‘Happy Are the Poor’: An Austrian Tale of Roundabout Production*
By Hiroshi Ohta,
Professor of Economics, Aoyama Gakuin University, Tokyo
For Presentation at National Taiwan University Workshop on International Trade and
Industrial Organization, November 1, 2008
ABSTRACT
Economic development may not require capital-intensive methods of production. Using a
simple Austrian model, we show that if labor productivity in the primary sector is low
enough, roundabout methods of production become feasible. Moreover, it is the more
labor-intensive methods that yield the greater final output. A higher labor productivity may
appear to be geared to greater final output, all the more if combined with a higher output
elasticity of capital under more capital-intensive roundabout methods of production. But
when primary labor productivity is high enough (relative to the total factor productivity of
final output) the roundabout methods become strictly inferior to the direct method. Thus,
whenever roundabout methods are feasible, the more labor-intensive methods become
superior to the more capital-intensive methods. If output elasticity of capital is large enough,
labor-intensive methods become infeasible, and roundabout methods fail.
*Preliminary draft not to be cited without author’s permission. The author is indebted to
Martin McGuire for his invaluable ideas and comments, and also Katsuhiko Akiba for his
technical assistance. This paper is indeed a part of separate work under collaboration with
them.
1
‘Happy Are the Poor’: An Austrian Tale of Roundabout Production
(A Neo-Classical Synthesis of Karl Marx and Boem Bawerk)
“The standard of living in Hong Kong had multiplied more than tenfold in forty years
(since 30s), while the standard of living in Calcutta has improved hardly at all.”
--- John Templeton, quoted by Mark Skousen (2002)
The quote above accounts for striking differences in economic growth that the two Asian
nations in extreme poverty under the British control fared in the 1970s upon the observer’s
second visit with them. The difference is attributed to the absence (in Hong Kong) and the
abundance (in India) of government planning and regulations. If Hong Kong is poorly
endowed with natural resources, it was also even pitied as being over-populated. That is,
what they have in abundance was deemed as of more liability than asset.
Motivated by Hong Kong’s miraculous experience, the present article ponders
conditions under which a very poor strictly inferior nation can not only catch up with, but
also beat its apparent superior. Here we focus our attention on conditions more
technological than institutional. We thus assume away government involvement in the
present inquiry.1
This paper integrates, albeit in a very limited sense, Karl Marx and Boem Bawerk
into a simple neo-classical synthesis. Addressed herein are the questions of production, the
structure thereof, distribution, and related welfare. We start with the basic departure point
thought that labor is the sole primary factor of production as is the case in both Marx and
Boem Bawerk. We denounce Marxian labor theory of value insofar as it predicts
exploitation of labor. This is not to say that we support Boem simply because he criticized
Marx on grounds that rent on capital does not derive from exploitation of labor. It does not
in fact. But that does not answer who, if anybody, is to be exploited by whom. We show
1
Even if government-planned, hasn’t such a heavily capital-intensive development as in
India had some positive impact on economic growth? If the Austrian roundabout methods
of production require a more capital-intensive than labor-intensive structure of production,
then the government plan along that line itself may not be to blame at least? If private
sectors are let alone, then their roundabout methods freely chosen could have been more
labor intensive than capital intensive. The present paper shows indeed the technological
conditions under which the roundabout methods of production require more labor-intensive
methods than capital-intensive methods, while assuming away government.
2
that any seeming rent on capital shall be dissipated, and fully paid out to workers. The same
wage payment made in the final consumption goods sector must be paid out elsewhere, too,
so that it can even exceed average labor productivity in the capital good sector. So, we
clarify what is the "surplus accruing to the capitalists" that is positive, but nevertheless no
"exploitation" is possible. We show that if anyone is to be exploited, it is the producers of
capital goods who are doomed in effect to self-exploitation inasmuch as all the output is
distributed among the workers as their wage incomes.
With no prior endowment of capital all the fruit of production in the capital goods
sector must go to the workers under perfect competition either directly or indirectly. If
directly, then the workers in the capital goods sector become the owners of the output. The
owners of output can either consume it or offer (supply) it as capital to be used together
with the remaining labor for the production of the final consumption goods. If they choose
to consume it, then they choose not to become capitalists, not going roundabout in
production. Only if it is worthwhile an effort, the working owners may choose to become
capitalists.2
In a primitive Boem Bawerkian economy there exists no capital to begin with,
however. The economy is endowed with nothing but labor. If labor alone is used to produce
something, then the whole produce must be theirs provided that production is subjected to
technological conditions of constant returns to scale and market conditions, if any, of
perfect competition. If output is the workers’, then they may as well consume it directly,
being their property. The whole product is theirs with no other factor inputs being used to
produce it. But even if it is theirs, they need not consume it directly. They may save it for
consumption later, if deferred consumption is expected to exceed direct consumption.
If consumption is to be deferred, then output of labor can be used as an intermediate
input. It may accordingly be considered as the workers’ property that the producers may
2
What if the workers choose not to become capitalists even if all the products are theirs to
keep and even if going roundabout pays? They can let the producers behave as capitalists
on behalf of them. However, the capitalist producers cannot rake in their rent on capital
because it must be distributed to the workers in the capital goods sector as deferred wage
payments anyway. But if wage rates in the two sectors get equalized in equilibrium, the
workers in both sectors must get paid strictly higher than the average product of labor in the
capital good sector. And who pays that extra? The capitalist producer does. Thus, the
workers as the sole owners of labor endowment get everything, and the capitalist producers
none. The latter’s economic profits disappear as they must under perfect competition.
3
utilize on behalf of the workers. The producers thus defined are accordingly in position to
allocate both the intermediate capital (produced by primary labor alone) and the primary
labor (remaining for employment at a later stage) to the production of final consumption
goods. The producers' choice accordingly is between producing output directly with labor
alone for final consumption and producing intermediate goods (called capital goods) first,
followed by the second process of producing the final goods. In any case, all the fruit of
production is the workers, nothing being left for the competitive producers. Here the
producers may or may not be the capitalists as well. If the producers are not the capitalists,
then it must be the primary stage workers who will be transformed to the capitalists in the
second stage of production provided that they set aside their wage income to be used as an
intermediate input. In any event, rent on capital, if any, is not the producers’ to keep. It is
the workers’, or the worker capitalists’. The rent on capital that workers may own must be
the workers’ to keep.
Within the confines of such a simple model of production and distribution we also
ponder and show whether or not the more capital-intensive method of production is
tantamount to the more roundabout method of production. We will show in particular
conditions under which the roundabout method of production requires its structure to be of
more labor-intensive than capital-intensive, not the other way round.
Greenhut and Ohta (1976, 1979) address impact of vertical integration of related
industries or stages of production (physical distribution in effect) upon profits and welfare.
Their strictly positive impacts (of lesser roundabout production in effect) are predicated,
however, upon the assumption that unit costs of production/distribution are fixed, and
independent from the number of intermediate stages of production/distribution. But if
reducing the number of stages (or periods a la Boem Bawerk) of production raises the
remaining sectors’ transactions costs large enough to cancel the benefits of ‘double
marginalization’, then vertical integration may provide no rational grounds.
It is such a supply side effect of either a more or a less roundabout method of
production that the present article is concerned about. In what follows we set forth our
simple model of production along with the underlying assumptions. Two sections 1and 2 of
assumptions and the model are followed by the derivation in Section 3 of a critical set of
technological parameters that tend to promote roundabout methods of production. Section 4
provides a formal proof that there exists a nonempty set of parameters that can help an
apparently poor nation to surpass a strictly richer nation by going roundabout in production.
4
Section 5 concludes the paper.
1. Assumptions
Assume that an economy is endowed with a given primitive factor of production, namely
labor. The economy faces two alternative processes of production, one direct and the other
indirect or roundabout method of production a la Boem Bawerk. The direct method of
production requires labor input to catch fish, pick pecan, etc. or produce bundles of these
and other necessary goods for direct consumption. Average labor productivity of such a
composite product is assumed as given constant . The indirect method, by comparison,
requires output of labor to be used as an intermediated input, called capital, which is
combined with labor to produce the final consumption goods. The production technologies
in either sector or stage of production, represented by production functions, are linear
homogeneous. The consumption sector’s labor and capital inputs are further assumed as
perfectly divisible and mutual complements as well as substitutes. For simplicity we further
assume the production function in the final good sector is of the Cobb-Douglas type.
2. The Model
Pursuant to those assumptions above our simple model of Austrian roundabout methods of
production can be set forth as the following basic system of equations.
1) K = LK,

