should be noted again that neither Mrs. Robinson nor her
immediate critics seem to have realized the importance of
the "perverse" case. For this the credit must go to
Pasinetti. See Luigi L. Pasinetti, "Changes in the Rate of
Profit and Switches of Techniques," ibid., pp. SO3~17.

”3David Levhair, "A Nonsubstitution Theorem and
Switching of Techniques," The Quarterly_Journal g£_Economics,
LXXIX (February, 1965), pp. 98~lOS.

huDavid Levhari and Paul A. Samuelson, "The Non-
switching Theorem is False," The Quarterly Journal 3;
Economics,.LXXX (November, 1933f, pp. 1 -l9.

 

 

26
theory. In the next chapter we shall explore in greater

detail the neoclassical reaction to the Cambridge criticism.

 

 

CHAPTER II

SWITCHES OF TECHNIQUES AND STEADYmSTATE
CONSUMPTION LEVELS

Io Introduction

The simple neoclassical theory of the preceding
chapter is based on the two fundamental assumptions of
(a) instantaneous production (i.e., a timeless production
function or the absence of "roundaboutness"), and (b) a
single homogeneous real capital good. But a roundabout
method of production may be viewed as an instantaneous
production process by treating goodsuin~process of differ-
ent ages as different goods (1.609 as so many fictitious
heterogeneous capital goods); hence it is the assumption of
a homogeneous capital good that is the more basic assump~

tion.”5 As we shall see below, once the heterogeneity of

 

uSAt least insofar as reswitching is concerned, see
Michio Morishima, "Refutation of the Nonswitching Theorem, "
The uarterl Journal of Economics9 LXX (November, 1966),
p. “52E. Actually, there are two aspects of a production
process with a time lag that are important for reswitching.
One is the element of compound interest; the other is the
diatribution of labor in time. It is the latter element
that corresponds most closely to the heterogeneity aspect of
capital goods in an instantaneous production process. On
the importance of the distribution of labor in time see
Maurice McManus, "Process Switching in the Theory of Capital, "
Economica, XXX (May, 1963), PP. 109 21. See especially
p. 112 for some necessary conditions for switching that
reflect this point.

27

 

28
capital goods is allowed for reswitching can be shown to be
a definite possibility.u6
It is this instantaneous production model with

heterogeneous capital goods that dominates the neoclassical
reply to the Cambridge criticism.u7 After some preliminary
discussion, we will employ such a model to bring out the
relation between switches in techniques and steady-state

consumption levels.
Ievhari}s Nonswitching Theorem

It is necessary to begin by commenting briefly on the
argument of David Levhari, whereby he is led to conclude
that the "Ruth Cohen Curiosum" " . . . may indeed be observed
in the production of a single good. But . . . it is impossi-

ble with the whole basis of production."LL8 Our concern is

 

uéRecall that this is precisely what Mrs. Robinson
was attempting to do when she came forward with her hybrid
production function. But9 as we emphasized, it was the
element g£_time in her measure of capital that was critical
in revealing the reswitching possibility. As we shall see
below, the most recent discussions do not employ agy_con-
cept of capital, i.e.9 they focus exclusively on capital
goods.

A7David Levhari, "A Nonsubstitution Theorem and
Switching of Techniques," The uarterl Journal g£_Economics,
LXXIX (February, 1965), ppo 98~10§. Also "Paradoxes in
Capital Theory: A Symposium," The Quarterly Journal.g£
Economics, LXXX (November, 1966): pp. 503-83. Levhari uses
a one period circulating capital model. It is here classi-
fied as an instantaneous production model since it cannot
be said to involve the element of time in any essential
way.

uaLevhari, 2p, cit., p. 99.

 

29

not with Levhari's proof per se but with the meaning of his
”whole basis of production.”9 This he defines as an (n x n)
matrix of commodity-input coefficients together with a
(l x n) row vector of labor-input coefficients. Each column
vector is then interpreted as one method for producing a
unit of output in one industry.50 Thus the whole basis of
production appears as the aggregate of the methods of pro-
duction (one for each industry) when operated at unit level.

‘What Levhari tried to prove is that once any given
matrix is left behind as the rate of interest falls (wage
rate rises) it can never be readopted as the interest rate
falls still further. Once this is proven it may be said
that the matrices are unambiguously ordered with respect to

the rate of interest. That, however, is all one could say.

 

H9His proof, being in erro r9 served as the catalyst
for the recent reswitching discussion. On the error in his
proof see P. Garegnani, "Switchin of Techniques," The uar-
terl Journal 2£_Economics, LXXX November, 1966), pp. ggfi-

l. Levhari has attempted " . . . to demonstrate that a
semi-positive vector Xx = (x? , . . . x*) of levels of pro-
duction of the various industries alwa s exists, such that,
at those levels of activity eircezr system A requires more of
all kinds of commoditywinputs than B9 or the contrary is
true, or, finally, A and B require the same quantities of
all those inputs." p. 556. Italics in original. The error
concerns the existence of this vector. I am not qualified
to comment on the mathematics of Levhari's proof, but others
have shown that this vector can be of no use to him. The
emphasis, however, has been placed on numerical demonstra-
tion and it is that level of argument which dominates this
paper.

SOLevhari, pp, cit., pp. 99 100. "Let there be lkl
activities which can be used to produce good 1,

. . , a1k l; k2activities to produce good 2,. . ., and kn
for good n. Each activity is composed of n + 1 elements.
The first element . . . gives the labor requirements, and
the remaining n components give the requirements of inputs
of goods to produce one unit of gross output of the given
commodity."

 

30

But, as Levhari clearly recognizes, the Cambridge

 

criticism concerns the capital intensity of production.51
Even if his matrices were unambiguously ordered with.respect
to the rate of interest, would this demonstrate that the
rate of interest is a single-valued function of the capital-
labor ratio? Perhaps. The point is, before anyone could
say, it would first be necessary to spell out the exact
sense in which these matrices could be said to be comparable
to the aggregate capital-labor ratios of the neoclassical

model.

 

Yet this was not done, either by Levhari or those
who have subsequently shown that his proof is in error.52
This seems most unwarranted in view of the fact that it is
the capital intensity of production that is the critical

variable in the Cambridge criticism.
Capital Intensity in a Heterogeneous Real Capital Model

let us here set out the equivalence between the

capital intensity of production in a heterogeneous

 

51In referring to the Cambridge criticism he says,
"This would have the unfortunate consequence that we could
no longer say that the lowering of the interest rate brings
about a.process of 'deepening' and each process is more
capital-intensive than its predecessors." Ibid., p. 99.

52The tendency in the literature is to associate

directly the "nonwcomingback" of a matrix with the capital
intensity of production. This, however, is not warranted.
It is not obvious that when a matrix Comes back it neces-
sarily represents the same aggregate capital-labor ratio.
This is the question of capital-intensity along the grand
factor-price frontier that will be fully explored in the
next chapter.

31

capital-goods model and the aggregate capitalwlabor ratios
of the neoclassical model.53 To begin, we assume a given
technology and a large number of firms each producing a
single product.

Now suppose each single-product firm is confronted
with a production constraint of the form:

eraLrY
(15) y = min (_2.: r

Kr = aKry

Where y, L, and K represent output and the services of labor

ti
179

(r = 1,2, .. .n)

 

and capital respectively. There are n different processes
available to the firm, and air (i = L,K) represents the
additional assumption of fixed minimum unit production
requirements for labor and capital. Less of either input
will yield less output and more will be superfluous.
Strictly speaking we do not have §_production function
but a number of production functions, one for each of the n
different processes. This complete set of production func-
tions may be viewed as the technology of the firm. Let us
refer to each production function as one method of produc~
tion. Then, as above?"L we may speak unambiguously of m

distinct industries, each of which selects one method of

 

53The following discussion is based on the treatment
in Robert Dorfman, Paul A. Samuelson, and Robert M. Solow,

Linear Pro rammin and Economic Analysis (New York: McGraw-
Hill, 1938), Chapter 11. See especially pp. 286-88 and

pp. 300-020
EuSee Chapter I, pp. lOwlZ.

32

production. The aggregate of these methods is referred to
as a technique of production.55

Thus, except for the additional assumption of fixed
coefficients, a technique of production in a heterogeneous,
real-capital model is nothing but the aggregate capital-
labor ratio of the neoclassical model. As long as at least
one of the methods of production is different between any
two arrays of industry methods (i.e., production functions),
the techniques will be distinct, and we may think of a
large number of such discrete capital-labor ratios as
representing the technology of the economy.

As an extreme illustration consider a two-sector
economy-—a consumption sector (0) and a capital-good
sector (M)--each of which has the following method of
production:

C=min(-§—C- ,KC)=<LC KC)

LC aKC T 9 T‘—
(16)

M=min(I-”-M- .3514.) (Twila)
am am 2 LL

The technique of production is then the aggregate capital-

labor ratio given by.3§9 or.$§fl, each of whidh equals 2.56

are aLM

 

SSThus Levhari's matrix is one technique of produc-
tion (supra p. 29). The same meaning attaches to all of the
so-called technology matrices to be found in the reswitching
literature. See Michael Bruno, Edwin Burmeister, and Dytan
Sheshinski, "The Nature and Implications of the Reswitching
of Techniques," Thg_ uarterl Journal 2£_Economics, LXXX
(November, 1966), pp. §28-3l.

56Notice that 3L0 = aLM and aKC = aKM. As explained

 

33
The simple but important point to be drawn from this

 

discussion is that a technique of production must be
thought of as the aggregate capital-labor ratio for the
economy. This will be the case regardless of the capital
model that one chooses for discussing the reswitching
phenomenon or any of its implications.

Thus when Levhari argued that his matrices were

unambiguously ordered with respect to the rate of interest,

 

he really meant to say that the aggregate capital-labor
ratio was so ordered.57 In the reswitching literature that
has appeared since his article, many writers have demon-
strated that these matrices are not unambiguously ordered
with respect to the rate of interest. The Cambridge
criticism was subsequently reasserted by these writers:
the aggregate capital-labor ratio is not a single-valued
function of the rate of interest. Furthermore it was
shown that this implied unorthodox behavior of steady-
state consumption levels. But~~and this is why we have
taken so much time with this introduction-~notice that
these propositions concern the aggregate capital-labor

ratio. In what follows it will be seen that the aggregate

 

below, Chapter III, pp. 7l~78, this assumption on the
coefficients means that our so-called two~sector economy
is in fact a single-sector economy. Thus the aggregate
capital-labor ratio is the sector capital-labor ratio.

57Supra,rr.51-

3h

capital-labor ratio has, in fact, been lost in the arguments

surrounding the switches of techniques.

II. Switches of Techniques in a Two-Sector

Heterogeneous Real Capital Model

The switch and reswitch in production techniques may
be most easily discussed within the framework provided by
Samuelson in his paper on the Surrogate Production Func-
tion.58 The model we wish to employ consists of the follow-
ing set of assumptions:

a) All firms are single-product firms (i.e., absence of
joint production).

b) Two goods are produced, a consumption good and a capital
good (denoted C and M respectively).

c) There is one primary unproduceable homogeneous input,
Labor.

d) Each industry uses labor of a given amount together with
a specific and fixed complement of the capital good to
produce a unit of output (i.e., fixed coefficients of
production).

6) Labor Employedz Labor Supply, Capital Employed?

Capital Supply
(i.e., either factor may be redundant).

f) Each industry has a finite number of production methods
as defined above.

g) Competition is perfect in all markets.
This model is then represented by the following price

equations:

 

58Paul A. Samuelson, "Parable and Realism in Capital
Theory: The Surrogate Production Function," Review 3;
Economic Studies, XXIX (June, 1962), pp. 193—207.

 

 

35

1 = aLCW + aKCrP ,
(17)

P = aLMW + aKMrP ,
where P is the equilibrium price of machines in terms of
consumption goods, w is the real wage rate, and rP = q is
the quasi-rent of the-capital good. There are two equa-
tions in the three unknowns; P, w, and r. We may elimi-
nate the relative price of the capital good and in terms
of the consumption good as numeraire obtain a single

equation in the two unknowns w and r:

 

(18) w = 1 - aKMr ,

where we have taken
aLM aLC

A = (aKCaLM _aLCaKM) = det. , and we recognize
aKM aKC

this matrix to be one page from the economy's ”book of blue-

prints" as discussed above.
The Factor-Price Frontig;

It is this trade-off between the feasible long—run
equilibrium wage and interest rates59 that stands at the
center of the recent reswitching discussion. Every tech-
nique yields a unique trade~off between these two variables.

Let us suppose that A = O, which amounts to assuming that

59Under our assumptions the rate of interest is
equal to the profit rate and we shall ask that the reader
keep this in mind as we will make explicit reference to the
rate of interest only.

36
both sectors use factors in the same proportions (i.e.,
aKC = a'KM). We then have
HLM

(19) w=l-9‘KMI-.

This linear relation between the real wage and the rate of
interest is illustrated in Figure h.

The maximum real wage that may be obtained under this
technique is seen to be determined by the unit production.
requirements in the consumption-good sector. The more labor
required to produce a unit of the consumption good, the
lower will be the maximum long-run wage rate. Thus the
intercept is the average productivity of labor in this
sector (this is an important relation that must be used in
interpreting the graphs appearing in the remainder of this
Chapter).

The maximum rate of interest with this technique is
likewise seen to be completely determined by the technical
production requirements of the capital-good sector. The
less capital needed to produce a unit of itself, the higher
the rate of return to the capital good.

Finally, note the inVerse relation that must hold
between the real wage and the interest rate under these
assumptions.60 When capital is redundant, the total

product obtainable with this technique is Lt x.1_ ,
aLC

 

60For a more rigorous treatment of this relation see,
for example, John Hicks, Ca ital and Growth (New York:
Oxford University Press, 19%3), pp» 317-18.

 

81H

Figure h.

 

37

 

The Factor—Price Frontier

5|“

38
a constant; and the total product goes to the wage earners.
If the wage rate falls, some of the product is freed from
the wage bill and goes to pay for the now relatively
scarce capital good. Since any movement in the wage or
interest rate takes place within the context of an un-
changed production technique, either factor gains only what
the other loses. If the rate of interest is to rise, the
real wage has to fall.

By arbitrarily varying the production coefficients,
we may generate any number of such trade-offs between the
real wage and the interest rate, called a "factor-price
frontier" by Samuelson and a "wage curve" by Hicks.61

bKC, bLM, bKM) such that

Consider another technique (bLC,
bLC > 81L0 and bKM < aKM and B = O. The factor—price
frontier is given by

(20) w = l - bxmr .
ch

Equations (19) and (20) are brought together in Panel a,
Figure 5.

It is assumed that given the wage rate, the technique
that allows the greater return to capital is "chosen" by the
economy. The critical wage rate is w*, with Alpha technique
chosen for higher real wage rates and Beta chosen for rates
below w*. Consider the wage falling monotonically toward

zero: the economy is said to switch from Alpha technique

 

61Hicks, ibid., p. 1&0, and Samuelson, 9p, cit.,
p. 196.

