Robert Vienneau
2004-06-26 09:38:29 UTC
"The marginal product of capital will in equilibrium be equal to the
real interest rate adjusted for market perceptions of risk. If one
investment offers a rate of return higher than others, capital will
flow to that investment and away from others, lowering the marginal
productivity of additional investment in the former and raising it
in the latter."
-- Don Dale making a mistake, 10 February 1997
"Neither let us forget that the real interest rate is equivalent
to the marginal product of capital."
-- Edward Flaherty making a mistake, 4 February 1995
"Suppose the interest rate were less than the marginal product of
capital. Borrowing and building capital would earn a positive return
so simple arbitrage would drive them together. Suppose the interest
rate were above the marginal product of capital. How could loans to
finance capital purchases then ever be repaid? So investment would
fall, allowing depreciation to raise the marginal product of capital
until it reached the interest rate."
-- Mark Witte making a mistake, 10 February 1995
"Then Buddha said: ... But tell me, Subhuti, do you really
believe that having only one homogeneous capital good will
permit you to derive a rate of profit purely from the
technical relationship between homogeneous capital and
output?
Subhuti replied: Thus it is said in some venerable books.
Buddha said: Revere them, Subhuti, but trust them not.
Suppose you do get the value of the marginal product of
capital in terms of output of consumer goods. In what units
will it be expressed? Physical units of additional consumer
goods per unit of additional homogeneoues capital. But the
rate of profit is a pure number. Surely you will need something
more in going from the first to the second to reflect the
relative price of the capital good vis-a-vis the consumer good.
But the equilibrium price of capital in units of consumer goods
depends on the rate of profit used for discounting, and a
variation of the rate of profit can involve a variation of the
value of the same physical capital in units of consumer goods.
This difficulty is not eliminated by having one homogeneous
good."
-- Amartya Sen, "On Some Debates in Capital Theory". Economica.
V. 41, August 1974.
1.0 INTRODUCTION
This essay demonstrates that the existence of "price Wicksell effects"
can lead to the inequality of the marginal product of capital and the
interest rate. The equality being challenged here should be understood
as it is used in macroeconomic models with aggregate production
functions. That is, macroeconomic modeling with aggregate production
functions is inadequately grounded in microeconomic theory. I conclude
with some rather far-reaching possibilities.
I have explained this before. Several economists have mistakenly
asserted this argument has a simple technical flaw, although they don't
all agree where that flaw lies. In fact, the length of my exposition
here results from my attempting to clarify several points of confusion
exhibited by economists responding to previous versions.
This argument is well-established in the literature ([1], [2]). I
suggest that those who think this argument mistaken should take a look
at some of my references. If my argument were mistaken, demonstrating
the mistake would be worthy of a paper.
I claim this argument is not about index number problems or the
aggregation of capital [3]. I also do not see how it relates to the
aggregation of production functions. Those who believe otherwise are
encouraged to be explicit about the connections. Perhaps, the question,
from a neoclassical perspective, is how the services of capital goods
are related to the quantity of "waiting" they supposedly represent.
2.0 SOME RELATIONSHIPS AMONG AGGEGATE VARIABLES
Consider a very simple capitalist economy in which the value of all net
output is distributed as wages or profits:
Y = W + P (1)
where Y is net national income, W is total wages, and P is total profits
(or interest charges). The term "profits" is used in some economic
traditions to mean what neoclassical economists think of as "interest."
If this causes confusions, read "profit" as "interest" throughout this
essay.
If there is some homogeneous unit in which to measure the labor force
(person-years), the wage w is related to total wages as in Equation 2:
W = w L (2)
where L is the number of person-years employed. Similarly, if the capital
stock used up in a year, K, can be valued in the same units as output,
total profits relate to the interest rate r as in Equation 3:
P = r K (3)
Equations 1, 2, and 3 are accounting identities, true by definition. No
assumptions have been made yet about how any of these variables are
determined.
Continuing with manipulation of accounting identities, we can transform
Equation 1 to Equation 4:
Y = w L + r K (4)
Or
y = w + r k (5)
where y is net output per head and k is the value of capital per head.
Note that the value of output per head, the wage, and the value of
capital per head are all measured in the same units, say bushels wheat.
The interest rate is a percentage rate with no units attached (other
than, perhaps, an implicit time dimension).
Some neoclassical economists relate net output to inputs of labor and
capital by means of an aggregate production function, which, when written
in per capita form looks like Equation 6:
y = f( k ) (6)
The function f is supposed to satisfy certain assumptions. Given these
assumptions and perfect competition, cost minimization (or the
maximization of *economic* profit) is supposed to ensure the
equilibrium conditions given by Equations 7 and 8:
r = f'( k ) = dy/dk (7)
w = f( k ) - k f'( k ) (8)
Equation 7 shows the interest rate is equal to the marginal product of
capital, while Equation 8 shows an equality between the wage and the
marginal product of labor [4]. I intend to challenge Equation 7 in any
truly multicommodity framework that includes Equations 5, 6, 7, and 8.