KL 1
2) X =
X
3) L = LX + LK
The basic system of equations above is yet to be fully complete, however. It consists of
only three equations in four endogenous variables: K, X, LK and LX, to be explained below.
To begin equation 1) defines the production function for an intermediate output K,
called capital, in terms of labor input LK allocated to the production of K, and a parameter
to represent average product of labor. Given , equation 1) represents a linear
homogeneous production function with respect to labor input LK alone. This will be referred
to as the primary sector (or first stage) production function.
Equation 2), by comparison, defines the production function for the final output X in
terms of two inputs K and LX. Note here that K is a capital good produced in the primary
5
sector and LX labor input allocated to the final consumption good X.
Equation 3) requires a given labor endowment L to be allocated to the primary sector
LK or the final consumption-good sector LX. Assumed here is that employment at either
sector of production precludes employment at the other sector, the two sectors being
separated consecutively in two stages such that the first stage production operates during
the daytime, say, and the second stage during nighttime. Those who work daytime to
produce capital goods are to be replaced by those who work after, say, 5 o’clock to produce
final consumption goods using the intermediate goods produced by the daytime labor.
Any producer/firm to employ workers in the intermediate capital-good sector must
promise the same wage rates to be offered in the consumption good sector under the
pressure of free competition. With such promise the producers must offer their intermediate
output K as capital on behalf of the primary sector’s workers. The producers in the final
consumption-good sector must employ both the capital available from the primary sector
and the remaining labor, not used in the primary sector, to produce the final consumption
goods. Their optimization problem is given by:
4) Max X  K  L1X 
subject to 1) and 3)