 

 

39

gofipcoam oOflnmlhopomm Gamma 039 .m opsmfim

p Hocam
\% s» s‘ s!

an. Ed w Hocwm

”o
:1:

 

 

 

 

ho

to Beta technique at the wage w*. A number of switches are
shown in Panel b of Figure 5. The envelope produced by
such switching of techniques (i.e., the heavily shaded
outermost line called the grand factor-price frontier) will
in this case possess properties identical to those of any
individual factor-price frontier. In particular our conclu-
sion concerning the relation of the real wage rate and the
rate of interest is unaltered by such switching. We will
always be operating on the envelope of the individual
factor-price frontiers by virtue of our profit maximiza-
tion hypothesis. In addition, with given money wages and
perfect competition, market prices will ensure that a fall
in the rate of interest is accompanied by an increase in
real wages.62

Reference to Panel b of Figure 5 is sufficient to
demonstrate that as the wage rate falls monotonically from
its maximum to zero, the economy can never "readopt" a
technique once it has been left behind. The techniques
are unambiguously ordered with respect to the wage rate
(interest rate). This is all we need concern ourselves

with at present for it is precisely the question of the

62Paul A. Samuelson, "A Summ ming Up," The uarterl
Journal of Economics, LXXX (November, 1966),_ p. §7§.
”Under perfect competition, either workers can hire capital
goods or capitalists hire workers. At a lower interest
rate, or cost of capital, workers can always pay themselves
a higher real wage even without changing techniques; so
capitalists will have to match.up." With perfect competi—
tion, the result is the same even if capitalists do all
the hiring.

hl
"readoption" of a technique once it has been left behind

that constitutes the issue of reswitching.

 

Reswitching on the Grand Factor-Price Frontier

It is possible that a technique which is the most
desirable of all feasible techniques at some relatively
high rate can once again become the most desirable at a
relatively low wage rate. As shown by numerous writers, a
"readoption" of techniques is a possibility both for the
individual firm and/or the economy as a whole. Consider
the two techniques of Figure 6.63

The critical wage rates are wf and w?. At any wage
rate above w?, the economy will adopt the Beta technique
because it yields the higher rate of return. As the wage

rate falls below w’tr the Alpha technique becomes the more

2’
profitable; but as the wage falls still farther, going

below wf, Beta technique again wins out in a competitive

market. Hence it is demonstrated that the economy may
switch back to the original technique before the wage rate
has fallen to zero.

The literature on reswitching contains several

numerical examples of the behavior depicted in Figure 6,

63Note that the Beta technique gives rise to a
nonlinear factor-price frontier. The following discussion
shows that reswitching is impossible if all techniques
give rise to linear factor-price frontiers. The frontier
will be nonlinear whenever the intrinsic factor intensities
of the capital and consumption good differ. See below,
Chapter III, pp. 86— 7

L12

an
h

 

 

Figure 6. Reswitching of Techniques

43

and we shall on occasion want to examine such examples in
great detail. The more immediate and interesting question,
however, is: granting the possibility of reswitching, what
are the conditions that must be satisfied if we are either
to ensure the behavior of Figure 6 or prevent it?

Conditions sufficient to prevent the return of a
technique have now been formulated by a number of writers.6u
Attention has centered primarily on the two-sector capital
model of this section. We have, for example, the obvious
conditions that for any two techniques, if there is one and
only one intersection in the positive quadrant, reswitching
is not possible (see e.g., Figure 5, Panel a). More pre-
cisely we require either that:65

bKM ) aKM while aLC > bLC ,
(21) or

bKM < aKM while aLC < bLC
That is, the technique with the lower (higher) labor-
consumption coefficient must also be the one with the

higher (lower) capital-capital coefficient.

6Ll-Piero Sraffa, Production Lf Commodities _1 Means
Lf Commodities (Cambridge: Cambridge University Press, 1963),
QE; Murray Brown, ”Substitution- -Composition Effects,
Capital Intensity Uniqueness and Growth, " Discussion Paper
No. 2, Department of Economics, SUNY at Buffalo, Buffalo,
N.Y. , 1967, pp. 1h- 18; Hicks, 2p. cit. , p. 15h; Morishima,
gfié- cit. , p. 523, and Bruno, et al_ , _E- cit., pp. 534 and

 

 

65This condition is stated by Sraffa, ibid., Hicks,
ibid., and Morishima, ibid., but most concisely by Bruno,
gt gl., ibid., p. 534

 

 

 

uh
In addition to such simple and direct formulations
as these, Bruno, Burmeister, and Sheshinski have attempted
to formulate similar conditions for a general n-capital

66 Unfortunately, as they emphasized, little

good model.
economic meaning can be attached to their results. Never—
theless, the method employed is worthy of comment for at
least two reasons. First, it appears that the authors
themselves have in the final analysis disregarded their

own warnings.67 Second, an examination of the method will,
I think, provide us with some justification for our own
preoccupation with numerical analysis.

Let the factor-price frontiers for the Alpha and

Beta techniques of Figure 6 be:

 

a — bKMr
(22) w=1*KMI’; w=l .
aLC 5L0 + Br

Set the two wage rates equal so as (hopefully) to determine
the two switch points. One then obtains the following
quadratic in r:

(23) r2 - (B + aLCbKM — chaKM) r + (aLC — bLC) = 0.
---1r---
B aKM B KM

The real positive roots of this quadratic will be the switch

 

points if there are any.68 Presumably we can impose

 

66Bruno, et al., pp. cit., pp. 538-u6 but especially
pp- Shh-MS.

67They tend to put too much weight on their results
(ibid., p. 527 and p. 545), though they are in agreement
with the points which follow (ibid., p. 5M9).

68For a complete technical analysis concerning the
possible number of switch points in a given model, see
McManus, loc. cit.

LLS
restrictions on the coefficients so as either to ensure the
behavior of Figure 6 or to prevent it. It is customary

69 and we shall

(but not necessary) to suppose that B ,> 0,
make the additional assumption that any solutions of equa-
tion (23) will be rea1.7o We turn first to the sufficient
conditions for reswitching.

According to Descartes‘ Rule of Signs, a condition
for two positive real roots is given by
(at) (am - ch) > 0 and (aLCbKM - bLCaKM) > 0 .
which requires either

bKM = aKM and aLC > ch ,
or bKM > aKM and aLC ’ bLC ,
or aLC ’ bLC by more than aKM >’ bKM .
It seems that we have three distinct sets of restrictions
that might be placed on the coefficients of equation (23)
so as to yield reswitching in this model. However, as
shown in Figure 7, only one set does, in fact, yield re-
switching: the case where aLC > bLC by more than aKM > bKM‘
Both Panel a and b demonstrate that as the wage rate falls

1
from its maximum of £36 to zero, the switch of techniques

69Thus requiring that production in the consumption-
good sector be capital-intensive relative to production in
the capital—good sector.

70This will be the case so long as the discriminagt
ofb (23) i3 positive. Thus we are assuming that (B +
- big axm) .> h(aLc)(aLC ' bLC) This condition is neces-KM
sary for reswitching in this model, and assuming its pres-
ence allows us to formulate some simple sufficient condi-
tions that will also be necessary. Without this assumption,
the conditions we formulate are necessary but not sufficient.

M6

 

mCMHm mo oazm .mophwomcn use mcflgopfizmom

 

 

 

 

EMm 2M9 on
ski H H e H as
E
H
mm
H
a
3
Home 55
s . IHI H h Hesse
mule
H.
mHle.
H
<

 

3

.Hh‘

Ema. Em
H H

o Hoswm

w Hocmm

 

.N omsmflm

E

la .4
,0

S
a

.43 .|
,0

 

’47
is from Beta to Alpha. The economy will not return to the

Beta technique before the wage rate reaches zero. Further—
more, as a comparison of Panel b with equation (21) makes
clear, we have involved ourselves in a contradiction. The
so-called obvious conditions for preventing the return of
a technique now appear as sufficient conditions fg£_the
return of a technique. What has gone wrong?

The difficulty is in our straight—forward applica-
tion of the method of Descartes to the problem of reswitch-
ing. Descartes‘ Rule of Signs states that the number of
positive real roots of f(r) = O is either equal to the
number of variations of sign in f(r) = O or less by
multiples of two.71 All our restrictions can guarantee is
that equation (23) will have two variations of sign.
Whether this yields reswitching cannot be determined with-
out additional information.

Descartes' Rule of Signs may be applied to the
problem of switching to determine the maximum possible
number of switches. But there may be none. Thus the
method that we have applied has an upward bias in the sense
that the number of variations of sign often exceeds the
actual switches of techniques. It is this important point
that is demonstrated in Panels a and b of Figure 7.

Turning to the sufficient conditions to prevent re—

switching, we must select our coefficients so as to allow

71See, for example, Leonard E. Dickson, New First
Course 13 the Theory 2; Equations (New York: Wiley, 1939),
pp. 75-9.

 

MB
at most one variation of sign in (23). We therefore let
(25) (am - ch) < O and (aLCbKM ' bLCQLKM) > 0 ,
which requires that

bKM > aKM by more than ch > aLo .

As shown in Panel d, Figure 7, we have restricted ourselves
to the irrelevant case in which one of the techniques is
always dominant. This merely points up the fact that one
variation of sign does Egg mean one switch point. One
variation of sign means there can be gt_m2§§ one switch
point. There may be none.

The point again to be emphasized is that the method
used to establish our sufficient conditions is biased toward
the reswitching result. We cannot, of course, say anything
about the magnitude of the bias; nevertheless, we ought not
lose sight of this bias when attempting to evaluate the
stringency of our conditions. This point is especially
important if we are to apply the method to a more general
capital model. For then we are almost certainly going to
overstate the case for reswitching.72

Now recall that we are investigating these suffi-
ciency conditions in order that we might be in a position

to say something about the possibility (likelihood?) of

72There is, I think, a very real danger of putting
too much weight on generality. See, for example, Bruno,
23131., 22. 233., p. 527. I for one would be reluctant to
give much weight to the realization that switch points are
related to the roots of n — degree polynomials. Unfor-
tunately, there just isn‘t much one can say, in general,
about these polynomials. McManus, 122. £13.

(+9
reswitching. Should we not then question the significance
of an exercise which, when all is said and done, leaves us
with the rather vacuous statement that " . . . reswitching
is, at least theoretically, a perfectly acceptable case in
the discrete capital model."73 After all, everyone will
agree that this possibility is demonstrated by the simple
and more direct method of numerical example.

But, let us not be too hasty. We had better check
to see if the conditions derivable by the method of
Descartes (or any similar method such as Budan's Theorem),
although shown above to be overly strong so far as reswitch-
ing is concerned, are not in fact of some importance when
we come to a consideration of the implication that has been
of much interest since the demonstrated possibility of the
return of a technique for the economy as a whole.7u I
V'refer, of course, to the behavior of the permanently sustain-
able consumption stream available to society as the rate of

interest falls.
III. Steady—State Consumption Levels

Our attention has been centered on the "choice" of
production technique as the interest rate is allowed to
fall (rise). The "choice" concerning Which of several

techniques is "best" carries with it society's decision

 

73Bruno, pp_§;,, pp, cit., p. 5H5.

7"This is apparently the position of Bruno, 23 g;.,
ibid., p. 5M9.

50

concerning which consumption stream is preferred over all
that are technologically feasible. We may either suppose
that society chooses directly between alternative consump-
tion streams, or we may suppose that the choice is indirect-
ly made through a decision to produce more or less "pro-
ductive stuff", i.e., to add to or to subtract from the
capital base.

The point of immediate interest is that a very
definite relation has long been thought to hold between
what we call the economy's technique of production, the
interest rate, and steady-state consumption.75 The latter
refers to a plateau of consumption that may be attained
and maintained indefinitely given an unchanging population
and technology. The traditional triad will be briefly
indicated by use of the neoclassical production model. We
then spell out the logical equivalence between the economy's
grand factor—price frontier and its steady-state consump-

76

tion curve. Finally the reswitching discussion is related

to the behavior of steady’state consumption levels.

The Traditional Ordering of Consumption and the Interest
Rate

Consider the smooth, twice differentiable, linearly

homogeneous production function of the neoclassical

75$ee, for example, Dan Usher, "Traditional Capital
Theory,” Review 23 Economic Studies, XXXII (1965), pp. 169-
86. Our steady—states will be stationary states as the
next sentence will make clear.

76This equivalence is noted by Usher, ibid.,
p. 173, n. 2.

51

model:77
(26) c = F(K,L) 5 LF(%) .
Thus
c _ K
(27) i — mi) ,

where we have taken the national product (Y) to be net
income (NNP) and thus equal to consumption (C) inasmuch as
we are interested in the stationary state relations.

The operation of diminishing returns ensures that

(28) F"(%) <0,

whereas the economic operation of enterprise requires

(29) wig) > 0

We further require competitive imputation, so

221:2!1: é — E 'E
(30) W bL 5L F(L) (L)F (L) > 0 ,

(31) r:fl=.b_C.=F'(.IS)>O.
The relation of interest to us may be obtained from (26)
and (31), parametric equations for the steady-state curve.

Differentiate (31) with respect to (é) to obtain

77The following argument is offered as a simplifica—
tion of the link between consumption flows and the changing
augmentable capital factor as discussed by Usher, ibid.,
pp. 177-82. The basic difference is that his analysis
allows for the element of time. It centers on what is
called the "marginal input-output equation" whereas ours
centers on the economy-wide production function. As he says,
"The former is an equilibrium condition of a system describ-
ing the passage of the economy through time. Valid in a
wide variety of circumstances, it will be shown to be the
link between the production process and the determination
of the rate of interest. The latter is one equation in a
complete system. It requires for its existence a much more
restrictive set of assumptions than does the marginal
input-output equation, . . . .", p. 177.

52
dr K
(32) = F"(—) < O.
(SJ-K) L
L
Next, differentiate (26) with respect to (%), obtaining

dC = r K

Dividing (33) by (32) yields the desired result:

K
dr =§= LF'(r>
d(%) dr Ftl(%)

d0 1
(3H) d(—_ ,

 

< 0.

we

The relations (32), (33), and (3h) are shown in Panels a,
b, and d of Figure 8. Panel d depicts what is called the
steady-state consumption curve; it shows the relation
between the rate of interest and the level of consumption
for alternative stationary states. The steady-state curve
has been drawn on the assumption that there is both a
positive minimum consumption stream obtainable (CM), and

a finite level of consumption available at bliss (CB).
(The latter defined as the maximum consumption attained in
any feasible stationary state).

Panel 0 depicts several alternative time paths of
consumption. Starting at a common consumption level at
time zero (Co), each is assumed to end at a different sta-
tionary state. As society foregoes consumption today in
favor of more consumption tomorrow, resources are diverted
into the production of additional quantities of the homo-
geneous physical capital good. This capital is said to

return to society not only the original consumption sacri—

——————-—————__—

53

umohchH Mo spam esp paw COHBQESmcoo cpmpmuhpmepm

 

 

 

 

 

 

 

.w chsmflm
A A. v Hocdm p(\ o Hocmm
llllllll :0

\ 0.
me ”Willie

i H

o o
D Honmm

Mil—l

 

m Hecmm

Mlsq

‘

 

 

Sh
capital h§p_§_ppp_productivity. Each additional sacrifice
of consumption will, therefore, move society to a new con—
sumption plateau, the height of which will depend on the
behavior of the net productivity of capital. With the
assumption of diminishing returns to capital, the greater
the level of the consumption stream, the larger the sacri-
fice of present consumption that is necessary to bring
about a given increase in the consumption plateau.

It is now clear that the behavior of C(r) is a
consequence of the net productivity of capital as society
decides to trade present for future consumption. This net
productivity is seen to depend, in turn, on the ratio of
our physical capital to labor, i.e., the aggregate capital-
labor ratio. But this is what we have referred to as the
economy's technique of production. We may therefore
directly relate the choice of technique and the consumption
plateau implied by this choice.

Panel a, Figure 9 is the Surrogate Frontier derived
above;78 Panel b is the steady-state consumption curve.
The slope of the Frontier is the capital-labor ratio; and
every point on the Frontier represents a technique of pro-
duction open to society. If we now think of the interest
rate as falling from its maximum to zero, the economy may
be said to ”choose" successive techniques each of which

displays a higher ratio of capital to labor. This will, so

 

780hapter I, pp. 7—10.

 

55

 

 

 

 

 

 

r
>7r
Panel b
Figure 9. The Surrogate Frontier and Steady-State

Consumption

   

56
long as capital can be said to have a pp; productivity,
move the economy to higher steady—state consumption levels.
But the operation of diminishing returns tells us that the
rate of interest associated with higher consumption streams
will be lower.