The argument for the aggregate production function, when written in
per capita form, is the value of capital per head. How can the value of
capital per head vary? Consider a multi-commodity model in a steady
state. Suppose the same technique is adopted at different interest
rates. The corresponding price structure will vary with the interest
rate. Even though the same capital goods may be used at different
interest rates, the value of capital per head will differ with the
interest rate. This variation in the value of a given set of capital
goods with the interest rate is known as a "price Wicksell effect."
Typically, though, the cost-minimizing technique will also vary with
the interest rate. Consider the prices ruling at a given interest rate,
where that interest rate is a switch point. That is, at least two
techniques are cost minimizing at the given interest rate. We can
then consider variations in capital goods resulting from a variation
in the usage of two cost minimizing techniques. The resulting variation
in the value of capital per head at the given prices is known as a
"real Wicksell effect." The chain-rule for differentiation shows how
the price and real Wicksell effects combine to determine the total
variation in the value of capital per head with the interest rate [5].
If only one fixed-coefficients (Leontief) technique is known, the
real Wicksell effect will be zero. But the price Wicksell effect may
be non-zero. So the assumption of a Leontief technique is no obstacle
to finding a nonzero variation in the value of capital per head with
the interest rate.
For completeness, I note there is a third manner in which the value
of capital per head can vary, namely if the composition of final output
varies, for example, due to a difference in the rate of growth. This
possibility is not important to my argument.
Now I want to prove a theorem by some simple formal manipulations.
Given Equation 5, the marginal product of capital is equal to the
interest rate (Equation 7) if and only if Equation 9 holds [6]:
k = - dw/dr (9)
Proof:
The total differential of Equation 5 is Equation 10:
dy = r dk + k dr + dw (10)
Thus, the interest rate is equal to the marginal product of capital
(Equation 7) if and only if Equation 11 holds:
k dr + dw = 0 (11)
Equation 9 follows. Q.E.D.
A demonstration that the value of capital per head need not be
equal to the additive inverse of the slope of the factor price
frontier (Equation 9) will demonstrate that the interest rate need
not be equal to the marginal product of capital (Equation 7).
3.0 A SIMPLE TWO-GOOD COUNTEREXAMPLE
The question to be investigated here is whether Equation 9 is
an implication of neoclassical microeconomics. I claim Equation 9
is not an implication of neoclassical microeconomic.
It is sufficient to demonstrate this negative conclusion by
describing an example compatible with neoclassical microeconomics, but
in which Equation 9 does not hold. The existence of such a
counterexample demonstrates the use in macroeconomics of models in which
the marginal product of capital and the interest rate are equal cannot
claim full generality. It is up to the users of such models to state
their assumptions and justify their use of these special cases.
How is the counter-example constructed? Assume we observe that
in our economy only two goods are produced, steel and wheat, each
measured in their own physical units, tons and bushels, respectively.
We also observe the physical quantity flows in each industry, which
I am going to write in a somewhat cryptic manner. Define
d( 0 ) = a12 a01 + ( 1 - a11 ) a02 (12)
Suppose the physical quantity flows are as in the following table
on a per worker basis:
INPUTS STEEL INDUSTRY WHEAT INDUSTRY
Labor [a01 a12/d(0)] person-years [(1 - a11) a02/d(0)] person-years
Steel [a11 a12/d(0)] tons steel [(1 - a11) a12/d(0)] tons steel
OUTPUTS [a12/d(0)] tons steel [(1 - a11)/d(0)] bushels wheat
where all quantities are positive and
0 < a11 < 1 (13)
Notice that the sum of person-years in both industries is unity, as
promised. Also notice that the sum of the inputs of steel is equal to
the steel produced by the steel industry. As a further clarification,
we observe that these inputs are purchased at the beginning of the
year, and the outputs become available at the end of the year.
Furthermore, the steel input is totally used up in these production
processes. The output of the steel industry just replaces the
steel used up in the economy. The net output consists solely of
wheat, which we observe to be a consumption good.
Assume constant returns to scale. This means we can express the
observed quantity flows as follows:
a01 person years & a11 tons steel PRODUCE 1 ton steel
a02 person years & a12 tons steel PRODUCE 1 bushel wheat
This explains the puzzling notation above. The parameters reflect
unit (gross) outputs in both industries.
Notice nothing has been assumed about the available technology other
than constant returns to scale and that these proportions are possible.
Based on our observations of the quantity flows actually used in this
little model economy, we can draw no conclusions about what outputs
will be produced when the inputs of either industry are in different
proportions. It might even be the case that wheat can be used as a
capital good for some other technology, or that copper or some other
capital good might be used at a different set of prices. We certainly
haven't assumed a Leontief fixed-coefficients technology.