Solving the optimization problem 4) above requires the following first-order condition:
4)’ 
K

LX
The system of equations above consists of four independent equations in four
unknown variables, now readily solved as follows.

K* = L



LX* = (1  L
LK* =   L
X* = ()(1 – )1 –  L
6
L. Figure 1 below illustrates how all the four endogenous variables, K, LX, LK and X, are
determined within the confines of the simple, yet intrinsically Austrian roundabout methods
of production. 3 Table 1 correspondingly presents the reduced form solutions to the
variables in terms of the key parameters  and , given L normalized as L =1.
K
X(LX, K), K = LK
max K= L
= L
K

LX
LK
L
Figure 1 Going Roundabout in Production
only if X > max K  L

Table 1 Primary Factor Input, Intermediate Input and Final Output in Equilibrium
Negishi (1989, 94) apparently considers this to be a ‘simplified’ representation of Boem
Bawerkian model. K. Shibata (1959), by comparison, would consider such a model to be
more intrinsically Austrian than Walrasian. According to Shibata the Walrasian model is
predicated on multiple primary factors of production as well as multi-final products
whereas the Austrian model starts with labor alone as the sole primary factor of production,
with which the primary sector production is to be carried out. Thus, nothing can be
produced without an initial input of labor, not even capital goods that may be subsequently
used along with labor to produce at a later stage in an intrinsically Austrian structure of
production. For purposes of the present paper, however, changing the structure of
production to the one that is more general Walrasian than simple Austrian does not change
the basic results of the paper, so we apply an Occam’s razor to simplify our analysis.
3
7
First Stage
Primary Factor Input

LK
Second (Final) Stage
LX* = 1
Intermediate Output/Input
K* = (as Output)
K* =  (as Input)
Final Output
None
X*=()(1a)1a
Underlying Parameters