It thus appears that the unidirectional movement of
our consumption plateau is based on two elements: a) the
unambiguous ordering of techniques over the range of
interest rates and b) the operation of diminishing returns
to capital. Let us assure ourselves that this traditional
behavior of C(r) does not depend upon our choice of a neo-
classical production model.

The Consumption Plateau in a Heteroggpeous Real Capital
Model

Consider the heterogeneous capital good model of
Section II. First we need a way of directly relating our
factor-price frontier and the steady-state consumption
curve in this model. We know that Y = C = NNP in the
stationary state. Let the labor constraint be given by
(35) L = aLCY .

The maximum real wage is known to depend only on the pro-
duction coefficients of the consumption sector;79 and a
typical price equation for the consumption good is

(36) PC = aLCW + aKCrP .

If we now let r = O we have

79Supra, p. 36.

 

 

57
(37) PC = aLCw .
Then with the consumption good as our numeraire, i.e.,
PC 5 l, substituting (35) in (37) yields
(38) 13.: w,
i.e., the maximum wage rate is attained when the marginal
and average products of labor are equal. Thus we may
observe directly the steady-state behavior of per capita
consumption over the range of relevant interest rates as
the economy's technique of production varies.80 Consider
Figure 10.

Panel a, Figure 10, contains three discreet tech-
niques Alpha, Beta, and Gamma all possessing properties
identical to those of the techniques discussed and illus-
trated above.81 Panel b is the steady—state consumption
curve. It shows the per capita consumption that can be
attained and maintained indefinitely under each of the
three techniques.

At very high rates of interest, r > rfi, the Gamma
technique is chosen as it is the most profitable. With
this technique in use, the maximum per capita consumption
that can be attained and maintained indefinitely is given
by(%?) Thus for very high interest rates steady—state
consumption is shown in Panel b as(%r

New let the interest rate fall and wages rise, so

80Essentially the same result, taking a slightly
different route, is contained in Bruno, p£_gl., pp. cit.,
pp- 546-49.

81Su ra, pp. 35-38.

 

58

 

 

 

 

 

C = w
I:
A
31d
(L)
C 0
(E)
C i
(E)
l I
l I _
rf r? '— r
| Panel a
| l
| l
l
% ' ‘
n | I
| l
GE)“ I ‘
C 9 _ _. _ _ |——l
(f)
(9)" 1 —————
Panel b r r

Figure 10. The Factor—Price Frontier and Steady-State
Consumption

 

59

that rf< r< rg. Entrepreneurs "switch" to the Beta
technique; and steady-state consumption rises to(%)e

Finally, let the rate of interest fall to very low
levels, O< r< rf. There is now a "switch" to the Alpha
technique that raises the steady-state consumption to the
even higher level of(-%):d

What this demonstration tells us is that the question
of the homogeneity of capital is of no consequence so far
as an investigation of steady—state consumption patterns is
concerned. In this model (as indeed for any model) it is
the movement of the net productivity of capital that is
paramount. Here we have merely submerged this relation.
The variable that attracts our attention in this model is
the average productivity of labor (the ordinate intercept).
As the rate of interest falls we switch to techniques that
raise the average productivity of labor (in the consumption
sector), which means that steady—state consumption is raised
as well. However, with constant technology, the average
productivity of capital is usually taken to be simultaneous-
ly falling.82 In other words, this model contains the

assumption of diminishing returns to the capital factor.83

82Actually, all we can say here is that the average
productivity of capital in the capital— goods sector is
falling.

83Thus diminishing returns is, as Samuelson has long
maintained, independent of the nature of the capital stock
present in the model. But, and there seems to be some con-
fusion on this point, the reason is not due to any "tech—
nical fact." It is simply a matter of assumption. See
Paul A. Samuelson, Economics (6th ed.; New York: McGraw-Hill,

 

60
The conclusion to be reached is that the unidirec-
tional movement in the steady-state consumption pattern is
a consequence of the unique ordering of techniques, to-
gether with the assumption of diminishing returns to
capital.

Unorthodox Behavior of Steady-State Consumption

But now consider the reswitching case of Section II,
as illustrated in Figure 11. When a technique chosen at a
relatively high rate of interest is replaced by another
technique as the interest rate falls and then returns at
an even lower rate of interest, the traditional ordering of
C(r) is lost. One can no longer say that a fall in the
rate of interest generally leads to a higher permanently
sustainable consumption stream. As shown in Panel b,

Figure 11, when entrepreneurs "switch" from Beta technique
to Alpha technique at the interest rate r?, the new steady-
state consumption level will fag; precisely because the
average product of labor in the consumption sector decreases
(as indicated by the ordinate intercepts).

This unconventional behavior of C(r) is not dependent
on the actual return of any given technique. This may be
seen in Figure 12. In Panel a, Figure 12, we have three
discreet techniques Alpha, Beta, and Gamma.84 These
l96h), pp. 595-96. (As Samuelson says, the situation is more

complex than we are accustomed to supposing. Samuelson,
"A Summing Up," pp, cit., pp. 579-82.)

8Ll-Notice that none of these techniques gives rise to

 

 

61

 

 

i: W
A
c D
(1:)
(9)“
L
|
|
I
l | i
rf r: '7 r
| Panel a
l I
| l
s g l
L.
A) I I
I |
l
C 0
(L)

 

 

Panel b
Figure 11. Reswitching and Steady-State Consumption

 

 

62

 

 

 

\ filo

.9.

A
v

V

 

t'IO evto HO
'5

A
V

 

Panel b

Figlxre l2. "Perverse" Ordering of Steady-State Consumption

e3
techniques are unambiguously ordered with respect to the
rate of interest. Thus at “the relatively high. interest
rates r > r95, Gamma technique is chosen by entrepreneurs.
For intermediate rates of interest, rf< r<r§, Beta tech-
nique will be chosen; and when the interest rate is very
low, O< r<r§f, Alpha technique will be the most profitable
technique. There is no question of reswitching here.

The maximum per capita consumption that can be ,
attained and maintained indefinitely under each technique
is measured by the ordinate intercepts in Panel a. This is
then related directly to the various rates of interest by
the steady-state consumption curve of Panel b.

Panel b illustrates the important possibility that,
even in the absence of reswitching, a lower rate of interest
may be associated with a lower steady-state consumption
plateau. This occurs as entrepreneurs "switch" from Gamma
technique to Beta technique at the interest rate r5.

It is this generalization of the reswitching result
that makes the reswitching literature more than a mere
curiosum itself. If we take the rateof interest as
reflecting the net productivity of capital, the net pro-
ductivity of capital as reflecting the technique of

__

a linear factor-price frontier. We have discussed above
(ft. 19) the reason for convexity (concavity) of the
frontier. There is the further question, now raised by
the lack of.linearity, of whether this has any bearing on
the validity of our measure of the capital'intensity of
PPOduction. Yes it does, but we will delay discussion of
13115-5: until the next chapter.

6A
production, and the choice of technique as reflecting the
decision by society to consume more or less today in favor
of tomorrow, then we are faced with this startling proposi-
tion: the consuming public will step up consumption today
in order to disaccumulate capital (select another technique)
so that they may have tomorrow a permanently lower consump-
tion stream! Surely something has gone wrong.85 Otherwise
stated, what characteristics of the model might help explain
this unorthodox movement in the consumption plateau?

For Bruno pp p;., it is the fact that the individual
factor-price frontiers cross below the outer envelope.86
Figure 12 on page 62 parallels theirs, and it is clear that
the movement of the consumption plateau in the vicinity of
%

r2 results from the fact that the two factor-price frontiers

do indeed intersect below the outer envelope, at r'. But it

9.-
O

is important to note that this intersection occurs to the
left of the relevant switch point.
Both Panels a and b of Figure 13 illustrate the

crucial point that it is the intersection of the factor-price

85Mere demonstration is hardly sufficient, but see
Samuelson, "A Summing Up," pp. cit. , p. 578. For a numeri-
cal example of such disaccumulation, see ibid. , p.
Bruno, Lt al., pp. Lit. , pp. 552- 53, have suggested that
such behavior is explainable in terms of the capitalist

"splashing" himself with consumption today by an amount just

equal to the then permanently lower consumption he will
experience tomorrow. The crucial assumption here is that
wage earners spend all their income on consumption.

 

86Their statement is not altogether clear. Presum—
ably they mean no two individual functions may cross twice
if one of their intersections is on the outer envelope.
Bruno, pp p;., ibid., p. 549.

 

 

65

 

 

 

QOHpQEswcoo cpwpmuhvwoum mo MCHAopno HMQOHuco>coo .MH casmfim
p Hosmm o Hccem

HUN

 

 

FA

9 Hcswm w Hecwm

 

 

 

 

 

66
frontiers below the outer envelope to the lpfp of their
intersection on the outer envelope that is, at least osten-
sibly, the culprit in all this. Notice that the conven~
tional ordering of C(r) is maintained (Panels c and d) even
though the relevant individual factor-price frontiers do
intersect below the outer envelope. Thus the most we may
say is the following:

a. If production techniques can be unambiguously ordered
in terms of "capital-intensity", reswitching is not
possible.

b. If we can interpret the ordinate intercepts as an
accurate reflection of the "capital—intensity" of
production techniques, the requirement that no two
factor-price frontiers intersect below the outer
envelope to the lp£p_of their intercept on the
envelope is to require that (a) be met.

Thus the conditions previously said to be sufficient for
preventing the return of a technique (so long as interpret-
able as "a") may be said to be necessary and sufficient for

the preservation of the traditional ordering of C(r).87

IV. Conclusion

The basic conclusion emerging from a consideration

of the reswitching discussion is that the "perverse"

 

87Notice, however, that we are not saying the tradi-
tional ordering of C(r) will always be found to contain at
most one intersection of any two frontiers (i. 6. this is
not a necessary condition for the traditional ordering of
C(r)). Compare with Bruno, pp pl,, ibid. , p. 53h.

67
behavior of C(r) is independent of the actual return to any
given production technique. Yet there is a single common
denominator present in these discussions that is, in fact,
the critical variable in much that is of importance so far
as reswitching and traditional capital theory are concerned.
This critical variable is the capital intensity of the
economy‘s production techniques.

The capital intensity of production must be taken to
refer to the ratio of capital (in some sense) to labor (in
some sense) embodied in the production techniques, i.e., to
the aggregate capital-labor ratio. Furthermore, it is the
ratio of capital to labor alggg the grand factor-price
frontier (i.e., the envelope of the individual frontiers)
that is of paramount importance. Clearly the questions
raised and discussed above concern the capital intensity of
production as we move alggg this frontier. Yet, as the
reader will have noticed, the reswitching discussion gggs
Egg make explicit reference to the aggregate capital-labor
ratio along the grand frontier.

This failure has been noted by Hicks and he has put
the issue as follows:88

It already appears that to distinguish techniques by
reference to the maximum rate of profit they would
permit, at a wage of zero, is not a very interesting
way of distinguishing them; but is labor intensity,
which amounts to distinguishing them on the basis of

profit being zero, so very interesting either? May
not the whole trouble have arisen because we have

88Hicks, op. cit., p. 166.

 

 

68

asked the wrong question? Instead of looking at what

might happen in extreme positions, at the 'ends' of

the curves ought we not to fix our attention on what

happens in the neighbourhood of the actual position

where the change takes place?
In other words, what we really need to look at is the move-
ment of the aggregate capital—labor ratio abnmages rise and
interest rates fall. In the next chapter we shall investi-
gate several real capital models to see just what can be
said about changes in the capital intensity of production

as the economy moves along its grand factor-price frontier

in response to the changes in the rate of interest.

 

 

CHAPTER III

CERTAINTIES AND AMBIGUITIES IN THE
RESWITCHING DISCUSSIONS

I. Introduction

In the preceding chapter emphasis was placed on the
switches of techniques in a two-sector, heterogeneous,

real-capital model. We saw that the movement of the aggre-

 

gate capital-labor ratio was the critical issue. In a
world of fixed proportions processes and heterogeneous
capital goods the simple neoclassical results concerning
the relation between production and input and output
markets may not hold. And it may not hold irrespective of
reswitching, although reswitching provides a stronger case
against simple neoclassical theory. Fundamentally, the
simple neoclassical results hold if, and only if, the
capital-labor ratio varies directly with the wage-interest
ratio.

It is clear from the reswitching discussions that
one cannot in general order "activities" (whether this is
taken to mean a single activity or a whole matrix of acti-
vities) with respect to the wage—interest ratio. However,

it is not clear that this means that the relation between

69

70
the aggregate capital-labor ratio and the wage-interest
ratio is similarly affected.

We saw that the propositions surrounding the re-
switching phenomena are based on the ordinate intercepts
of the so-called canonical model introduced by Samuelson.89
This means, first, that the labor-output ratio in the con-
sumption sector is being used as an index of capital
intensity (i.e., as the aggregate capital-labor ratio),
and, second, that a technique of production is being
identified with a unigue aggregate capital-labor ratio.
(Unique in the sense that it is a constant along the
process frontier.)

There is no doubt that a technique of production
(i.e., matrix of activities) is an aggregate capital-labor
ratio.90 What is uncertain is whether, in a multi—sector
capital model, the ratio is unique; and, if it is, whether
the labor-output ratio in one sector serves as an adequate
index of it.

To help answer these questions three different
heterogeneous capital models are analyzed below. It will
be seen that the labor-output ratios prominent in the
reswitching literature are less capable of measuring the
capital intensity of techniques the closer one approaches

a bona fide two-sector capital model.

8C)Paul A. Samuelson, "Parable and Realism in Capital
Theory: The Surrogate Production Function," Review 23
Economic Studies, XXIX (June, 1962), pp. 193-207.

9OSupra, Chapter II, pp. 30-33.

 

71
II. Samuelson's "Parable"91

In 1962 Samuelson set out to show that the simple
neoclassical theorems are valid in a world of fixed pro-
portions and heterogeneous capital goods. To that end, he
retained the neoclassical assumptions of constant returns
to scale, competitive imputation, and one primary non-
reproducible factor of production. However, he assumed
that there are many, but not an infinite number of differ-
ent kinds of capital goods, each of which, when combined
with a very definite amount of labor (i.e., fixed produc-
tion coefficients), can be used to produce either a unit of
final output or a unit of itself. His model may be repre-

sented by the following price equations:

P = a w + a rP
C LC KC M ’

(39)
PM = aLMw + aKMrPM .
All terms in the expressions are as defined above.92
Samuelson now makes the crucial assumption that aLc =
aLM and aKC = aKM (notice that the assumption is much
stronger than that of equal proportions production).93
For each of the finite number of different capital

goods there is a set of price equations as above. Each

capital good thus gives rise to a separate factor-price

91Ibid.
928u ra, Chapter II, p. 35.

93Although this assumption contains the less strin-'
gent one of equal proportions, the reverse is not true (i.e.,
Egg = EEK does not require, for example, aKC = aKM).
aLC aLM

 

72
frontier represented by an equation of the form

_ a
((4-0) w_aiL—EE(I') 1

where we have dropped the sector notation in view of the
above assumption. The envelope of these individual fron-
tiers is the grand factor-price frontier whose characteris-
tics we wish to investigate.

First, take for the unit of output the net national
product or national income of the technique and use this
as numeraire, thereby adding an additional equation to the
system but no new unknowns. We may then represent the
frontier by

= 1 _ £2
(#1) w XE AL(r)

The AK and AL now represent the total requirements of
capital and labor under this technique.

Let us call equation (Al) the Alpha factor-price
frontier. It is plotted in Panel a, Figure 1h, as the
straight line whose ordinate intercept is g: and whose
abcissa intercept is ii. It is important to notice that

A
(L12) —% =f§=§

That is, the slope of the Alpha factor-price frontier is
the aggregate capital-labor ratio.