We have observed that [a12/d(0)] tons steel per worker are used in the
economy, but this is not the value of capital per worker, k, used in
Equation 5. The units are different. If aggregate output per head, y, is
measured in units of bushels wheat per capita, capital per head, k, must
also be measured in bushels wheat per capita in the aggregate production
function framework. We have to figure out how many bushels of wheat this
quantity of steel represents. But that's what prices are for.
Suppose we observe that prices are unchanged over the year in
which we are making our observations, and that the wage w is paid at
the end of the year. Suppose that we also observe that competition
has brought about the same rate of interest in both industries. Let
wheat be numeraire. Then the following price equations obtain:
a11 p (1 + r) + a01 w = p (14)
a12 p (1 + r) + a02 w = 1 (15)
I have not specified enough equations to fully define the price
system. Thus, we can solve for two of the price variables in terms
of the third, say the rate of interest. Define:
d( r ) = a12 a01 (1 + r) + [ 1 - a11 ( 1 + r ) ] a02 (16)
The price of a ton of steel as a function of the interest rate is
then given by Equation 17:
p = a01 / d( r ) (17)
Hence, the quantity of steel, when measured in bushels wheat, is:
k = [ a12 a01 ] / [ d( 0 ) d( r ) ] (18)
This value quantity of steel varies with the interest rate.
The wage can also be found as a function of the rate of interest:
w = [ 1 - a11 (1 + r) ] / d( r ) (19)
Equation 19 expresses the factor price curve [7] associated with the
observed technique. A different technique may be preferred at a
different rate of interest. All these curves can be graphed on the
same diagram with the wage as the ordinate and the interest rate
as the abscissa. The cost-minimizing technique(s) at any interest rate
will correspond to the technique(s) with the highest wage at that
interest rate [8]. The factor price frontier is thus formed from
the outer-envelope curve of the factor price curves corresponding to
each individual technique. Points on this frontier that lie on two or
more curves for individual techniques are known as "switch points."
The optimal cost-minimizing technique is unique at interest rates
for non-switching points.
Assume the observed technique is a non-switching point in an
interval in which that technique is selected. Then the factor price
curve for the selected technique will be tangent to the factor
price frontier at this rate of interest. The desired derivative, dw/dr,
in Equation 9 is the slope of the factor price frontier at the observed
rate of interest. From this tangency relationship one can see that the
slope can be found by differentiating Equation 19, the factor price
curve for the observed technique.
It seems useful to provide an aside on a correct understanding of
marginal productivity relationships before proceeding with this
differentiation. The analysis of the choice of technique in long
run equilibrium through the construction of the factor price frontier
is completely general in circulating capital models. It applies to a
Leontief technique, a choice among several Leontief techniques, or
continuously differentiable micro-economic production functions in
which all inputs are specified in physical units (e.g. tons, bushels,
person-years). In the last case, all points along the frontier will
be non-switching points, although the chosen technique will vary
continuously with the interest rate [9]. Price Wicksell effects, as
explained in this essay, result in the difficulty in defining a unit
of "capital" in any case. Marginal productivity is another method of
analyzing the choice of technique in the continuously differentiable
case. When correctly applied, marginal productivity will not determine
the distribution of income, and no equation analogous to Equation 7 will
arise [10]. If my example is supplemented by the needed assumptions, one
can show that the price of a ton of steel is equal to the value of the
marginal product of (ton) steel in both sectors. Since time discounting
is used in this relationship, the interest rate will appear in the
mathematical statement of these equalities. But these equalities clearly
differ from Equation 7, for capital is measured in the same units as
output in Equation 7. The two good model has other properties that can
differ from the simple one good model.
Now we can return to our problem of examining Equation 9 for this
simple two-good model. From Equation 19, the slope of the factor price
frontier is given by Equation 20:
dw/dr = - a12 a01 / [ d( r ) d( r ) ] (20)
We can now compare the value of capital with the additive
inverse of the tangent to the factor price frontier.
The right hand sides of Equations 18 and 20 do not look like additive
inverses of one another. As a matter of fact, assuming a positive
interest rate, the interest rate is equal to the marginal
product of capital at any given interest rate if and only if
Equation 21 holds:
a01 / a11 = a02 / a12 (21)
So if neoclassical theory is compatible with a steady state in
which Equation 21 does not hold, macroeconomic models in which the
interest rate is equated to the marginal product of capital are
not the general case.
Equation 21 is quite interesting. It implies that equilibrium
prices are proportional to labor values, as defined in classical
economics. As a matter of history, the reliance of the labor theory
of value on this sort of extremely special case was thought to be a major
weakness. If neoclassicals find this condition too extreme for the
labor theory of value, they can hardly find it general enough as a
defense of neoclassical macroeconomics [11].
Perhaps the solution lies in adopting another method of evaluating
the physical quantity of capital in the same units as net output.