Note from Table 1 that the capital-labor ratio k* in the second stage (final
consumption-good sector), defined as k* = K*/LX* = /(1 – ), increases with , given .
3. The Parameters Set ( , 
Enables Roundabout Methods to Pay
Given the Cobb-Douglas type production technology as of 2), it is now straightforward to
derive the conditions under which roundabout methods of production become superior, i.e.,
X* > LNote the feasibility condition is defined as follows.
5) X*  K * L *1X   
Also note that substituting equilibrium values of K and LK from Table 1 into the

left-hand-side of 5) yields 5)’ (also shown in the table):
5)’ X*=()(1a)1a
Combining this with 5) in turn yields the following set SR of parameters  and :
6) SR  {(,; ) |   
  /(1 ) (1  )}
1/(1 )
This set defines technological parameters  and  that make roundabout methods
ofproduction feasible. Note that this set relation shows that for any given  within the
assumed domain of 1 > > 0, the labor productivity  in the capital good sector is required
to be strictly less than (1), which can be shown to be a strictly downward
sloping curve (function) of , given . This relation of  to  is illustrated by the shaded
area of ( ) in Figure 2 below. The shaded area asserts that, for any given within the
8
assumed range of 1 >  > 0, the output elasticity  is required to be strictly less than a
critical level given by the relation SR.
Thus, for example, if labor productivity in the capital good sector is low, say, with
 small enough to approach zero, the maximum output elasticity of capital  that makes
roundabout methods of production feasible is required to be large enough to approach unity.
In this case, the method of production in the final consumption good sector becomes highly
capital intensive. Conversely, if the labor productivity  in the primitive stage of
intermediate production happens to be large enough, the maximum  that is required to
justify roundabout production becomes low enough, and the related method of production
must be highly labor intensive.
Figure 2 illustrates the feasibility set SR (shaded in pale green), assuming for
simplicity  = 1.4 The following observations on the set are in order.

(; ) = (1)

B
B
1/4
A
1/2
1

Figure 2. Country A’s (, ) ‘In’ the Feasibility Set SR, Country B’s ‘Not In’
i) If direct output productivity of labor  is high enough to exceed the vertical
intercept value of the curve (; ) in Figure 2 above, then going roundabout
becomes infeasible regardless of .5 If a country has a high  thanks to, say,
If  is treated as a parametric variable instead of a given constant, then the set SR above
1/(1 )  /(1 )
may be redefined as SR  {( ,; ) |   

(1  )} , where  is now treated
as a parametric constant. Akiba (2004) assumes  = 1/2 to derive the set SR as  < 2/4.
5
What if labor productivity is less than unity, but high enough to approach unity? Going
roundabout does become feasible even in such a case if only output elasticity of labor in the
final sector
is high enough so that the method of production becomes sufficiently more
4
9
rich natural resources, then there is no need to go roundabout in production.
ii) If going roundabout pays, then direct output productivity of labor is required
to stay below or unity when = 1.6 However, a low  does not necessarily
call for roundabout methods,. Unless  is also low enough, going roundabout is
likely to fail.7 Consider, for example, two countries with an identical , say,  =
1/4, but suppose Country A’s  = A is lower than 1/2 and B’s = B, by
comparison, is higher than 1/2. Then Country A’s parameter combination (A
<1/2, A=1/4) must be included in the shaded area of Figure 2, but B’s
counterpart (B>1/2,  B = 1/4) is not. This implies that A will go roundabout,
but B will not. A related observation, which is straightforward and also
intriguing, is that A will produce more X by going roundabout than B will. Not
going roundabout, B will end up in producing no more than L, i.e., XA > XB =
L; or strictly less if they choose to go roundabout.
iii) More generally, for any given , the lower the output elasticity of capital in
the final consumption good sector, the more labor intensive the roundabout
process of production, and the higher the final output will be provided that (,
) is included in the set SR.8
iv) To be more precise as well as general (as a corollary in effect to footnote 8), it
can be shown, for any given , that final output X* under conditions of
roundabout methods of production becomes a strictly U-shaped convex function
of . Given an output elasticity of labor that is either smaller or greater than a
labor intensive than capital intensive.
6
However, even if direct labor productivity is very low, if output elasticity of labor is also
low enough, then going roundabout may not pay. For roundabout methods to prove feasible
under low enough , marginal product of labor in the second (final) stage of production
must be high enough relative to marginal product of capital.
7
For an intuitive interpretation of this relation consider the following. Given low labor
productivity in the primary sector, if the labor productivity in the second (or final) stage is
also low relative to the productivity of capital, then the second stage production would
require large enough input of high-powered capital. But in order to produce the needed
capital a large input of labor in the primary sector is needed with little labor left for input to
be combined to produce the final consumption goods. A large input of labor to produce
miniscule capital to be combined with all the more negligible input of labor is most likely
to yield unimpressive final outcomes.
8
For confirmation of this last result take partial of the left-hand-side of 5)’ with respect to
. The result, being X*ln /(1), is strictly negative as long as  < 1/(1+).
10
critical level of  = 1/(1+), it then follows that the greater the deviation thereof,
the greater the factor intensity of either input, i.e., capital or labor, will be and
the greater the final output tends to be. (See Appendix Figure.)
v) Getting back to the question of how a poor country A can surpass a rich
country B via roundabout methods of production, note for any given B <  for
Country B there always exists A for Country A that is small enough to make
roundabout methods of production superior to B’s direct counterpart methods.9
Put more generally, for any given B < , there exists a non-empty set of
parameter combinations (A, A) that yields an outcome of XA(A, A) > B.
4. Proof of v):
Consider an arbitrary B =  < 1. Then for a roundabout output for County A to exceed this
output level assumed for Country B the following inequality relation must hold.
7)
(A  ) (1  )1   B   0
A  1/0   1 (1  )( 1)/ 
This combined with 6) provides a set {, } that enables roundabout methods of production
both feasible and strictly more productive than the counterpart direct methods with any B