Now add many capital goods, Alpha, Beta, Gamma,
Delta, et cetera. Their factor-price frontiers are shown
in Panel b as o< a] , flfll’ r X], and JOCI. The heavily
shaded outermost line comprises the grand factor-price

frontier. Since the slope of each individual factor-price

 

73

Hope: prfimwu Hmfioomm m_comaossdm .da ensmfim

n Hocwm w Honwm
mm 5

.W {A \‘ b0 H 4

1 .HA

.H

 

 

 

 

7h
frontier is necessarily the aggregate capital-labor ratio
as well.

Thus, with the exception of the kinks, we have here

a striking similarity to the Frontier yielded by the J. B.
Clark neoclassical capital model.91L In particular, we have
a special validation of the proposition that a fall in the
rate of interest will raise the capital-intensity of the

economy's production techniques.

Now consider the relation between the capital intens-
ity of techniques and the labor-output ratio in the consump-
tion sector. Look at the ordinate in Panel a, Figure l#%

113 > Eli so BL) AL . Since A1, = 1%, we can unequivocally say
that the labor~output ratio in the consumption sector is
greater when Beta is used than when Alpha is used. This
result is the same as that yielded by equation (#2); there-
fore, one may, without hesitation, use the labor—output
ratio in the consumption sector as an index of the capital
intensity of techniques in this model.

But notice why this is so. The assumption of
identical unit production requirements in each industry
means that AL =4% = fi. That is, the labor-output ratio in
the capital sector is the same as that of the consumption

sector. Therefore the labor-output ratio in the consumption

guThough potentially important, the kinks in the
grand frontier are of no real consequence in this model.
The discontinuity means that the aggregate capital-labor
ratio falls discreetly as the interest rate rises. The
meaning of this discontinuity for more general capital
models is discussed below in Section III, Chapter IV.

75

“(r
ml

 

 

 

Figure 15. A Switch of Techniques in Samuelson's Model

76
sector i§_the labor-output ratio in the economy.95 In this
model, no distinction is made between the two sectors or
between either of them and the economy as a whole. Other-
wise stated, this is a single-sector capital model.

We may talk about a consumption—good industry and a
capital-good industry but the distinction is one of labels
only. In fact, the economy is characterized by a single
production function.96 The level of aggregation is the
same as that of the variable-proportions neoclassical model.
Of course, the level of aggregation does not itself validate
the use of the labor-output ratio in the consumption sector
as an index of the capital intensity of techniques. The
labor-output ratio must be a single-valued function of the
aggregate capital-labor ratio. Thus there is implicit in
this model the assumption that increases in the aggregate
capital—labor ratio raise the average productivity of labor
in the economy.97

 

9SThe labor-output ratio in the economy is a weighted
arithmetic mean of the labor-output rfitio inL each sefitor
considered separately. Thus we have cec( L) +£om(M

= C(OC +6014), and since (LDC +M-EQM

BY)

assumption so

-——-'—, —
wehavelfi 5.9.}! Q
Q
96Or to be more precise, the economy is characterized
by a single production function given some equilibrium wage-
interest ratio. For then only one of the capital goods is
ever relevant.

97In the reswitching literature, this aspect of
simple neoclassical theory is taken for granted by critics
and exponents alike. However, for an example of switching
that casts some doubt on this relation see below p. 83,
and the comment in footnote 103.

77

A numerical example brings out clearly the special
nature of Samuelson's model. Let the production relations
for a particular capital good (say Alpha) be given by
K),

V.

c = min(%L
(Ll-3)

M = min(%L K) .

‘0

Thus we see that the production functions are really 3
production fUnction; the two goods are economically identi-
cal in that their equilibrium price must be identical.

New add a different capital good (e.g., Beta) with

its production relations

2K) ,

V.

C min (fiL

min (%:L

The corresponding factor-price frontiamsare, respectively,

(LLLL)

M 2K) 0

V0

2 2
The frontiers are shown graphically in Figure 15. As the

(t5) w=-:-L---l-(r), w=ff-§<r>.

wage falls from its maximum toward zero, the economy
switches from the Alpha technique to the Beta technique at
the real wage of two-twelfths. The labor-output ratio in
the consumption sector rises and the aggregate capital-labor
ratio falls from one-half to one~eighth in accordance with
the neoclassical parable.

Samuelson's model, like the J. B. Clark neoclassical
model, leaves no doubt about the capital intensity of pro-

duction as the wage-interest ratio varies.98 And it is

 

98Perhaps it should also be mentioned at this time
that Samuelson's model, like the J. B. Clark neoclassical

‘ 78
equally clear that, in his model, the labor-output ratio
(in the consumption sector) is an adequate index of the
capital intensity of techniques. Fundamentally, these
results flow from what is essentially a single-sector

assumption.99

III. Hicks' Two Rulesloo

Let us now consider a model that has been emphasized
by Hicks. It differs only slightly from Samuelson's model;
but there is a significant difference in results. Hicks
extended Samuelson's model by relaxing the assumption on
the fixed coefficients so as to require equal-proportions
production only. That is, the production relations are

described by

model, leaves no doubt as to how entrepreneurs will behave
as the wage-interest ratio varies. Any "perverse" behavior
is completely ruled out by the assumption on the production
coefficients. That is, it is impossible for entrepreneurs
to adopt more capital-intensive processes as the rate of
interest rises.

99The above suggests that Mrs. Robinson's irrita-
tion with the neoclassical production function might
equally well have been directed toward the level of aggre-
gation implied by it. This appears to be the really
crucial" assumption so far as the results of simple
neoclassical theory are concerned.

lOOJohn Hicks, Capital and Growth (New York: Oxford
9 PP: B NEH

University Press, 196 l 3-

79

3
ll

(us) 0 = min(3;l{EKc ,ElfiLC) ,

8.
KM _ aKC a a
T‘r’ KM" KC
LM LC

Each set of relations (46) describes one technique. Again
it is easy to show that each technique yields a factor-

price frontier whose equation has the general form

__.1_ m
U47) w—aLC - aLLC(I')
Hence

QK_E_@_E
(“8) ‘ dr aLc Lc

The slope of the frontier does not give the aggregate
capital—labor ratio; and hence it is possible that the neo-
classical relation between factor proportions and relative
factor prices will not hold. Let us see what can be said
about the capital-intensity of techniques in this model.

Consider Figure 16 in which two linear processes are
shown. Visually, EIE'>-B%5' Hence ch > aLC' Since 8L0 =
g, we can unequivocally say that Beta is labor intensive

relative to Alpha in the weak sense that the labor-output

 

ratio in the consumption sector is greater when Beta is used
than when Alpha is used. This is the first of Hicks' two
rules.
0 l 1
Now look at the abscissa in Figure 16. FEM > iii,
which implies that aKM > bKM. The capital-output ratio in

 

 

80

as

EEIH

 

 

1 1;?

am 13104

Figure 16. An Equal-Proportions Production Capital Model

81
the capital-good sector is greater with Alpha technique
than withBeta. So again in 3.3222 sense we may say that
Alpha is capital intensive relative to Beta. This is the
second rule.

In this model, the technique that is labor intensive
in one sector is always less capital intensive in the
other. Hence as the wage-interest ratio rises the tech-
niques become more capital intensive ip.§h£hwg§k_§gp§2_
that the capital-output ratio in the capital-good sector
increases and/or the labor-output ratio in the consumption-
good sector falls.

Thus the two rules are consistent with one another
and it makes absolutely no difference which of them is used
to classify techniques. In the reswitching literature the
first rule is implicitly adopted.101 We shall now show
that even in this highly simplified model, a classification
of techniques according to this rule may be inconsistent
with the associated aggregate capital-labor ratios; and
these are really the proper criteria for classification.

First a word about the aggregate capital-labor ratio

itself. In a multi-sector capital model the aggregate

 

101In the sense that statements about capital in-
tensity and/or steady-state consumption rest on the ordinate
intercepts which measure net final product per man. This,
of course, is the reciprocal of the labor-output ratio in
the consumption sector. See, for example, Luigi L. Pasi-
netti, "Changes in the Rate of Profit and Switches of Tech-
niques,” The uarterl Journal gnyconomics, LXXX (November,
1966), pp7_§l3-1K, or Michael Bruno, Edwin Burmeister, and
Eytan Sheshinski, "The Nature and Implications of the
Reswitching of Techniques," ibid., pp. 548-h9.

 

 

- 82

capital-labor ratio is a weighted arithmetic mean of the
proportion of capital to labor in each sector considered
separately. Thus we are interested in

(#9) 11,2: COG (2%)+®M (2%")

The weights are simply the value shares of each sector in

the value of total output

=..__(.3__.. o.) = .314...

These weights will, in general, depend on both the relative
prices of the two goods and the level of income under the
various techniques. As the price of machines falls relative
to the consumption good "workers" may become "capitalists"
and thereby substitute the relatively cheaper for the
relatively more expensive good. Thus there is a market
substitution effect that will accompany a change in the
wage-interest ratio. In addition, there may be an income
effect due to a lower (higher) income accompanying the
adoption of less (more) capital-intensive techniques by
individual entrepreneurs.

In view of the additional relations that one must
consider in a multi-sector capital model, there is little
reason to believe that the capital intensity of techniques
is adequately indexed by a.rule that fails to account for
them. However, it is precisely these considerations that

are eliminated by the equal-proportions production

83

102 Hence one might reasonably expect that the

assumption.

rule will work for this model. But now look where it takes

US.

Let the production relations for two different tech-

niques be given by

c = min(L ; ix) ,
M = min(lL 3 ELK) 9

2 6

(51) l 1
C = min(-2-L 3 EEK) ,

M = min(L ; EK)

Each individual factor-price frontier is shown in Panel a,
Figure 17. The Alpha frontier has a slope of six, the Beta
frontier a slope of one. The two techniques are equally
profitable at a real wage of four-tenths. As the real wage
falls from one to zero entrepreneurs switch from Alpha to
Beta. The labor-output ratio in the consumption sector
increases and the aggregate capital-labor ratio falls from
three to two. The rule does appear to work.

Now let the production relations be

C = min(L ; -]2=K) ,

l
MK) ,

éK),
min(L ;.%K) .

‘00

= mint-LL
(52) 2

‘00

M
C = min(%L
M

 

102By assumption (ES—)0 = k
write equation (h9) of text as.%.=

inasmuch as 63C + (Q)! E 1. Thus 3% =

M5 hence one may
k

)+(l-—‘-b)(k)

8L!-

. DI . .
w
B
as
m
up
a
5 &
p
H

a Honwm
comm
9H

.0
mm

 

 

 

 

 

 

85

The factor-price frontiers are shown graphically in Panel b
Figure 17. The Alpha frontier has a slope of four and the
Beta frontier of one and one-half. The techniques are
equally profitable at the real wage of two-tenths. As the
wage-interest ratio falls, entrepreneurs switch from Alpha
to Beta and the labor-output ratio in the consumption
sector increases. But in this example the aggregate
capital-labor ratio increases also, going from two to three.
Beta is definitely capital intensive relative to Alpha.
Nevertheless, the rule classifies Beta as labor intensive
relative to Alpha because the labor-output ratio in the
consumption sector is greater for Beta than for Alpha.

This model is consistent with either an increase or
a decrease in the capital intensity of techniques as entre-
preneurs switch techniques in response to changes in the
wage-interest ratio. The above examples clearly show that
the result depends on the assumed set 2; technical alterna-
tizgg, Hicks' two rules may be used to classify techniques
so long as the technology is consistent with neoclassical
switching. But as Mrs. Robinson and Mr. Sraffa have made
clear, it is possible that the technology is as indicated
in equations (52). This is sufficient, by itself, to prove
that one cannot, in general, infer anything about the aggre-
gate capital-labor ratio from the labor-output ratio in the
consumption sector. Thus the implicit adoption of the first
of Hicks' two rules as the basis for a) generalizing the

reswitching result, and b) drawing inferences about the

86
behavior of steady—state consumption levels is definitely

unwarranted.103

III. The Implications Of A Nonlinear

Process Frontier

Let us now change Hicks' assumption slightly so as
to allow the intrinsic factor intensities of the capital
and consumption good to differ. Specifically, assume that
production in the consumption-good sector is more capital
intensive than production in the capital-good sector. Thus

the production relations are described by

C = min(§—-KC , EICLC)

(53) M = aim-3541M , a—lflfin .

aKC aKM
aLC ELM

9

Each set of relations (53) describes one technique. Each
technique yields a factor-price frontier whose equation has
the general form

(st) w=l ‘aKMr,
aLC+ AI”

 

103The equations (52) also point up what is perhaps
the most fundamental question raised by the reswitching
discussions. In the analysis of steady-State Consumption

levels, the ordinate (El_) measures net consumption per man.

Thus, referring to Panel b Figure 17, an increase in the
rate of interest raises the aggregate capital-labor ratio
and steady-state consumption falls. This suggests a sur-
prising possibility. It implies that even if the aggregate
capital-labor ratio is a single-valued function of the
wage-interest ratio, a higher steady-state consumption
level may be associated with a higher rather than a lower
rate of interest.

87
where A = aLMaKC - aLcaKM .

The frontier is negatively sloped inasmuch as

(55) W = imam + A =.aLMaKC
a? (are + Ar)2 ”Loki”?

 

It is convex to the origin by virtue of our assuming that
A > 0. To see this first write the price equation for the

capital good:
_ P
(56) PM — aLMW + aKMr M .
Now divide both sides by PM to obtain

(57) 1-aKMr >0.

Then from equation (Sh) it is seen that

(58) aLc+Ar20.

Hence the frontier is convex or concave to the origin

according as A Q 0 inasmuch as
(59) fl ___ 2A(aLc 4- AI') aLM9-KC

dr2 (aLC + AHA

The fact that the frontier is nonlinear is very
important. Assuming continuous full employment of capital
and labor, it implies that the economy's endowment of
capital and/or labor is changing as we move along the pro-
cess frontier.lOLL The frontier is based on given, unchang-
ing proportions of capital to labor in each of the two
sectors considered separately. These sector ratios are
'what they are regardless of any variation in the wage-

interest ratio.

 

10LLIn the absence of this assumption it implies that
capital and/or labor is in various stages of idleness.

88

However, as the wage-interest ratio falls the price
of the relatively labor-intensive good falls relative to
the price of the relatively capital-intensive good.105
This implies a change in the composition of output in favor
of the relatively labor—intensive sector. Thus, referring
to equation (M9), it is clear that the aggregate capital—
labor ratio must fall. The reason is that if the relative-
ly labor-intensive sector is to absorb all the capital
released by the relatively capital-intensive sector, the
economy‘s endowment of labor must rise. Or, if the labor
force is taken to be a constant, the change in output
composition implies that capital has been disaccumulated.
Either way, the point is that strict adherence to the full
employment assumption implies that the factor endowment
ratio is a variable along any given nonlinear, fixed~
proportions process frontier.106

Now consider Figure 18 in which two nonlinear
frontiers are shown. Visually, 3%5 > 5%6 and 5%E > EHMI'
Hence alg < bLC and aKM > bKM. Following the rules of the
preceding section, technique Beta is classified as labor
intensive relative to technique Alpha. But again it would
be easy to construct an example in which, at any wage-
interest ratio, the aggregate capital-labor ratio implied

by technique Beta is greater than that implied by technique

lOSA more rigorous proof of this statement is given
in the next chapter.

106Ibid.

Figure 18.

 

89

 

_1_
8LKM

A Nonlinear Capital Model

 

90

Alpha (below we offer such an example). In such a case one
would classify Beta as capital intensive relative to Alpha.