Champernowne has proposed a "chain index" measure of capital that will
restore the macroeconomic equality [12]. However, this measure only works
under special cases, too. Burmeister has shown that the macroeconomic
equality can be established with this chain-index if and only if
real Wicksell effects are always negative. However, as was shown
by the Cambridge Capital Controversy, this assumption of "negative
real Wicksell effects" is not a general case either. In fact, nobody
has determined what are necessary assumptions on technology to
ensure the desired conclusion will follow [13]. Finally, if this index
is used to express an aggregate production function in per capita
terms, the wage is no longer equal to the marginal product of
labor as that marginal product is typically expressed in such
functions [14].
But, some may object, aggregate production functions work
empirically. So if economists cannot even state their assumptions, they
may say, this empirical success justifies the continual use of aggregate
production functions. This is an extremely weak defense. Income
distribution has been stable over much of the period in which
macroeconomists have been using aggregate production functions. Franklin
Fisher has shown through simulation that the supposed empirical success
of aggregate production functions can arise under these conditions even
in cases where the needed assumptions do not hold. Thus, this supposed
empirical success of aggregate production functions fails to test the
models with the unstated assumptions of aggregate neoclassical theory or
to test among alternative theories. In fact, economists who rely on this
defense seem to be confusing their empirical results with another
accounting relationship [15].
4.0 CONCLUSION
This article has presented a simple explanation of the
nonequality of the interest rate and the marginal product of
capital, as that equality is understood in macroeconomic models.
Thus, neoclassical microeconomics does not imply that equality.
Various attempts to defend the macroeconomic models considered
here have been examined and have been found wanting. An interesting
aspect of this criticism is that it does not seem to be about index
number problems [16]. Nor has this argument depended on the
phenomena of reswitching at all, and it depended on capital reversing
only in criticizing Champernowne's chain index [17]. If the
demonstration of the theoretical possibility of these phenomena
are taken as central to the Cambridge Capital Controversy, then
that controversy should have been about something other than
aggregate production functions [18]. As far as I can see, this argument
is fairly well understood on the Cambridge, England side.
I conclude that the CCC was indeed about something else. I happen to
think the topic under debate was a confrontation of two rival paradigms
of value and distribution, namely the Classical theory and the so-called
Neoclassical theory [19]. In particular, it was shown, I think, that
Neoclassical economists cannot consistently maintain in their equilibrium
framework that owners of capital goods make any contribution to
production. Hence interest cannot be a payment for such a
contribution [20].
Finally, it is curious that economists continue to use aggregate
production functions despite the clear warnings of this traditional
argument. Many of those economists who follow Solow or Lucas do not
seem concerned about their inadequate concept of capital. Although these
researchers may be interested in technical improvements in their models,
capital-theoretic problems do not seem high on their agenda. Furthermore,
much of graduate education in economics seems to leave newly emerging
economists unaware of capital-theoretic problems with their models. These
young economists do not seem to possess the analytical tools that were
forged in the CCC, such as the correct analysis of the choice of
technique, a correct analysis of the relationship between the theory of
rent and income distribution, or even how to analyze depreciation and
the economic life of machines in a framework of joint production.
What explains this apparent continuation of the miseducation of
economists that Joan Robinson decried over forty years ago [21]? My
hypothesis is partly ideological. Any advanced treatment of capital
theory and the appropriate analytical tools [22] would expose the
student to the Cambridge Capital Controversy. The student would then
learn about some serious questioning of the internal consistency of
many claims of neoclassical economists. There are obviously normative
overtones to this controversy, for example, over the exploitative
nature of profits and the capitalist system as a whole. Neoclassical
economics might be claimed to currently fill the social role of "hired
prize fighters" for capital, what Marx characterized as "vulgar
economics" [23]. This social role is threatened by the CCC.
5.0 FOOTNOTES
[1] Elements of this argument can be found in Joan Robinson's article
"The Production Function and the Theory of Capital," _Review of Economic
Studies_, 1953-4 and Geoff Harcourt's _Some Cambridge Controversies in
the Theory of Capital_, 1972. The closest formulation to my argument is
in the following papers:
Amit Bhaduri, "The Concept of the Marginal Productivity of Capital
and the Wicksell Effect," Oxford Economic Papers, XVIII, 1966,
pp. 284-88
Amit Bhaduri, "On the Significance of Recent Controversies on Capital
Theory: A Marxian View," Economic Journal, LXXIX, 1969, pp. 532-9.
But the argument was also in Sraffa. See, for example:
Piero Sraffa, "Production of Commodities: A Comment," _Economic
Journal_, V. LXXII, June 1962, pp. 477-9.
[2] "Now a major problem existed because capital, unlike either labor or
land, is a produced means of production and cannot be measured
unambiguously in purely physical terms: the amount of capital can be
measured only in value terms. The problem was to establish the idea of
a market for capital, the quantity of which could be expressed
independently of the price of its service (i.e. the rate of profit)...
The basic deficiency with this approach is in its treatment of capital,
which cannot be measured independently of the rate of profit. As
observed above, the value of capital, like that of all produced
commodities, depends on the rate of profit, or interest."