not greater than 1/(1).10
Figure 3 below obtained by superimposing 6) over Figure 2 above, illustrates
how the poor Country A’s primary labor productivity can be compared to B’s (assumed
to be as large as B =1/2 for simplicity) under which the poor A nevertheless can surpass the
rich B in final output per capita. In the Figure a circular area A is a subset of (, ) in blue
and stays strictly below the dotted line with  = 1/2. This implies that Country A is
endowed with strictly lower (A, A) than Country B’s counterpart (B, B), i.e., both A <
B and A < B, which in turn implies that A’s
finaloutput
exceeds B’s.
7) with
B = 1/2
9
It goes without saying, however, that even if Country A chooses a roundabout method,
and its production methods become more labor intensive, it does not guarantee its final
output and welfare to exceed B’s under the direct methods. Suppose A’s  is strictly lower
than B’s, i.e., A < B. Then, it is possible that even if A goes roundabout, its final output
remain below B’s if  A remains not small enough.
10
This critical value of 1/(1) becomes unity when  = 1, as assumed in Figures 2 and 3.
11

1
B
1/2
A
B = 1/2
6)
1/2
1

Figure 3. The Set {, } that Helps the Poor A Surpass the Rich B with B =1/2
Table 2 summarily presents certain particular combinations of technological
parameters and  that may yield contrasting impacts upon factor intensities and final
outputs (per capita). Even if the poor A’s primary stage labor productivity is half B’s or
even much lower, their final output per capita can be strictly higher than the rich B’s
provided that the poor’s output elasticity of capital is low enough to induce its second (or
final) stage of production highly labor intensive.
Table 2. Impacts of Low  cum Low  upon
Capital Intensity k and Final Output X/L