Suppose it is known that for each technique con-
sidered separately the aggregate capital-labor ratio is
always greater for Beta than it is for Alpha. Does this
mean that a switch from Alpha to Beta raises the capital
intensity of techniques? In an equal-proportions produc-
tion model it certainly does. However, the interesting
aspect of this model is that the outcome is uncertain. It

To illustrate, consider the techniques whose produc-

tion relations are
_ l
C“min(L,§K)’
min( L , 2:1!) ,
2

(60)

and

0
ll

.1. _3L
min(2L ’lZK) ,

l
M = min(.'§L, K)

 

107This is an important proposition as it highlights
What is perhaps the basic fallacy contained in the rules
enumerated by Hicks and implicitly adopted in the reswitch-
ing literature. Under the best of circumstances, the rules
would classify techniques in a manner that is consistent
With a classification according to the associated aggregate
capital-labor ratios in the sense that, if Beta were known
to imply a higher aggregate capital-labor ratio than Alpha
for all rates of interest, the rules would classify Beta
as the relatively capital-intensive technique. But any
such classification is based on the implicit assumption
that a switch of techniques does not itself affect the
aggregate capital-labor ratio. The basic point developed
in this section is that such an assumption is unwarranted
in a fixed-proportions, heterogeneous capital model that
allows the intrinsic factor intensities of the goods to
differ.

91
The two techniques are plotted in Figure 19. Both process
frontiers are convex to the origin because the consumption
sector is assumed to be the relatively capital-intensive
sector. The two techniques can be shown to be equally
profitable at a real wage of three-tenths.

As the real wage falls from its maximum toward zero,
entrepreneurs switch from Alpha to Beta and both sectors 1
become more capital intensive. The question is whether the I
total proportion of capital to labor has increased.

Look at Table I. Notice that the aggregate capital-
labor ratio is greater for Beta than it is for Alpha.
Demand is assumed to shift from the capital-intensive
sector to the labor-intensive sector as the wage-interest
ratio falls. Hence the aggregate capital-labor ratio is
falling for movements down each frontier considered sepa-
rately. New consider the point ($1p’) in Figure 19. As
entrepreneurs switch to the Beta technique, the weight
going to the relatively labor-intensive sector falls,
remains unchanged, or increases depending on the movement
of relative commodity price §£.the switch pgigt.

If, as in this example, the switch to Beta causes
relative commodity price to fall,108 the economy's tech-

nique need not become more capital intensive even though

‘.

 

108As the reader may easily verify, relative com-
modity price (fig) falls from .58 to .12 as a result of the

switch from Alpha to Beta. A simple expression for rela-
tivi fiommodity price is given in the following chapter,
p.1.

92

 

 

 

0.50 1.00

Figure 19. A Switch of Techniques in a Nonlinear Model

.n. _.-.‘_-..‘~_'__,,

93

TABLE I
AGGREGATE CAPITAL-LABOR RATIOS FOR TWO TECHNIQHES

 

Aggregate Capital- Aggregate Capital-

 

Output Composition: £abfiriRatigi h gablolriRatig:t
«)0 03M ec n %%? p a ec n q;: e a
1.0 0.0 5.0 6.0
0.9 0.1 h.6 5.7
0.8 0.2 h.h* 5.5
0.7 0.3 4.1 5.2
0.6 0.h 3.8 4.9
0.5 0.5 3.5 h.7
0.1; 0.6 3.2 his?
0.3 0.7 2 9 h.1
0.2 0.8 2.6 3.9
0.1 0.9 2 3 3.6
0.0 1.0 2 0 3.3

 

*Referred to in text, p. 9h.

9h _

all entrepreneurs adopt more capital-intensive techniques.
For example, referring to Table I, suppose the fall in
relative commodity price implies a shift in the composition
of output from (DC = .8360M = .2) to (log = .14. 3wM = .6).
Then the aggregate capital-labor ratio is unchanged at the
switch point (Sig ) though the proportion of capital to
labor is higher in each sector considered separately. The
explanation for this apparent paradox is contained in
equation (A9). At the same time that entrepreneurs are
switching to the Beta technique, relative commodity price
is changing so as to automatically increase the weight
going to the relatively labor-intensive sector.109 The
aggregate capital-labor ratio need not rise so long as
substitution in consumption is sufficient to overcome
"perverse" substitution in production.

Thus it is demonstrated that the aggregate capital—
labor ratio may fall continuously for movements down the

grand factor-price frontier even though all entrepreneurs

 

adopt more capital-intensive processes. The outcome

depends on the movement of relative commodity price and the

readiness of individuals to substitute goods in consump-

tion.110 The relation between factor prices, factor

109For an interesting graphical demonstration that
the aggregate capital-labor ratio may remain unchanged when
both sectors become more capital intensive, see Wolfgang F.
Stolper and Paul A. Samuelson, "Protection and Real Wa es,"
Review 2; Economic Studies, XI (November, 1941), pp. 6 -69.

110Our discussions implicitly assume that the income
effect of a change in techniques is of the "right" kind.
Thus, in the above example the income effect is assumed to

95
endowments, and relative commodity price in both the
variable and fixed coefficient two-sector production model

is examined in greater detail in the following chapter.

favor the relatively labor-intensive sector. But even if
the income effect favors the capital-intensive sector, the
above statements are valid so long as the market substitu-
tion effect dominates the income effect. Thus it is the
market substitution effect that is most fundamental.

 

CHAPTER IV

INTERRELATIONS BETWEEN FACTOR PRICES,
FACTOR ENDOWMENTS, AND RELATIVE
COMMODITY PRICE

I. Introduction

The economy's technique of production is the prevail-
ing aggregate capital-labor ratio. This ratio, however, is
a value-weighted arithmetic mean of the proportion of
capital to labor utilized in each sector separately. Thus,
in a disaggregated capital model, the capital intensity of
techniques depends in general on (1) the technology of the
economy, and (2) the distribution of demand between sectors.
When one makes the two fundamental assumptions of full
employment of both factors and perfect competition, switches
of techniques imply: (a) factor prices are changing,

(b) the proportion of capital to labor in one or more sec-
tors is changing, (0) relative commodity price is chang-
ing, and (d) the composition of output is changing.

The reswitching discussions implicitly assume that
the distribution of demand is fixed. In addition the
”normal" switch is taken to mean that there has been a

change in the proportion of capital to labor 13 one sector

96

 

97
2211.111 Thus when a switch occurs it implies that the
economy‘s endowment of capital and/or labor is changing.
Below it is shown analytically, for both the Clark and
heterogeneous capital models, that in general movements
along the factor-price frontier imply a change in the
economy's endowment of capital and/or labor. When the
labor force is taken as a constant (the usual procedure in
these discussions), any movement down the frontier implies
that capital is being disaccumulated. However, switches of
technique in a heterogeneous capital model involve a move-
ment from one frontier to another. Thus, as will be seen
below, a switch of techniques need not mean that the factor
endowment ratio has changed. When it has, it may have
moved in either direction.

With technology specified, changes in the factor
endowment ratio have a predictable impact on relative com-
modity price. In general, it can be shown that, for move-
ments down the frontier, the price of the relatively
labor-intensive good falls relative to the price of the
relatively capital-intensive good. But in a world of
fixed-proportions processes and heterogeneous capital goods,

the question is what happens to relative commodity price

lllLuigi L. Pasinetti, "Changes in the Rate of
Profit and Switches of Techniques," The uarterl Journal
of Economics, LXXX (November, 1966), p. Ell. See also,
Eichael Bruno, Edwin Burmeister, and Eytan Sheshinski, "The
Nature and Implications of the Reswitching of Techniques,"
ibid., p. 5h2, where this proposition is presented as a
theorem.

 

98

at th£_pgipt where entrepreneurs are switching techniques?
A priori, little can be said about the behavior of relative
commodity price at this point. It may fall, remain un-
changed, or rise. An example of each is offered below.

Finally, the reswitching phenomenon is reconsidered
taking into account the behavior of relative commodity
price at the switch point. It is shown that the neoclassi-
cal relation between the aggregate capital-labor ratio and
the wage—interest ratio is virtually impossible to destroy
so long as reswitching occurs after the economy has passed
some critical rate of interest and/or relative commodity
price falls as a consequence of entrepreneurs returning to

some previous technique.

II. The Model: Variable Coefficients112

The technology is described by the columns of the
A matrix
aLM aLC
(62) A =
aKM aKc
where the aij denote the quantity of factor 1 required to

produce a unit of commodity j. The input coefficients are

assumed to vary neoclassically with the wage-rental ratio:

112For convenience the complete model is reproduced
here. A similar model using machine rental rather than the
interest rate can be found in Ronald W. Jones, "The Struc-
ture of Simple General Equilibrium Models," Th2 Journal 23
Political Economy, LXXIII (December, 1965), pp. 557-72.
The present chapter, especially parts II and III, is heavi-
1y indebted to this very instructive article by Jones.

99

W
(63) aij = aij(§) , i = L, K, j = c, M.
By definition

(éu) q = rPM .
That is, the quasi-rent of the capital good equals the

 

product of the rate of interest and the price of the capital
good.

The technology is assumed to exhibit constant re-
turns to scale so factor demands are given by the product

of the unit production requirements and the level of output

 

in each industry. Assuming full employment of both factors
gives
0+9. M

(65) L aLC LM 3
(66) K = aKCC + aKMM .

The sum of the sector demands for each factor must absorb
the total supply of each factor.

Assuming perfect competition, the equilibrium prices
of the two goods are given by
(67) p = aLMW + aKMq ,
(68) l = aLCW + aKCq ,
where p is the consumption-good price of machines (i.e.,
PC a 1). Equations (67) and (68) state that the equilibrium
market prices of the goods reflect their unit costs of pro-
duction. Finally, assume that the ratio of commodities

demanded depends solely on relative commodity price.113

113In this paper we are primarily interested in the
implications of changing relative commodity price for the

100
(69) g = f(p)
The basic structure of the model is contained in

equations (65) - (69), and the immediate problem is to con-
vert these into equations of change. The latter can then
be used to bring out the nature of the relation between
factor prices, factor endowments, and relative commodity
price along the grand factor-price frontier.

Let A denote the relative change in the variable x,

i.e., i = %%. Taking the proportionate rate of change of

both sides of equations (65) - (68) we obtainzllLL

reswitching argument and its derivatives. Hence the above
demand specification. However, in general, there will be a
change of income associated with a change of techniques,
thus one could write equation (69) as M = f(p, ¢) where ¢ '

is to be interpreted as a shift parame er representing the
level of real income in the economy. Then in place of equa-
tion (8h) below we would have, after taking the proportion-
ateAchanges of both sides of equation (69), (M - C) =

~d’p + gyy, wharecfi’represents the change in demand compo-
sttion as a result of a one percent change in relative
commodity price:

M
seal-is

and 6y represents the change in the composition of demand
as a result of a one percent change in real income:

Y M
5y 412)“)
Thus (if in gegeral 87>0) demand is biased toward the M
sector or the C sector according as a change of techniques
raises or lowers the level of real income in the economy.
This more general demand specification follows closely that
of Akihiro Amano, "Determinants of Comparative Costs: A

Theoretical Approach,” Oxford Economic Papers, XVI (Novem-
ber: 196“): PP- 395-97.

 

llLl-The procedure used here has been set out in terms
of finite differences in Chapter Two of J. E. Meade, A_Npp-
classical Theory pf Economic Growth (revised edition;
London: George Allen and Unwin Ltd., 1962). One must take

101

(70) >\LMM + )LCC = i - (ALMQLM + Noam) ,
(71) )‘KMM + 7‘ch = I? - (>‘KMQKM + CKCaKC) ,
(72) 0m»? + GKMq =5 — (Gangs, + GKMaKM) ,

(73) excl? + 9K0?1 = - (GLCQLC + excaKc)

The Alcoefficients measure the proportion of factor i used
in sector j.115 Thus, for example,

(7A) 115 + )LM = l

The fraction of labor used in the consumption sector plus

the fraction of labor used in the capital sector must add

to unity by the full employment assumption. Similarly for
)KC and )KM-

These terms may be used to define a A matrix

ALM ALC

<75) A=
)KM AKC

whose determinant116

(76) W =)‘LM -)KM =XKC - )Lc .

the total differential and convert this into proportional
terms.

llSFor example, we have: L = aLCC + aLMM. Taking
the total differential gives dL = aLCdC + CdaLC + aLMdM +
MdaLm, which in proportional terms is
dL = (amC)dC + (aLcC)daLC + (aLMM)dM + (amM)daLM .
T L c L' am L M L aLM
Rearranging terms then gives equation (70) of text.

 

 

 

116Since each row sum is unity we have, for example,
PKG = (1 - hKM) thus:

\M = ALMAKC - )KMALC
ALMu- Am) — [Am/[(14:14)]
ALM - Am .

ll

102
is positive or negative according as production in the
capital-good sector is labor intensive or capital inten-
sive relative to production in the consumption—good sector.
The (9 coefficients, by contrast, measure the rela-

tive share of factor i in sector j.ll7 Hence we have, for
example,
(77) 9m +9Kc = 1
The share of labor in the consumption sector plus the share
of capital in this sector must add to unity by the zero
profit condition (equations 67 and 68). Similarly for SIM
and 9194. These coefficients serve to define a Gmatrix

GiM etc

ekM ETD

whose determinant is

(79) |0|== 6%14- etc = 5th - GKM .

Conditions sufficient to determine the aij may be formu-

(78) 9 =

lated,118 and substituting for the Sij in equations (70) -

 

117For example, p = aLMw + aKMq so that in propor-
tional terms
in = am as + ELM damp 31041 dq + qamclam),
P p i”) p (ELM p (I?) "ET”EE§
Rearranging terms gives equation (72) of text.

118The aaij may be determined in the following manner.

(ALMa LM + ALCa LC) shows the percentage change in the total
quantity of labor required by the economy as a result of
changing factor proportions in both industries at a given
level of output. The extent of this change depends on the
elasticities of substitution between factors in each in-
dustry. In relative terms theseAmay be defined as

/‘
CHM: aKM ‘ aLM , and 46 = aKC - aLC , Now consider the 9
A A A

w - q w - q

(73) yields:

(80) Amfi+xm8=i+éL(t-§-§
(81) Amfi + Ame = R - (ma—L5)
(82) em)? + em? =fa<1 -em) ,
(83) 9M + ext?" = - emf),

where 5L =?‘LM 9mm +7‘LC excfc ,

6K =}\KM SIM/M +?\Kc GLCO’C -

The¢775 denote the elasticities of substitution between

factors in each industry.

terms, as:

A A
a - a ,
a’j = _KJ_'_ LJ 3 J =

(iv-ii)

They are defined, in relative

One may interpret the<§’s as reflecting the aggregate per-

centage saving in labor (51) or capital (6K) inputs result-

ing from the substitution of one factor for another in

response to changing factor prices at unchanged outputs.119

&
wei hted average of the aij's. For
GTE KC) shows the percentage change
producing the consumption good as a
factor proportions at a given level

A
example, (EtgaLc +

in the total costs of
result of changing
of out ut. Thus for

bothAindustries we must have (SLMaLMI+ GEM KM) = Q,

(emaLc + QicaKc) = 0-

That is, the EBweighted average of

changes in the coefficients must equal zero in each in—

dustry. This result follows from equations (63) and (77)
of the text. Solving these equations in pairs then yields
the desired result:

A

aLj=-9Kjd’j(fi-$-fw)s j=M,C

/\ A

aKj= GLj/jm-r-fi); j=M,C

119Perhaps this is better seen by writing directly,

for example,

SL = -(>‘LMaLM + kmam)

(iv-(i)

 

10h

It is important to notice that these equations no longer
contain the quasi—rent of the capital good (a). In rela-
tive terms equation (6h) becomes 8 = E + 5. We have
substituted for 3 since it is the rate of interest that
will concern us in what follows.

Finally, taking the proportionate changes of both
sides of equation (69) serves to define the elasticity of

substitution between commodities on the demand side:

(at) a/D=g .