-- J. E. Woods, _The Production of Commodities: An Introduction to
Sraffa_, Humanities Press International, 1990, pp. 306-307.
[3] [the divergence between the marginal product of capital and the rate
of interest] "is attributable to the fact that it is impossible to
find an invariant unit in which to measure the social quantity of
capital."
"To put the matter another way, we may say that a change in the supply
of capital - arising, for example, from new voluntary savings - alters
the units in which all the previously existing capital is measured;
and it is therefore incorrect to say that the supply of capital as a
whole has increased by the amount of the voluntary saving. It is
important to emphasize that this problem of measuring the quantity
of capital is not an index-number problem. There are, to be sure,
numerous index-number problems of the greatest complexity in the
theory of capital. But the problem to which I now refer would exist
even in the simplest economy in which all output consisted of a single
type of consumer's good and firms were exactly alike."
-- L. A. Metzler, "The Rate of Interest and the Marginal Product of
Capital," _Journal of Political Economy_, Vol. 53, 1950, pp.
284-306.
Metzler provides a brief literature review of awareness of this problem
going back to Wicksell. My analysis is closest to his comments on
Knight's capital theory, though I think my presentation is clearer.
[4] A more general statement of these relations, abstracting from
price Wicksell effects, is given by Equations 7' and 8':
df+/dk <= r <= df-/dk (7')
f( k ) - k df-/dk <= w <= f( k ) - k df+/dk (8')
where df-/dk is the left-hand dervative and df+/dk is the right hand
derivative. In the discrete case without price Wicksell effects, the
neoclassical aggregate production function is supposed to consist of
positively sloped line segments connecting "kinks" where the left-hand
and right-hand derivatives are not equal. The line segments correspond
to switch points (defined below), while the "kinks" are non-switching
points.
[5] A good explanation of price and real Wicksell effects can be found
in:
Edwin Burmeister, "Wicksell Effects," _The New Palgrave_.
Burmeister writes:
"The value of capital, however, is not an appropriate measure of the
'aggregate capital stock' as a factor of production except under
extremely restrictive assumptions. Wicksell (1893, 1934) originally
recognized this fact, which subsequently was emphasized by Robinson
(1956)."
[6] One can show that Equation 9 follows from 7' at a non-switching
point in the discrete case. Consider two sets of values for y, w,
r, and k:
y2 = w2 + r2 k2 (I)
y1 = w1 + r1 k1 (II)
The difference is given by Equation III:
y2 - y1 = w2 - w1 + r2 k2 - r1 k2 + r1 k2 - r1 k1 (III)
Or, in obvious notation:
dy = dw + k2 dr + r1 dk (IV)
Assume 7'. There are two cases.
Case 1: dk > 0 in Equation IV. Thus, dr < 0. Ignoring higher-order
terms:
y2 = y1 + (k2 - k1) (df+/dk)(k1) (V)
Equation 7' gives:
y2 <= y1 + (k2 - k1) r1 (VI)
Or:
dy <= r1 dk (VII)
Thus,
dw + k2 dr <= 0 (VIII)
Rearranging and taking the left-hand derivative gives:
- dw/dr- <= k (IX)
Case 2: dk < 0 and dr > 0 in Equation IV. Once again, ignoring
higher order terms:
y2 = y1 + (k2 - k1) (df-/dk)(k1) (XI)
The inequality in Display VI follows once again from Equation 7'.
Thus, Display VIII holds here, as well. Rearranging and taking the
right-hand derivative yields:
k <= - dw/dr+ (XII)
Since the left-hand and right-hand derivatives of the factor-price
frontier are equal at a non-switching point in the discrete case,
k = - dw/dr (XIII)
which was to be shown.
[7] See:
Heinz D. Kurz, "Factor Price Frontier," _The New Palgrave_.
[8] Textbook treatments of the connection between cost minimization and
the factor price frontier can be found in (Woods 1990) or
Heinz D. Kurz and Neri Salvadori, _Theory of Production: A Long
Period Analysis_, Cambridge University Press, 1995.
[9] These properties of the factor price frontier in the "continuous
substitution" case were brought out by Luigi Pasinetti in correcting
a technical mistake by Robert Solow. See:
Luigi Pasinetti, "Switches of Technique and the 'Rate of Return' in
Capital Theory," _Economic Journal_, 1969, pp. 508-513.
It seems worth pointing out, since many may be confused on this point,
that reswitching and capital reversing are possible when the optimal
technique varies continuously with the interest rate. See:
P. Garegnani, "Heterogeneous capital, the Production Function and the
Theory of Distribution," _Review of Economic Studies_, v 37, June 1970,
pp. 407-36.
[10] See:
Frank Hahn, "The neo-Ricardians," _Cambridge Journal of Economics_,
V. 6, 1982, pp. 353-374.
[11] Hence, Paul Samuelson's defense of aggregate production functions
is inadequate. This defense can be found in
Paul A. Samuelson, "Parable and Realism in Capital Theory: The
Surrogate Production Function," _Review of Economic Studies_, 1962,
pp. 193-206.