k
X/L
Country A
Low
Low
Low
High
Country B
High
High
High
Low
Observations on the set SR yield more general propositions as follows.
First, it is not necessarily the more capital-intensive methods of production that
underscore the Austrian roundabout structure of production. Technological conditions exist
under which the more labor-intensive, rather than capital-intensive, method of production
proves to be an optimal roundabout structure of production.
Second, of the set of parameters that promote roundabout methods of production, a
12
substantially large part (more than double the other parameters set 11 ) requires more
labor-intensive than capital-intensive methods to be adopted.
Third, for roundabout methods of production to prove feasible the primary sector’s
labor productivity  is required to be small enough. Otherwise, if labor productivity  of
something happens to be high enough to exceed unity, the producer can forgo any such
roundabout production process as producing something first as a means of production, then
combining it with labor input to produce something else for final consumption, or for
further input.
Forth, if labor productivity happens to be quite high, approaching but not exceeding
unity, roundabout production can nevertheless be feasible, if only output elasticity of labor
in the final consumption good sector is high enough so that the method of production
becomes increasingly more labor-intensive than capital-intensive. Conversely, the lower the
, the higher the maximum output elasticity  of the good used as capital input so that the
roundabout method of production can be more capital intensive than labor intensive.
5. Conclusion
If economic development requires roundabout methods of production, it does not ipso facto
require capital-intensive methods of production.12 On the contrary the more labor-intensive
rather than capital-intensive methods could yield greater final output when the primitive
labor productivity, denoted by , is low enough. If, on the contrary,  were large enough,
say exceeding unity in our simplistic model, then a large enough output elasticity of capital,
Note from Figure 2 that the set {(,) | being a subset of the shaded set
SR, assures roundabout methods to pay off, and moreover shows that the lesser the output
elasticity of capital , the more labor-intensive the roundabout process of production
becomes, and the larger the final output will be. This rectangular area is strictly greater than,
more than double, the shaded area to the left of = 1/2 on the horizontal axis. The shaded
parameters set on the right-hand side of = 1/2, by contrast, shows that  greater than 1/2
and the related production process with relatively more capital intensive than labor
intensive methods assure roundabout methods to be feasible. But it does not guarantee the
greater  and the related capital-intensive methods to yield the greater final output.
12
Hong Kong may be an example of success stories of economic growth with
11
labor-intensive methods of production in stark contrast with India with a bitter experience
of development in heavily capital-intensive industries. See Mark Skousen (2002).
13
denoted by , could generate the greater final output than does the low enough . But the
roundabout methods are strictly inferior to the direct method when primary/primitive labor
productivity is high enough relative to the total factor productivity in the final sector, i.e., 
> 1/(1 – ). (Cf. footnote 10 supra.) Thus, whenever the roundabout methods are feasible,
within the confines of our intrinsically Austrian structures of production with the primary
factor of production being primitive labor alone, the more labor-intensive methods are
generally superior to the more capital-intensive methods. If, however,  happened to be
large enough so that no labor-intensive method becomes feasible, then roundabout
production itself should be abandoned.
The poor (or weak) can surpass the rich (or strong) by going roundabout in
production with more labor-intensive rather than capital-intensive methods. This is not to
say, however, that the poor is ipso facto advantaged over the rich. If the rich has got the
same output elasticity of capital as the poor does, the rich must be better off than the poor.
For the poor (in primary productivity) to be able to surpass the rich, its output elasticity of
capital is required to be strictly less, less enough than the rich’s. Indeed the poor’s chances
to be able to surpass the rich increase with output elasticity of labor. That is, the smaller the
, i.e., the larger the (1), the greater the final output for any given , average
productivity of labor at the primary/primitive stage of production, will be.
For any given , however, if  is not small enough, then going roundabout in
production tends to be hardly attractive. This is the case even if a higher  could generate a
greater final output by adopting a more capital-intensive than labor-intensive method of
production. More generally put, if any combination of (, ) is not included in the set SR,
Figure 2, then no country could benefit from going roundabout in production. Country B
with a higher  loses to Country A with a strictly lower  because while B cannot take
advantage of going roundabout in production A can, and can enjoy strictly greater final
output than does B.
The upshot: Happy are the poor provided that output elasticity of labor in the
higher stage(s) of production is large enough to justify large input of labor, albeit combined
with a small amount of capital. Even though the available capital is limited due to low
primary productivity of labor besides the low output elasticity of capital as labor’s own
output, it nevertheless is an indispensable factor along with labor in the production of the
final output insofar as both factors are mutual complements as well as substitutes as
14
intrinsically embodied in the Cobb-Douglass production function we assume.13
A byproduct of our present inquiry, albeit straightforward, may warrant mention
as a final note. Regardless of either more capital-intensive or labor-intensive structures of
production that may yield surplus beyond average products of labor any such surplus will
be distributed not to the capitalist producers, but instead to the workers alone. This follows
from our basic assumptions that labor is the sole primary factor of production and perfect
competition prevails in all the markets, labor as well as the product markets.
13
If the Cobb-Douglass production function is a warp of the present model, then
an intrinsically Austrian roundabout structure of production starting with labor alone as the
sole primary factor of production is a woof to weave our tale of the happy poor.
15
References
Katsuhiko Akiba (2004), “On the Superiority of Labor-Intensive (rather than
Capital-Intensive) Roundabout Methods of Production,” Ph. D. Dissertation, Submitted to
SIPEB, Aoyama Gakuin University.
M. L. Greenhut and H. Ohta, “Related Market Conditions and Inter-Industrial Mergers,”
American Economic Review, June1976, 66, 257 – 77.
M. L. Greenhut and H. Ohta, “Vertical Integration of Successive Oligopolists,” American
Economic Review, March 1979, 69, 137 – 41.
Takashi Negishi (1989), History of Economic Theory, Amsterdam: North Holland.
Takashi Negishi (1989), The History of Economics, The Collected Essays of Takashi
Negishi, II, Aldershot, UK and Brookfield, USA: Edward Elgar.
Kei Shibata (1959), Dynamic and Dialectic Theories of World Capitalism, Minerva Shobo,
Kyoto, Japan.
Mark Skousen, "Poverty and Wealth: India Versus Hong Kong," Ideas on Liberty, February,
2002, 52.
16
Mathematical Appendix
On X*(;,):
X*=(1a)1a
X*     (1  )1
 lim X*  
 0
X*  
lim