The comparative static properties of the model are
now exhibited in equations (80) - (83). Given the wage-
interest ratio, and hence relative commodity price, the )
matrix serves to link factor endowments to commodity out-
puts in the same way as the G} matrix links relative

commodity price to the wage~interest ratio given factor

endowments.120

120The link is a technological one inasmuch as the
rates of growth of outputs depends on the sector capital—
labor ratios whenever capital and labor grow at different
rates. See Jones, 23. 313., pp. 561—62. Also below,
footnote 114 Similarly, the relation of factor prices to.
relative commodity price depends on the technology speci-
fied. Solving equations (82) and (83) directly for VJ and
r gives:

A = GKC A = '(GLC + IQI )P which, in View of equation
(79) of the text, may be wnitten as
(W - p)= '9' , r (9L0 + (9| )p"

Thus if the capital sector is the relatively labor-intensive
sector, a rise in the price of machines relative to the
price of the consumption good raises the real wage by an
even greater relative amount and the rate of interest falls
That is, w > p > r where r < 0. One might say that the

 

 

105
The model may now be used to show: (1) with the
consumption sector relatively capital-intensive, relative
commodity price varies directly with the factor endowment
ratio (capital/labor) and (2) the factor endowment ratio
varies directly with the wage-interest ratio. Together
(1) and (2) imply that relative commodity price falls con-

tinuously as one moves down the grand factor-price fron-

tier.121

Solving equations (80) and (81) one may obtain the

change in the ratio of outputs produced:

8 A—A=(_£_:E c{1+ch "-"_".
( 5) (M 0) [Al +-__7fir__ (w r p)

Subtracting (83) from (82) gives the change in the wage-

interest ratio:

A

(as) (iv-s) =_1_:_|_Q|_p,
IE” 12
so that, by substitution, 2

change in relative commodity price has a magnified impact on
factor prices. Jones, 23. git. In terms of our discussions
this simply means that as we move along the factor-price
frontier relative commodity price behaves as shown in equa—
tion (86) of the text.

All of this implies that factor endowments have no
effect on factor prices (the factor-price equalization
theorem). Paul A. Samuelson, "Prices of Factors and Goods
in General Equilibrium,” Review 92 Economic Studies, XXI
(l953-5h), pp. 5-6. However, as Samuelson notes on page 7,
ibid., with commodity prices determined within the model,
factor endowments will influence factor prices. Hence the
qualification in the text. This last fact is also reflected
in equations (89) and (97) of the text.

121The relation of the wage—interest ratio to rela-
tive commodity price is given directly by equation (25) of
the text.

 

122Substituting (w - 9) for (w ~ 9 - p) gives

106

A A i _ I? A
(87) (M-C) =l—lw =0’sp ,
where 0’s = .1 __ (61 + 6K) a

BI N

may be interpreted as the elasticity of substitution between
commodities on the supply side.123 The change in relative
commodity price is then given by the mutual interaction of
supply (equation 87) and demand (equation 8h):

(88) A = _(fi - i)
p IM ((3%))

Equation (88) shows that an increase in the economy's endow-
ment of capital relative to labor implies an increase in the
price of the relatively labor-intensive good relative to the
price of the relatively capital—intensive good. With tech-
nology given, the extent of the change in relative commodity
price depends on the substitution properties of the system.
There are two essentially equivalent explanations for
this impact of changing factor endowments on relative com-
modity price. First, an increase in capital relative to
labor raises the output of the relatively capital-intensive
sector relative to the output of the relatively labor-

intensive sector.12u Thus relative commodity price moves

fl‘fi-psowehave 1 (3.
I94 1%!
123Along the transformation schedule. 43 is a com-
posite of the elasticities of substitution between factors
in each sector. Supra, p.

 

lauJones,pgp. cit. At constant commodity prices
equations (19) and (20) may be solved directly for M and 8.

The solutions may be written as (fi — i) =«lkfiLLi - R), and
l/\'

107
in the direction indicated by equation (88). Second, so
long as there is any substitution in consumption, the
increase in the factor endowment ratio implies that the
price of the relatively more abundant factor (capital) has
fallen relative to the less abundant factor (labor).125
Thus the equilibrium price of the relatively labor-intensive
good will be higher relative to that of the relatively
capital-intensive good.

Substituting equation (88) into (86) gives directly
the change in the factor endowment ratio that is implied by
a changing wage-interest ratio:

(89) (ii—i) =JEUALLM92m—e)
l + [9

I
Equation (89) shows that, regardless of the technology,126
the factor endowment ratio (capital to labor) declines for
movements down the grand factor-price frontier. Suppose

that the labor force is given and constant (the usual

 

.. .. “hi,....l_.-_......._——n

A
(6 - fi) = “KM (K - i). Thus the sign of IA] is crucial.

With lhl>'0 an increase of capital relative to labor has
what JonesAcalls aAmagpification effect on outputs. In
this case C > K ) L > M. That is, the output of the rela-
tively capital=intensive sector is growing faster than the
endowment of either factor alone. These magnification
effects are dampened by price changes but in no case can
the direction be reversed.

125With substitution in consumption, it will gen—
erally be impossible for all of the relatively greater
quantity of capital to be absorbed solely by an expansion
of the capital—intensive sector. Thus some alteration in
factor proportions would be indicated.

126Since, as reference to equation (79) in the text
shows, the denominator is necessarily non~negative.

108
assumption in much of the reswitching literature).127 Then
every fall in the equilibrium rate of interest (rise in the
real wage rate) implies that capital has been accumulated
by an amount that depends on: (a) the elasticity of sub-
stitution between commodities in production, (b) the elas-
ticity of substitution between commodities in consumption,
(c) the differential between the proportions of labor and
capital used in each sector, and (d) the differential
between the relative share of labor in each sector. These
magnitudes cannot be specified §.priori but it is clear
from equation (89) that, so long as there is substitution
between commodities either in production or consumption,
every fall in the rate of interest implies that capital

has been accumulated.128

 

127See, for example, Bruno, Burmeister, and
Sheshinski, 22. cit., p. 5&6.

128When the two sectors are equally capital inten—
sive, equation (89) of the text breaks down. We have
shown in Chapter III that when this is the case information
about the sector capitalwlabor ratios is all we need.
Supra, footnote 102. Under the above assumptions, an in—
crease in the wage-interest ratio raises the proportion of
capital to labor utilized in each sector separately. Hence
the aggregate capitalulabor ratio must have risen.

The relation between the wagewinterest ratio and the
factor endowment ratio implied by equation (89) does, of
course, presuppose a definite kind of mechanism whereby the
economy moves from one factor endowment configuration to
another. One possibility, consistent with the above and
intuitively reasonable, is that a rise in the real quasi-
rent from an equilibrium position induces further net .
investment while a decline results in net disinvestment Via
natural depreciation. This is one specification prominent
in the two—sector growth story. See, for example, Hirofumi
Uzawa, "On A Two-Sector Model of Economic Growth," Review
2; Economic Studies, XXIX (October, 1961), p. 43, equation
21. Or more recently, Akira Takayama, "On a Two-Sector

 

 

 

109

Together, equations (88) and (89) show, for the
Clark type capital model, that every fall in the wage-
interest ratio lowers the capital intensity of techniques.
The ratio of real capital to real labor is falling as one
moves down the frontier, and the weight going to the rela-
tively labor-intensive sector is rising as the price of
the relatively labor—intensive good falls relative to the

price of the relatively capital-intensive good.129
III. The Model: Fixed Coefficients

With fixed coefficients of production, equations (80)
- (83) are greatly simplified as every aij and, therefore,

the éJs reduce to zero. The structure of the production

model is then exhibited in the following equations:130

Model of Economic Growth; A Comparative Statics Analysis,"
Review 2; Economic Studies, XXX (October, 1963), p. 100,
equation 2 .

129The change in the relative importance of the two
sectors may be related directly to the chan e in the factor
endowment ratio by substituting equation (88) into equa—
tion (8h) thus obtaining:

A A A A

(M—C) = JD (L—K)

I/H (ofi‘tJD)

Notice that while it is large values ofcfllthat serve to
increase the demandwcomposition effect of a chan e in the
wage-interest ratio, it is a small value of 43 %and the
underlying elasticities of substitution in each sector)
that accomplishes the same end.

130Notice that the price equations are identical for
both models. This is due to the fact that the equations
represent equilibrium points only. At an equilibrium point
the relation of factor prices to commodity prices must be
the same whether the 31 are fixed or variable. In between
such points, the variab e coefficients would allow lower
prices (costs) in response to changing factor prices than

110
(90) )mfi + m; -
(91) Mmfi + )KC
(92) 13(1 '9KM) ,
(93) 9108 ”'9ch =-9ch> -
With the coefficients of production fixed, it is necessary

9

p.» t">

S

5;
ii,

to distinguish between the process factor-price frontier
and the economy's grand factor-price frontier (the envelope

).131 Each process frontier cor—

of the process frontiers
responds to one technique of production. A switch of tech-
niques occurs as we move from one fixed-proportions process
frontier to another.

In the preceding chapter it was argued that with the
consumption sector relatively capital intensive both rela-
tive commodity price and the factor endowment ratio
(capital to labor) fall for movements down any non—linear
process frontier. We can now give a more rigorous demon-
stration of these propositions.

Solving equations (92) and (93) one may obtain the
relation between relative commodity price and the wage-
interest ratio for movements along the process frontier:
(9h) f»: to) (cps)

W

would be possible with coefficients fixed. But these points
are irrelevant anyway or as Samuelson puts it: "The Substi-
tution effects are of a higher order of smallness, influenc-
ing curvatures rather than first-order slopes." Samuelson,

"Prices of Factors and Goods in General Equilibrium,"

pp. 223., p. 5. This is an example of what Samuelson calls

the Wong-Viner envelope theorem.

l3lsupra, Chapter II for discussion of these points.

 

lll

 

For movements down any nonlinear frontier, equation (94)
shows that the price of the relatively labor—intensive good
must fall relative to that of the relatively capital—
intensive good.

Subtracting equation (91) from (90) yields the change

in the ratio of outputs produced:

 

(95) (fa-6) :QTXTQ'

Thus by using equation (8h), one may relate the change in

the factor endowment ratio to the change in relative com-

 

modity price:
(96) (fi-i)=wotf>.
Finally, substituting equation (9h) into equation (96) one
obtains the change in the factor endowment ratio that is
implied by movements down the process frontier:
(97) (fi-i)=fi|_|id_/ (cps),

l + )9)
Equation (97) clearly shows that the economy's endowment of
capital is falling relative to its endowment of labor as
one moves down the process frontier.

The factor—price frontiers that are central to the
reswitching discussions show the edtilibrium real wage and
interest rate combinations that are feasible under a speci-
fied technique. Equations (98) and (97) show how, gpggp g
specified technique, equilibrium commodity price and the
economy's factor endowment ratio must vary in response to
changes in the equilibrium wage-interest ratio. Thus, for

example, in Figure 20, relative commodity price is lower

112

W
A
_1_
3L0
.1...
bit

 

 

 

1 1
am bKM

Figure 20. The Nonlinear FactormPrice Frontier

113
at A' than it is at A, and it is lower at B' as compared
to B. The same holds for the factor endowment ratio. In
fact, as equation (96) shows, with the consumption sector
capital intensive, every fall in relative commodity price
implies that the economy's endowment of labor has risen
relative to its endowment of capital.

Now consider the switch.point Sip . The equations
of change do not (indeed, cannot) tell us how relative
commodity price behaves at the precise point where entre-
preneurs are adopting the Beta technique. In marked con-
trast to the production model with variable coefficients,
the grand factor-price frontier of Figure 20 is kinked at
the switch point. This kink is important as it means that
When the switch occurs relative commodity price may "jump."
If it does, there is no reason why it should not "jump"
upward. That is, relative commodity price may be higher
under the Beta technique than it is under the Alpha tech-
nique when the wage~interest ratio is that relevant to the
point 349 .

For any point on the grand factor—price frontier,

relative commodity price is given by132

 

132Write the price equations as
PM = aLMW + aKMrPM , and PC = aLCw + aKCI'PM , so that we

a

W , and PC :: WaLC 4' AW
1 " aKMr 1““ aim? ~ P
A = aLMgKC - aLCaKM . Thus relative commodity price (p%)
is given by equation (98) of text.

, where

 

 

have PM =

11).

am

(98) p = 3L0 + Ar .

Thus relative commodity price is lower, unchanged, or
higher under the Beta technique _a_Ԥ_ 3113 mp3 Sdfi accord-
ing as

(99) aLM(bLC + Br) a bLM(aLC + Ar) .

In the special case where the factor proportions are equal

between sectors (A and B equal zero), equation (99) can be

shown to reduce to133
100 o 2 .4
( ) (AID < ALC ,

The ALC refer to the proportion of labor employed in the
consumption sector under each technique. Thus if the pro-
portion of labor employed in the consumption sector rises
when the new technique is adopted, the price of machines
will rise relative to the price of the consumption good at

the switch point.

 

133When A = B = 0 relative commodity price is equal
under the two techniques if aLM z bLM or aLM = aLC By

EL"L"<3"l3'L—cz 311715170-

 

M 4 a M
definition XEM = 8%? and %LM =‘“%¥” = géfl.. Thus we
_me L”

e o L
_Zlil‘l = _ELE or .891 = MN . Adding one to both

may write XLC XLC XLC F)

p a 3
sides then gives XiM t1 XIC ~ ALM + 810 . By assumption
LC his

of full employment )LM + ALC

Air X1.

1

ll
1...:
m
0

H

II

 

115
_H- When we allow for different sector factor intensities,
expression (99) is more difficult to interpret. All one
can say is that so long as the new technique is no less
capital intensive in the consumption sector than was the
old technique, the behavior of relative commodity price at
the switch point depends on changes that take place in the
proportion of labor utilized in each sector as a result of
the switch.

However, the point to be emphasized is that §_priori
one cannot say anything about the behavior of relative
commodity price at the point Sdp in Figure 20. Relative
commodity price may fall, be unchanged, or rise due to the
switch of techniques. Thus all entrepreneurs may be adopt-
ing more capital-intensive techniques, but the aggregate
capital-labor ratio may remain unchanged or even fall.

To complete this section, a numerical example of
each type of price behavior is offered as definite proof
that relative commodity price can move in any direction
when a switch of techniques occurs in a fixeduproportions,

heterogeneous, real capital model.

I. Relative commodity price falls. Two possible techniques

of production are:

. l
Alpha-l C —' mln( 1’9 5K) 9

00

3

1 1
M=min-L -K
(b. . 2)

and
. 1 l .
C = min(EL , 3K) ,

M = min(-]3:L , K)

Bet a1

60

116
Some values of p“ and p9 are computed in Table II. Entre-
preneurs switch from the Alphal technique to the Betal
technique at r = 0.2500. Relative commodity price falls
from 1.600 to 0.800.

II. Relative commodity price is unchanged. Two possible

techniques are:

l
Alpha2 : C min(13:‘gK) 9
M =min(L, is) ,
min(é—L , %K) ’

min(%L , K)

and

C
Betaz o

M
Some values of p“ and p9 are computed in Table III. Entre-
preneurs switch techniques at r = 0.33 and relative com-

modity price remains unchanged at 0.h31.

III. Relative commodity price rises. Two possible tech-

niques are:

— .l
Alph83 3 C — min( L 3 K) .9
M = min(iL , ix) ,
and 3 2
C = min(lla,.lK) ,
Beta3 3 2 3

M = min(%L , K) .
Some values of p“ and pp are computed in Table IV. Entre-
preneurs switch from the Alpha3 technique to the Beta3 at
r = 0.135. Relative commodity price increases from 0.9h9
to 1.332.