[12] D. G. Champernowne, "The Production Function and the Theory of
Capital: A Comment," _Review of Economic Studies_, V. 21, 1953-4, pp.
112-35.
[13] See the reference in footnote 5.
[14] Salvatore Baldone, "From Surrogate to Pseudo Production Functions,"
_Cambridge Journal of Economics_, V. 8, 1984, pp. 271-288. Baldone also
shows Burmeister's claims are problematic when used to compare
quasi-stationary economies with a positive rate of growth, instead of
just stationary economies.
[15] Anwar Shaikh, "Humbug Production Function," _The New Palgrave:
Capital Theory_, Macmillan, 1990. See also John S. L. McCombie,
"The Solow Residual, Technical Change, and Aggregate Production
Functions," _Journal of Post Keynesian Economics, V. 23, Winter
2000-2001, pp. 267-298.
[16] See footnote 3.
[17] "The construction of the production function does not even require
this refutation via the phenonomenon of returning techniques
('reswitching'), because a production function for which the marginal
product equals the factor price already becomes impossible if the
wage curves of single techniques are not straight lines (except for
a few unimportant cases; see Garegnani, 1970, Hunt and Schwartz,
1972). Contrary to the usual interpretations today, the debate about
the possibility of returning techniques is important not only because
it proves that the production function with its marginal products is
nonsensical, but because, on a more general level, it can be shown
that a demand function for capital...cannot be defined."
-- Bertram Schefold, _Mr. Sraffa on Joint Production and Other
Essays_, Unwin Hyman, 1989, p. 292.
[18] "[O]ne should emphasize the distinction between two types of
measurement. First, there was the one in which the statisticians were
mainly interested. Second, there was measurement in theory. The
statisticians' measures were only approximate and provided a suitable
field for work in solving index number problems. The theoretical
measures required absolute precision. Any imperfections in these
theoretical measures were not merely upsetting, but knocked down the
whole theoretical basis. One could measure capital in pounds or dollars
and introduce this into a production function. The definition in this
case must be absolutely water-tight, for with a given quantity of
capital one had a certain rate of interest so that the quantity of
capital was an essential part of the mechanism. One therefore had to
keep the definition of capital separate from the needs of statistical
measurement, which were quite diffent. The work of J. B. Clark,
Bohm-Bawerk and others was intended to produce pure definitions of
capital, as required by their theories, not as a guide to actual
measurement. If we found contradictions, then these pointed to
defects in the theory, and an inability to define measures of capital
accurately. It was on this - the chief failing of capital theory -
that we should concentrate rather than on problems of measurement."
-- Piero Sraffa, Interventions in the debate at the Corfu Conference
on the "Theory of Capital", 4-11 September 1958.
[19] "Capital theory has been one of the most contentious areas of debate
in economic analysis. One reason for this is that it is the point at
which the classical theory of value and distribution, and the
neoclassical theory of price meet, so to speak, on the same ground.
Classical theory was devoted to explaining the determination of the
rate of profit and associated 'natural prices' in an economy using
reproducible means of production. In so far as neoclassical theory
attempts to determine the rate of profit and associated 'long-run
prices' it is offering an alternative explanation of exactly the
same things."
-- John Eatwell, Murray Milgate, and Peter Newman, "Preface,"
_The New Palgrave: Capital Theory_, Macmillan, 1990.
[20] "...while wages are paid for work, and one can (and in some
circumstances should) think of the wage bill...as reproducing the
power to work, *profits are not paid for anything at all.* The flow of
profit income is not an exchange in any sense. The Samuelson diagram is
fundamentally misleading; there is no 'flow' from 'household supply' to
the factor market for capital. The *only* flow is the flow of profit
income in the other direction. And this, of course, leads straight to
that hoary but substantial claim that the payment of wages is not an
exchange either, or at any rate, not a fair one. For Wages plus Profits
adds up to the Net Income Product; yet Profits are not paid for
anything, while wages are paid for work. Hence the work of labor
(using the tools, equipment, etc., replacement and depreciation of
which is already counted in) has produced the entire product. Is
labor not therefore exploited? Does it not deserve the whole product?"
-- Edward Nell, "The Revival of Political Economy," in _Economic
Relevance: A Second Look_, edited by Robert L. Heilbroner and Arthur
Ford, 1971, 1976.
[21] This miseducation is asserted on slightly different grounds in the
following quote:
"Observe that even in neoclassical theory full employment alone is
not enough to transform marginal-productivity relationships into a
long-run theory of distribution. In long-run neoclassical theory, the
capital:labor ratio is endogeneously determined, so that the wage rate
cannot be determined solely by marginal productivity of labor at full
employment - not even in Chicago. Instead, distribution must reflect
household preferences with respect to present and future consumption."