1
Two related notes are in order.

1) The X* is a strictly convex U-shaped function of .
2) The term (1a)1a remains to be strictly lesser than unity for any   (0,1).
These two observations combined yield the following diagrammatic illustration of X*.

X*(;  >1)
X*

X*(;  =1)
X*(;  = 1/4)
1/2
1/4
1/5
0
1/2
4/5
1

Appendix Figure 1: Roundabout methods of production feasible only if  is small
enough (<1/2 here) when  is small enough (say, = 1/4) regardless of 
Thus, for example, given a red curve above, note for any  < 4/5, the smaller the
, the greater the final output X*, and moreover it exceeds 1/4 when  becomes smaller
than 1/2. However, the greater the  exceeding 4/5, by contrast, the greater once again the
final output X*; however it will not exceed 1/4 even if  approaches unity.
Consider a blue curve, by comparison, that assumes  =1. It shows Roundabout
methods are barely feasible only at either  0 or Roundabout methods are inferior
17
to the direct method of production for any value of  in between. More generally it follows
that unless  remains strictly below unity the roundabout methods become inferior to the
direct method of production. Moreover, regardless of how small  may be the roundabout
methods prove superior provided that  happens to be small enough as X* approaches 
however large it may be as  approaches 0.
The green curve above shows that roundabout methods of production become
superior once again if only  happens to be either small enough or large enough provided
however that  exceeds . But note here that when  exceeds unity the roundabout method
with higher  proves to be even more superior than that with lower 
In a nutshell the larger the  relative to , the more likely the roundabout methods
of production become superior to the direct methods. But regardless of  the value of  is
crucial in determining whether or not the roundabout methods prove to be superior with
either a more or less labor-intensive method. Specifically, the following relations hold.
3 1) :   1  lim X * ( )    lim X * ( )
 0
 1
3  2) :   1  lim X * ( )  lim X * ( )  
 1
 0
From 3-1) it follows that when  < 1 the only roundabout methods of production
feasible and superior to the direct method is when labor-intensive methods of production
 available with small enough  approaching 0.
are
When  > 1 from 3-2), by comparison, the roundabout methods of production can
be seen to become superior when either labor-intensive or capital-intensive methods of
production are available with either small enough  approaching 0 or large enough 
approaching unity. However, in this case capital-intensive roundabout methods prove to be
superior to the labor-intensive counterpart methods regardless of  provided that  > 1/,
however. Without this proviso even capital-intensive roundabout methods become inferior
to direct methods since lim X*(=1) =  1(maximum possible output under roundabout
methods of production) not exceeding unity, falls below  assumed as > 1.
A final note on what  > 1 means may be warranted. This condition being
equivalent to K > L implies that 
the real value of an intermediate capital good produced by
labor is greater than unity in labor units. Put alternatively, output of labor is more valuable
than labor. Then the more capital intensive than labor intensive methods must make sense.
Moreover, the higher the output elasticity of capital, the higher the capital intensity, and the
greater the final output.
18