The factor~price frontiers implied by these examples
are shown graphically together with the trace of relative

commodity price over the range of feasible interest rates

117

TABLE II

COMPUTED VALUES 0F RELATIVE COMMODITY PRICE FOR
TECHNIQUE ALPHAl (p4) AND TECHNIQUE BETAl (p0 )

 

 

r p“ pp
0.00 h.000 1.550
0.05 3.076 1.276
0.10 2.500 1.111
0.15 2.105 0.983
0.20 1.808 0.882
0.25 1.600 0.800
0.30 1.428 0.732
0.h0 1.176 0.625
0.60 0.869 0.h8u
0.90 0.625 0.361
1.00 0.571 0.333

 

Note: Entrepreneurs switch techniques at r = 0.25 and
relative commodity price falls from 1.600 to 0.800.

118

TABLE III

COMPUTED VALUES 0F RELATIVE COMMODITY PRICE FOR
TECHNIQUE ALPHAg (pd) AND TECHNIQUE BETA2 (p9)

 

 

 

r p‘ p9
0.00 1.000 1.000
0.05 0.833 0.833
0.20 0.555 0.555
0.25 0.500 0.500
0.30 0 MBA 0.45t
0.33 0.831 0.h31
0.35 0.817 O.h17
O.h5 0.357 0.357
0.75 0.250 0.250
0.85 0.227 0.227
1.00 0.200 0.200
Note: Entrepreneurs switch techniques at r = 0.33 and

relative commodity price is unchanged at 0.h31.

119

TABLE IV

COMPUTED VALUES 0F RELATIVE COMMODITY PRICE FOR
TECHNIQUE ALPHA3 (p4) AND TECHNIQUE BETA3 (pa)

 

 

 

r p“ pp
0.000 3.000 2.50
0.055 1.596 1.82h
0.105 1.119 1.h88
0.135 0.9h9 1.332
0.205 0.701 1.073
0.h05 O.h01 0.689
0.505 0.330 0.588
0.705 0.2hh 0.hh8
0.805 0.216 0.h01
0.905 0.19M 0.363
1.000 0.177 0.333
Note: Entrepreneurs switch technique at r = 0.135 and

relative commodity price rises from 0.9h9 to 1.332.

120
in Figure 21. Panel a, Figure 21, shows case Iwith
relative commodity price falling when the switCh occurs;
Panel b, case 11 with relative commodity price rising at
the switch point. Case III, in which relative commodity
price falls continuously for movements down the grand
factor-price frontier, is shown in Panel 0, Figure 21.
Notice that the factor-price frontiers have common ordinate
and abscissa intercepts. The wage-interest ratio is, of

course, different for each Case at the switch.point.

IV. Reswitching and the Aggregate
Capital-Labor Ratio

An example of reswitching, discussed at length in
Chapter II, is shown graphically in Figure 22. Given a
choice of techniques, entrepreneurs are assumed to select
the technique that yields the greatest rate of profit.
Thus at very high wage rates, W2 < w < wB,,the Beta tech—
nique is selected. At intermediate wage rates, wl < w < W2,
the Alpha technique is the most profitable technique, and
for very low wage rates, 0 < w < wl,.the Beta technique is
again selected as it has once more become the most profit—
able of the two possible techniques.

The conclusion that has been drawn from examples
such as this is that, if the economy goes over to "less
capital" at the first switch.point (Spo(), it is neces-

sarily Coming back to "more capital“ at the second switch

121

coanm hpfioossoo o>fluoaom so mosvfianooa no monopfiSm mo pooumm

n Hoses

 

 

 

 

 

o-aonom-

 

as“

/

 

 

 
  

_dh ——_——.

 

raw

.Hm enemas

Figure 21.

 

122

 

 

 

 

(continued)

Panel c

I"

123

W3

W2

W1

 

 

 

Figure 22. Reswitching of Techniques

124

point (Sap).13u Thus it is said that, with a given,
fully-employed labOr force, the aggregate capital-labor
ratio first falls (at the point Spa! ), then rises (at the
point 544;), though the rate of interest is rising con-
tinuously.135

In the preceding section we saw that, for any tech-
nique giving rise to a nonlinear process frontier, both the
factor endowment ratio and relative commodity price are
necessarily lower at the point 8d}; than they are at the
point Sp“ . Hence it is clear that when the Beta tech-
nique "comes back" it does not represent the same ratio of

136 It

capital to labor as it did when it went out of use.
represents a smaller aggregate capital-labor ratio. Now
suppose that, at the first switch.point of Figure 22 (Spgs),
the economy goes over to a smaller aggregate capital-labor
ratio as a result of all entrepreneurs adopting less
capital-intensive processes. Suppose also that the new
technique is one that requires equal-proportions produc-
tion throughout the economy. The latter assumption ensures
us that the factor endowment ratio is a constant for move-
ments down the process frontier implied by the new technique.

The question then is whether the capital-labor ratio under

the Beta technique is greater at the second switch point

 

l3’4Pasinetti, 2p, cit., p. 51h.
13SBruno, 23.21,, PPo 5h8f49-
1360f. ante, Chapter II, footnote 8h.

125
.(Sdfl ) _thanwthat of the Alpha technique. It. may be. But
the point is that it need not be. All one can say for
certain is that, if all entrepreneurs adopt less capital-
intensive processes at fied , all entrepreneurs adopt more
capital-intensive processes at 84%;. Beyond this, one must
know how relative commodity price behaves at the second
switch point before any inferences about aggregate capital-
1abor ratios are warranted.137 Unfortunately, there is
little one can say a priori about how relative commodity
price behaves §§_thg_switch point. As we saw, given any
wage-interest ratio, relative commodity price may fall,
remain unchanged, or rise EE 3 result 2£_§Qtrepreneurs'
switching techniques.

Let us plot the relation between the aggregate
capital-labor ratio and the rate of interest that is implied
by movement down the grand factor-price frontier of Fig—
ure 22. we know from equation (97) that the aggregate
capital-labor ratio falls continuously for movement down
the Beta process frontier. Also from previous discussions
we know that the aggregate capita1~labor ratio is given by

the sector capital—labor ratios for any equalnproportions

 

137The behavior of relative commodity price at the
first switch point is of no consequence in view of the fact
that we are switching to an equal-proportions production
technique at this point. Su ra, Chapter III, footnote 102.
In the more general case of different sector factor intens-
ities under the Alpha technique and the Beta technique,
the behavior of relative commodity price at both switch
points would have to be considered.

126
138

production technique. Thus, with all entrepreneurs
adopting less capital-intensive techniques at theswitch
point (Sfl4 ), the aggregate capital-labor ratio is neces-
sarily lower under the Alpha technique than it is under

the Beta technique at the M (3,64 ). Finally, we know
that the aggregate capital-labor ratio is a constant for
movement along the Alpha process frontier.

Figure 23 shows how the aggregate capital-labor
ratio must behave as the wage-interest ratio falls and one
moves down the grand factor-price frontier of Figure 22.
We assume a given fully employed labor force. At a zero
rate of interest, the Beta technique represents a larger
aggregate quantity of capital per man than does the Alpha
technique. This fact may be inferred from the ordinate'
intercepts of Figure 22 alone.139 (One might say that
the Beta technique represents a larger (aggregate) plant
than does the Alpha technique). As the rate of interest
rises, the quantity of capital per man falls continuously
under the Beta technique as shown by the curve kb- The
aggregate capital-labor ratio for the Alpha technique is
given by the line ka. At some rate of interest the aggre-
gate capital-labor ratio implied by the Beta technique must
be equal to that of the Alpha technique. This critical

 

l388ee preceding reference.

l39Provided we accept the neoclassical parable that
says the real wage is higher the greater is the capital per
man. of; ante, Chapter III, footnote 97.

127

 

 

 

 

 

.K.
LI
5
fiat
B \
\c
l\
I \»D E ‘%a
A P
l |\
l I \
l | F
| |
I I l
I | '
I I :
l I k
b
I I I
0 l I“ l *1,
r1 1"“ P2

Figure 23. Effect of Reswitching on Capital Intensity:
Relative Commodity Price Constant

 

128
rate of interest is labeled (r*) in Figure 23. Furthermore,
at some interest rate below this critical rate, entre-
preneurs will have switched from the Beta technique to the
Alpha technique. Let the switch occur at the point (334)
in Figure 23. Increases in the interest rate beyond the
point (§p0() eventually result in a return to the Beta
technique. There are now three types of price behavior
that must be considered: relative commodity price is un-
affected by the switch of techniques, it falls, or it rises.
Case I: relative commodity price is unchanged at the point
where reswitching occurs.

This case is already implied in Figure 23. Let the
return to the Beta technique occur at one of the three
interest rates labeled r1, r*, or r2. If the switch takes
place at the critical interest rate r*, the capital-labor
ratio is, of course, unchanged by thereturn to the Beta
technique. In Figure 23, the aggregate capital-labor ratio
moves from B to A, is unchanged as reswitching takes place
at D, then falls toward F as the interest rate rises
further.

If the switch occurs to the right of the critical
rate r*, for example, at the rate of interest r2, the
aggregate capitalalabor ratio must fall as a result of
entrepreneurs returning to the Beta technique. Thus, refer-
ring to Figure 23, the aggregate capital-labor ratio falls
from B to A as the Alpha technique replaces the Beta tech-

nique. It is then unchanged until entrepreneurs return to

129
the Beta technique at r2. At re the aggregate capital-
1abor ratio again falls going from E (=A) to F.

 

Finally, if entrepreneurs return to the Beta tech-
nique before the critical rate has been reached, for
example at the interest rate r1, the aggregate capital-
labor ratio will rise when the switch occurs. This possi- i
bility is shown in Figure 23 by the dotted line rising from
ka to the point C at the interest rate r1.
Case II: relative commodity price falls as entrepreneurs

return to the Beta technique.

 

This case is shown in Figure 2h. The fall in
relative commodity price has the effect of shifting the kb
curve downward at the point of reswitching (from kb to kb').
The explanation for this is that more weight is automatical-
ly being given to the relatively labor-intensive sector
when the switch occurs. Again suppose that entrepreneurs
return to the Beta technique at one of the three interest
rates r1, r*, or r2.

The aggregate capitalmlabor ratio may now fall as
entrepreneurs return to the Beta technique even when the
switch occurs at rates below the critical interest rate
(r”").lLLO In Figure 2A, the aggregate capital—labor ratio

is shown as unchanged when the switch occurs at r1 <. r*.

lLtOThere is some ambiguity here as one could view
the fall in relative commodity price as meaning that the
critical rate has been lowered thereby reducing the range
of interest rates for which a return to the Beta technique
could imply a higher aggregate capital-labor ratio. How-
ever, the results are unaffected by this choice.

 

 

130

film

 

 

 

Figure 24. Effect of Reswitching on Capital Intensity:
Relative Commodity Price Falling

d

 

131
At the switch point (pr_), the aggregate capital-labor
ratio falls from B to A. As entrepreneurs return to the
Beta technique at the interest rate rl the fall in relative
commodity price causes the point C of Figure 23 to fall to
C' in Figure 2h. Thus the aggregate capital-labor ratio is
unchanged by the return to the Beta technique. Further
increases in the interest rate are associated with a fall-
ing capital-labor ratio as shown by the dotted line from
C' to F'.

We cannot, of course, be certain that the aggregate
capital-labor ratio will never rise for this case. However,
the possibility is greatly reduced when the switch itself
lowers relative commodity price. When reswitching occurs
at r* or any higher rate of interest such as r2, the aggre-
gate capital-labor ratio will always fall as a consequence

of returning to the Beta technique.

Case III: relative commodity price rises as the Beta
technique returns.

This case is shown graphically in Figure 25. Of the
three cases, it presents the greatest possibility that a

return to the Beta technique will raise the aggregate

141

capital-labor ratio. The rise in relative commodity

lLLlThis appears to be the case implied by the re-
switching literature. See, for example, Samuelson's dis—
cussion of the influence of the interest rate on relative
costs (pricea’as he offers an explanation of why reswitch-
ing can occur. His explanation is that relative costs
(prices) reverse themselves. Paul A. Samuelson, "A Summing
Up," The uarterl Journal g£_Economics, LXXX (November,
1966), pp. 71—73. "" "‘" "

 

 

132

rflm

 

 

 

 

 

Figure 25. Effect of Reswitching on Capital Intensity:
Relative Commodity Price Rising

133
price automatically increases the weight of the relatively
capital-intensive sector as the entrepreneurs return to the
Beta technique. Thus the cruve kb shifts upward at the
point of reswitching (from kb to kb' in Figure 25).

If the return to the Beta technique occurs at r* or
any lower rate of interest such as r1, the aggregate
capital-labor ratio must rise. For example, as the inter-
est rate increases from zero, we move along the curve kb

of Figure 25 and the aggregate capital-labor ratio falls.

 

At the switch.point Spd.’ entrepreneurs switch from the
Beta to the Alpha technique. The aggregate capital-labor
ratio falls from B to A. The aggregate capital-labor
ratio is then unchanged for further increases in the rate
of interest until the interest rate r1 is reached. At this
point, entrepreneurs switch back to the Beta technique,
relative commodity price rises as a consequence of the
switch of techniques, and the aggregate capital-labor
ratio increases from A to C'. The aggregate capital-labor
ratio then falls continuously (from 0' toward F') until
the interest rate reaches the maximum possible under the
Beta technique. The argument is similar when the switch
occurs at r*.

But notice that, even with relative commodity price
rising as a result of entrepreneurs returning to the Beta
technique, the aggregate capitalmlabor ratio need not-
necessarily rise. There may be a number of feasible inter—

est rates (such as r2 in Figure 25) where a return to the

13h
Beta technique is accompanied by a fall in the aggregate
capital-labor ratio.“ In Figure 25, return to the Beta
technique at the interest rate r2 causes the aggregate
capital-labor ratio to fall from E to F'. It is therefore
clear that even under the most unfavorable conditions the
aggregate capital-labor ratio may fall though entrepreneurs
are returning to techniques that were in use at some pre-
vious wage-interest ratio.

To sum up, the foregoing demonstrates that reswitch-
ing cannot be taken to mean that the aggregate capital-
labor ratio is necessarily reversing itself. For any two
techniques, there will generally be some critical rate of
interest for which the aggregate capital-labor ratio is
the same for both techniques.lu'2 Thus whether reswitching
means that the aggregate capital~labor has reversed itself
or not depends first of all on whether the economy has
passed this critical interest rate before reswitching
occurs. Once this critical rate is passed, it is virtually
impossible for the aggregate capitalelabor ratio to rise as
a consequence of a fall in the wage-rate of interest ratio.
When reswitching occurs before the economy has passed this
critical rate, but relative commodity price falls as a

result of reswitching, it is equally unlikely that the

 

lAZWhen the sector factor intensities differ under
both techniques, it will be necessary for the technique
that represents the larger aggregate quantity of capital
at a zero interest rate to react more strongly to changes
in the rate of interest than the technique that represents
the smaller quantity of capital.

 

7 13s
aggregate capital—labor ratio has risen. It is only when
reswitching occurs before the critical rate is passed and
relative commodity price is unchanged or rises as reswitch-
ing takes place that the aggregate capital-labor ratio is
necessarily higher.1)“'3

Whatever the behavior of relative commodity price
at the point of reswitching, it is clear from the above i
that the relation of the aggregate capital-labor ratio to
the wage—interest ratiq depends critically on the interest
rate itself. This is the one, single, unalterable fact
made clear by the reswitching phenomenon. And it is
important to record that though reswitching reveals this
dependence, it is itself independent of any actual re—

switching.luu

l’43'1‘he above demonstrates that reswitching may be
accompanied by a reversal of relative commodity prices
but it is important to notice that this need not be the
case. Thus reversal of relative commodity prices implies
reswitching but the reverse is not true.

lbrLtWe have concentrated on reswitching because it
presents the strongest case against simple neoclassical
theory. If the neoclassical link between the aggregate
capital-labor and the wagemrate of interest ratio can be
shown to hold in the face of reswitching, it will hold
g_fortiori for more general cases. For example, consider
again Figure 12, page 62 of the text:

136

In an Austrian capital model, a model that does not

 

 

 

 

At ($94 ) entrepreneurs switch from the Beta technique to
the Alpha technique and at (Sag) from the Alpha technique
to the Rho technique. Plotting the relation between the
aggregate capital-labor ratio and the interest rate as r
rises from zero to its maximum under the Rho technique we
have;

K

I I

 

 

l k‘b kc.