"Thus, it is fair to conclude that there are two marginal-
productivity theories. The first is a relatively innocuous, general
theory that involves nothing more controversial than competitive
profit maximization - and provides correspondingly little
contribution to the theory of growth and distribution under
capitalism. The second is more powerful, and very special, providing
by itself a theory of distribution, for the short run at least,
whose 'only' defects are (1) that it assumes full employment
and (2) that it begs the question of accumulation. The wonder is that
it is precisely this theory that so many students come away with from
their study of economics. Only slightly more wondrous is that by and
large they believe it!"
-- Stephen A. Marglin, _Growth, Distribution, and Prices_, Harvard
University Press, 1984, pp. 330-331.
Neither version of marginal productivity theory need include the equality
of the marginal product of capital and the interest rate in an aggregate
production function framework.
[22] Syed Ahmad, _Capital in Economic Theory: Neo-classical, Cambridge,
and Chaos_, Edward-Elgar, 1991.
[23] "In summary I believe that Marx's sociology of economic knowledge
was quite an impressive achievement, in spite of being flawed by its
reliance on functional explanations and the labor theory of value...
The recent 'capital controversy' shows that these are not dead issues.
Surely some cognitive confusion lay at the origin of the idea that
'capital' can be treated as a homogeneous 'factor of production,' for
instance an inference from the fact that *capitalists* form a fairly
homogeneous class. And conceivably the tenacity with which the
neoclassical economists stuck to the notion of aggregate capital has
something to do with non-cognitive interests. This, admittedly, is
sheer speculation, and I may be quite wrong. Vested intellectual
interests may suffice to explain the resistance. Be this as it may,
the sociology of economic conceptions and economic theory is a field
worth cultivating, if proper attention is paid to the many
methodological pitfalls in this domain."
-- Jon Elster, _Making Sense of Marx_, Cambridge University Press,
1985, p. 504.
--
Try http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/Bukharin.html
To solve Linear Programs: .../LPSolver.html
r c A game: .../Keynes.html
v s a Whether strength of body or of mind, or wisdom, or
i m p virtue, are found in proportion to the power or wealth
e a e of a man is a question fit perhaps to be discussed by
n e . slaves in the hearing of their masters, but highly
@ r c m unbecoming to reasonable and free men in search of
d o the truth. -- Rousseau
real interest rate adjusted for market perceptions of risk. If one
investment offers a rate of return higher than others, capital will
flow to that investment and away from others, lowering the marginal
productivity of additional investment in the former and raising it
in the latter."
-- Don Dale making a mistake, 10 February 1997
"Neither let us forget that the real interest rate is equivalent
to the marginal product of capital."
-- Edward Flaherty making a mistake, 4 February 1995
"Suppose the interest rate were less than the marginal product of
capital. Borrowing and building capital would earn a positive return
so simple arbitrage would drive them together. Suppose the interest
rate were above the marginal product of capital. How could loans to
finance capital purchases then ever be repaid? So investment would
fall, allowing depreciation to raise the marginal product of capital
until it reached the interest rate."
-- Mark Witte making a mistake, 10 February 1995
"Then Buddha said: ... But tell me, Subhuti, do you really
believe that having only one homogeneous capital good will
permit you to derive a rate of profit purely from the
technical relationship between homogeneous capital and
output?
Subhuti replied: Thus it is said in some venerable books.
Buddha said: Revere them, Subhuti, but trust them not.
Suppose you do get the value of the marginal product of
capital in terms of output of consumer goods. In what units
will it be expressed? Physical units of additional consumer
goods per unit of additional homogeneoues capital. But the
rate of profit is a pure number. Surely you will need something
more in going from the first to the second to reflect the
relative price of the capital good vis-a-vis the consumer good.
But the equilibrium price of capital in units of consumer goods
depends on the rate of profit used for discounting, and a
variation of the rate of profit can involve a variation of the
value of the same physical capital in units of consumer goods.
This difficulty is not eliminated by having one homogeneous
good."
-- Amartya Sen, "On Some Debates in Capital Theory". Economica.
V. 41, August 1974.
1.0 INTRODUCTION
This essay demonstrates that the existence of "price Wicksell effects"
can lead to the inequality of the marginal product of capital and the
interest rate. The equality being challenged here should be understood
as it is used in macroeconomic models with aggregate production
functions. That is, macroeconomic modeling with aggregate production
functions is inadequately grounded in microeconomic theory. I conclude
with some rather far-reaching possibilities.
I have explained this before. Several economists have mistakenly
asserted this argument has a simple technical flaw, although they don't
all agree where that flaw lies. In fact, the length of my exposition
here results from my attempting to clarify several points of confusion
exhibited by economists responding to previous versions.
This argument is well-established in the literature ([1], [2]). I
suggest that those who think this argument mistaken should take a look
at some of my references. If my argument were mistaken, demonstrating
the mistake would be worthy of a paper.