és >’r

Following the notation and analysis of the text the aggre-
gate capital-labor ratio falls from B to A at the switch
point (Spa). At the second switch point (30(1) the aggre-
gate capital-labor ratio may again fall, remain unchanged,
or rise in accordance with the analysis of the text.
Notice that the aggregate capitalalabor ratio under the
Rho technique is greater than that of the Alpha technique
when r = O. The interest rate that will be critical in
this scheme is again labeled r* and corresponds to the

 

137

allow reswitching,l)+5 this dependence is revealed by the
fact that the "average period" (the "independent" measure
of capital) can be smaller or larger depending on whether
the interest rate is relatively low or relatively high
even though production processes are themselves unchanged.

In a heterogeneous capital model such as the one
used in our discussions, this dependence is revealed by the
fact that for any two techniques of production there is
some viable interest rate at which the two techniques

represent the same aggregate quantity of capital.

 

Precisely what this new (old) dependence between
the interest rate and the quantity of capital implies for
simple neoclassical theory is difficult to say. We have
seen that it does not mean that the neoclassical relation
between the aggregate capital~labor ratio and the wage-
interest ratio is necessarily broken. There is a possi-
bility that at some wage-interest ratio the neoclassical
relation will not hold. However, when one considers the
interrelations between factor prices, factor endowments,
and relative commodity price the possibility does not
appear to be a strong one. Mrs. Robinson's paradox is more

likely to be to neoclassical production and/or capital

 

interest rate at which the aggregate capital-labor ratio
under the Rho technique is equal to that of the Alpha tech-
nique. At constant relative commodity price, the aggregate
capital- -labor ratio must fall when the switch to the Rho
technique occurs after the economy has passed the interest
rate r*.

luSFor the analysis of this model and its relation to
reswitching, see Samuelson, ”A Summing Up,"o _p, cit.,

568- 83

138
theory what Giffen's paradox is to neoclassical demand
theory. The relation is certainly more complicated than
the simple parables would have us believe. But that, I
think, is all one is entitled to conclude.

 

CHAPTER V
SUMMARY AND CONCLUSIONS

The chief characteristic of the neoclassical,
real-capital model is the assumption that there exists a
function Q = F(K,L) that is homogeneous of degree one in
a single homogeneous capital good (K) and a single homo-
geneous labor force (L). As we saw, this implies, (a) that
the real wage varies inversely with the interest rate, and
(b) that the real capital-labor ratio varies directly with
the wage-interest ratio. Hence, an increase in the real
wage leads to an increase in the capital intensity of
techniques, and permanently sustainable net national
product increases as the rate of interest declines.

The critics of simple neoclassical theory (es-
pecially Mrs. Robinson) have long maintained that the
above assumption is "crucial" in the sense that a very
elaborate set of propositions are being made to stand on
what is in some respects a very dubious assumption. The
trick, of course, was to show exactly how the results of
simple neoclassical theory depended sensitively on this
assumption.

Initially, much attention was given to the "physical

jelly" that went into the production function under the

139

 

 

1hO
label of capital. or course, no one (especially J. B.
Clark) denied the existence and/or importance of hetero-
geneous capital goods; the question was how best to quanti—
fy them. For Mrs. Robinson, the answer was that it could
not be done but if one insisted on doing it anyway, the
element 2£_timg in the production, construction, and
service life of these goods should be taken into consider-
ation. Thus Mrs. Robinson replaced the "physical jelly"
of the neoclassical production function with past labor
time, compounded at interest, embodied in the stock of
capital goods. The most significant result of this modi-
fication was that the ratio of real capital to labor was
made to depend on the interest rate. Each technique (i.e.,
each particular set of capital goods) now appeared as a
greater or lesser amount of real capital according as the
interest rate was relatively high or relatively low.

A surprising possibility was revealed by this
dependence. It implied that a decrease in the rate of
interest may be accompanied by a change from more to less
capital-intensive techniques. This was contrary to what
had always been taken for granted and so there developed a
tendency in the literature to label the possibility as
"perverse," "anomalous," or "exceptional.” Ultimately the
fallacy of composition was invoked, and it was agreed that
the "curiosum" (as Mrs. Robinson called it) may indeed be
found at the level of the industry but never for the economy

as a whole.

‘ lhl

Subsequently the proof offered to support this
contention was shown to be in error, and the "reswitching
of techniques" became a "fact of life." Pasinetti,
Morishima, Bruno-Burmeister-Sheshinski and others defi-
nitely proved it is possible for a technique that is the
most profitable of all feasible techniques at some rela-
tively high wage rate to also be the most profitable at
some relatively low wage rate. The technique "comes back"

as the wage falls monotonically toward zero, with.the con-

 

sequence that an unambiguous ordering of techniques is
impossible. Thus, it seemed a simple technological fact
that there could exist a "set of blueprints" for which
there is no one-to-one correspondence between the capital-
labor ratio and the wage-interest ratio. If so, it is not
necessarily true that a fall in the rate of interest
entails the adoption of more capital-intensive techniques,
with the consequence that the permanently sustainable
consumption stream is greater.

The problems posed by the critics of simple neo-
classical theory concern the relation between the aggregate
capital-labor ratio and the wage-interest ratio. There-
fore, it is a serious criticism of recent discussions that
a quantity of capital has come to be identified with a
matrix of activities with no attempt to demonstrate how the
columns of the matrix relate to the aggregate capital-labor
ratio of the neoclassical model. That omission has had two

unfortunate consequences: first, it has resulted in an

lh2

incomplete appreciation of the basis of the criticism, and
second, it has allowed the role of relative prices to be
completely ignored. By letting the aggregate capital-labor
ratio represent itself in the Cambridge criticism, the
present investigation demonstrates that the criticism is
less important than recent discussions would have one
believe.

As we saw, recent discussion has focused exclusively
on capital goods. If a particular set of capital goods

(i.e., matrix of activities) "comes back" as the wage-

 

interest ratio falls monotonically toward zero, the aggre-
gate capital-labor ratio is said to reverse itself regard-
less of how one might choose to measure capital. The only
basis for this assertion is the labor-output ratio in the
consumption sector relevant to the various techniques at a
zero rate of interest (i.e., the labor-consumption
coefficient). And this comes down to using the labor-
output ratio of the consumption sector as an independent
index of capital intensity.

There are two objections that one may raise against
this index. First, and most obvious, is the fact that the
labor-output ratio of the consumption sector is relevant
only at a zero rate of interest. This is the major
implication of a nonlinear factor-price frontier. Second,
and perhaps more important, the labor-output ratio relevant
at a zero rate of interest may not be a single-valued

function of the aggregate capital—labor ratio. As we saw,

 

1LI3
the relation between the ratios depends on the assumed set
of technical alternatives. If entrepreneurs switch neo—
classically, the two ratios are inversely related. However,
as the reswitching literature makes clear, entrepreneurs
may behave "perversely." In this case the ratios are
directly related. It is possible for an increase in the
labor-output ratio to be associated with either an increase
or decrease in capital intensity. Hence, the index is
inconsistent.

Beyond this, there is a rather subtle confusion
present in the preoccupation with the labor-output ratio of
the consumption sector. To suggest that the capital
intensity of techniques is reflected in the labor-output
ratio of the consumption sector at a zero rate of interest,
is to implicitly assume that the aggregate capital-labor
ratio is unique for a given technique (i.e., matrix of
activities). That is, it implies that a technique embodies
a given amount of real capital irrespective of the level
of the 'wage-interest ratio. This, of course, presupposes
an independence between the aggregate capital-labor ratio
and the wage-interest ratio. Thus the basic relation on
which the Cambridge criticism rests is lost in the litera-
ture devoted to the switches of techniques.

Reswitching and its implications are theoretical
possibilities precisely because this interdependence
exists. That is why the reswitching possibility was

revealed only after the interdependence was incorporated

luu
into the analysis of alternative, steady-state, capital-
labor ratios. And it is also why reswitching is not
necessary for the unconventional behavior of steady-state
consumption. Reswitching and its so-called implications
are all rooted in the fact that hhe aggregate capital-labor
ge§;e_;e_hee independent 22.322 wage-interest £3322-

As Mrs. Robinson says, this point, previously made
by Wicksell in connection with Bohm-Bawerk‘s "average
period," does not seem to have been properly digested into
the teaching of the neoclassical doctrine.lu6 I think it
is fair to say that this point has not been properly
developed in the most recent debates concerning the doc—
trine.lu7 Perhaps this is due to the reluctance of recent

writers to employ overtly the notion of a quantity of
1h8

capital. Whatever the reason(s), the fact remains that

lué"Wicksell points out that the length of the

period of production does not by itself determine the ratio
of capital to labour, because the value of capital required
for a given method of production depends on the real-wage
rate. This is a much more fundamental criticism of Bohm-
Bawerk's theory than the objection that the length of the
period of production is an overmsimplified way of represent-
ing the real capital ratio." Joan Robinson, The Accumula-
tion of Ca ital (2d. ed., New York: St. Martin's Press,

W66 ): P 39 Also ibid., p. 391.

lLL7For the only explicit recognition of this point
in the recent Symposium, see P. Garegnani, "Switching of
Techniques," The Quarterly Journal of Economics, LXXX
(November, 1933), p. 562, n.

lLLBIt should be recorded that the failure to develop
this point cannot be attributed to the type of model chosen
to demonstrate the reswitching phenomena. As others have
rightly pointed out (Bruno, eh'el., ”The Nature and Impli-

cation of the Reswitching of Techniques," ibid., pp. 528—
31), the instantaneous production model is the more general

 

 

 

 

lLIS‘
the technology matrix is an aggregate capital-labor ratio.
The point that needed to be developed is that the aggre-
gate capital-labor ratio is not unique for a specified
technique. This, of course, is the point contained in
Mrs. Robinson's shifting productivity curve.

In Mrs. Robinson's analysis, the quantity of real
capital embodied in a specified technique is dependent on
the wage-interest ratio because of the element of interest
in the cost (in terms of labor time) of capital goods. An
increase in the rate of interest (fall in the real wage)
raises her real-capital ratio. In the analysis of this
paper, the quantity of real capital embodied in a speci-
fied technique is dependent on the wage-interest ratio
because of a strict adherence to the full employment
assumption. A fall in the wagewinterest ratio lowers the
price of the relatively labor-intensive good relative to
that of the relatively capital~intensive good. And so
long as demand is sensitive to relative commodity price,
the fu11 employment assumption requires either an in-
crease in the economy's labor endowment or a decrease in
its capital endowment. The result is a fall in the ratio

of real capital to real labor embodied in the technique.

 

model. And as we saw, a careful consideration of the
implications of the nonlinear factor-price frontier of this
model reveals the interdependence. On the other hand, as
Samuelson has shown (ibid., "A Summing Up," pp. 568-83)

it is quite evident in an Austrian-type production model
that reswitching is in some sense an interest rate phe-
nomenon. However, as Wicksell realized, one must go beyond
the "average period" to get at the interdependence.

 

 

 

 

1&6

When the number of techniques exceeds one, the
dependence of the factor endowment ratio on the wage-
interest ratio implies that, for a given labor force, there
is generally some interest rate (wage rate) at which any
two techniques embody the same quantity of real capital.
As we saw, this interest rate is critical insofar as the
reswitching argument is concerned. Whether reswitching
means that-the aggregate capital-labor ratio has reversed
itself or not depends in the first instance on whether, in
the process of accumulation, the economy has passed the
particular rate of interest that equates the real capital
of the displaced technique with the real capital of the
technique being adopted.

lh9 one cannot

Finally as Mr. Sraffa has made clear,
examine the ratio of capital to labor independently of
either the wage-interest ratio 2£_relative prices. The
latter are important to the problems posed by reswitching
because the aggregate capitalmlabor ratio is a value-
weighted mean of the ratio of capital to labor utilized in

each sector of the economy considered separately. In a

multi-sector capital model it is necessary to examine the

 

1H9"The reversals in the direction of the movement
of relative prices, in the face of unchanged methods of
production, cannot be reconciled with any notion of capital
as a measurable quantity independently of distribution
and prices. Italics in original. Piero Sraffa, Produc-
tion of Commodities by Means of Commodities (Cambridge:
Cambridge University Press, 1533), p. 38.

11L?
interrelations of factor endowments, factor prices, and
relative commodity price.

In a simple twoesector J. Bo Clark neoclassical
capital model, the relations between the wage-interest
ratio, the factor endowment ratio, and relative commodity
price are quite straightforward. The most important charac-
teristic of the model is the continuous factor price fron-
tier. As the wage-interest ratio falls one moves down the
frontier, the ratio of real capital to real labor falls,
and the price of the relatively labor-intensive good falls
relative to the price of the relatively capital-intensive
gbod. Hence, the aggregate capital-labor ratio is directly
related to the wage-interest ratio and the neoclassical
results follow°

In a fixed-proportions, heterogeneous capital model,
one must distinguish between the process factoraprice
frontier and the economy's grand factormprice frontier
(the envelope of the process frontiers). For an individual
process frontier, the relations between the wageminterest
ratio, the factor endowment ratio, and the relative come
modity price are precisely those implied by movements along
the continuous factorwprice frontier of the J. B. Clark
neoclassical capital model. Thus.9 for each.process frontier
considered separately, a decrease in the wagewinterest ratio
implies: (a) that the ratio of real capital to real labor
has fallen and (b) that the price of the relatively labor-

intensive good has fallen relative to the price of the

lh8
relatively capitalwintensive good. Hence, the aggregate
capital-labor ratio declines for movements down the
process frontier.

, When entrepreneurs switch techniques in a hetero-
geneous capital goods model, they move from one fixed-
proportions process frontier to another. The result is a
kink in the grand factor-price frontier at the point of
switch. The kink is important because it implies that a
switch of techniques has an independent influence on rela-
tive commodity price. §_priori there is little one can
say about the behavior of relative commodity price at the
switch point° It may fall, remain unchanged, or increase
as a result of entrepreneurs' switching techniques.

The introduction of relative prices into the analysis
of reswitching substantially affects the Cambridge criticism
insofar as that is directed toward the neoclassical link
between the aggregate capitalwlabor ratio and the wage-
interest ratio. It means that, contrary to the assertions
in the literature, reswitching does not immediately imply
a reversal in the capital intensity of techniques. If
individuals are quite ready to substitute goods in consump—
tion, and if reswitching causes relative commodity price
to fall, the aggregate capital—labor ratio may be a single-
valued function of the wagewinterest ratio though indi-
vidually entrepreneurs are adopting more capital-intensive

processes.

 

149

In conclusion, one can only record that the
neoclassical relation between factor proportions and fac—
tor prices depends in a complicated way on both the
interest rate and relative commodity price. This, I think,
is the simple but important kernel in the Cambridge criti-
cism of simple neoclassical theory. The quantity of real
capital embodied in any specified technique is not insensi-
tive to the rate of interest, and relative prices are not
insensitive to switches of techniques. It is possible that
these interrelations may work so as to increase the capital
intensity of techniques subsequent to a fall in the wage—
interest ratio. I have shown how these interrelations may
work so as to bring about a decrease in the aggregate
capital-labor ratio in full accordance with the simple
J. B. Clark neoclassical fairy tale. The invariant neo-
classical relation is an empirical question that depends
ultimately on the readiness of individuals to substitute
goods in consumption. Unfortunately, it is an empirical
question that has no empirical answer because economies do

not proceed along smooth exponential growth paths.

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