I claim this argument is not about index number problems or the
aggregation of capital [3]. I also do not see how it relates to the
aggregation of production functions. Those who believe otherwise are
encouraged to be explicit about the connections. Perhaps, the question,
from a neoclassical perspective, is how the services of capital goods
are related to the quantity of "waiting" they supposedly represent.
2.0 SOME RELATIONSHIPS AMONG AGGEGATE VARIABLES
Consider a very simple capitalist economy in which the value of all net
output is distributed as wages or profits:
Y = W + P (1)
where Y is net national income, W is total wages, and P is total profits
(or interest charges). The term "profits" is used in some economic
traditions to mean what neoclassical economists think of as "interest."
If this causes confusions, read "profit" as "interest" throughout this
essay.
If there is some homogeneous unit in which to measure the labor force
(person-years), the wage w is related to total wages as in Equation 2:
W = w L (2)
where L is the number of person-years employed. Similarly, if the capital
stock used up in a year, K, can be valued in the same units as output,
total profits relate to the interest rate r as in Equation 3:
P = r K (3)
Equations 1, 2, and 3 are accounting identities, true by definition. No
assumptions have been made yet about how any of these variables are
determined.
Continuing with manipulation of accounting identities, we can transform
Equation 1 to Equation 4:
Y = w L + r K (4)
Or
y = w + r k (5)
where y is net output per head and k is the value of capital per head.
Note that the value of output per head, the wage, and the value of
capital per head are all measured in the same units, say bushels wheat.
The interest rate is a percentage rate with no units attached (other
than, perhaps, an implicit time dimension).
Some neoclassical economists relate net output to inputs of labor and
capital by means of an aggregate production function, which, when written
in per capita form looks like Equation 6:
y = f( k ) (6)
The function f is supposed to satisfy certain assumptions. Given these
assumptions and perfect competition, cost minimization (or the
maximization of *economic* profit) is supposed to ensure the
equilibrium conditions given by Equations 7 and 8:
r = f'( k ) = dy/dk (7)
w = f( k ) - k f'( k ) (8)
Equation 7 shows the interest rate is equal to the marginal product of
capital, while Equation 8 shows an equality between the wage and the
marginal product of labor [4]. I intend to challenge Equation 7 in any
truly multicommodity framework that includes Equations 5, 6, 7, and 8.
The argument for the aggregate production function, when written in
per capita form, is the value of capital per head. How can the value of
capital per head vary? Consider a multi-commodity model in a steady
state. Suppose the same technique is adopted at different interest
rates. The corresponding price structure will vary with the interest
rate. Even though the same capital goods may be used at different
interest rates, the value of capital per head will differ with the
interest rate. This variation in the value of a given set of capital
goods with the interest rate is known as a "price Wicksell effect."
Typically, though, the cost-minimizing technique will also vary with
the interest rate. Consider the prices ruling at a given interest rate,
where that interest rate is a switch point. That is, at least two
techniques are cost minimizing at the given interest rate. We can
then consider variations in capital goods resulting from a variation
in the usage of two cost minimizing techniques. The resulting variation
in the value of capital per head at the given prices is known as a
"real Wicksell effect." The chain-rule for differentiation shows how
the price and real Wicksell effects combine to determine the total
variation in the value of capital per head with the interest rate [5].
If only one fixed-coefficients (Leontief) technique is known, the
real Wicksell effect will be zero. But the price Wicksell effect may
be non-zero. So the assumption of a Leontief technique is no obstacle
to finding a nonzero variation in the value of capital per head with
the interest rate.
For completeness, I note there is a third manner in which the value
of capital per head can vary, namely if the composition of final output
varies, for example, due to a difference in the rate of growth. This
possibility is not important to my argument.
Now I want to prove a theorem by some simple formal manipulations.
Given Equation 5, the marginal product of capital is equal to the
interest rate (Equation 7) if and only if Equation 9 holds [6]:
k = - dw/dr (9)
Proof:
The total differential of Equation 5 is Equation 10:
dy = r dk + k dr + dw (10)
Thus, the interest rate is equal to the marginal product of capital
(Equation 7) if and only if Equation 11 holds:
k dr + dw = 0 (11)
Equation 9 follows. Q.E.D.
A demonstration that the value of capital per head need not be
equal to the additive inverse of the slope of the factor price
frontier (Equation 9) will demonstrate that the interest rate need
not be equal to the marginal product of capital (Equation 7).
3.0 A SIMPLE TWO-GOOD COUNTEREXAMPLE
The question to be investigated here is whether Equation 9 is
an implication of neoclassical microeconomics. I claim Equation 9
is not an implication of neoclassical microeconomic.
It is sufficient to demonstrate this negative conclusion by
describing an example compatible with neoclassical microeconomics, but
in which Equation 9 does not hold. The existence of such a
counterexample demonstrates the use in macroeconomics of models in which
the marginal product of capital and the interest rate are equal cannot
claim full generality. It is up to the users of such models to state
their assumptions and justify their use of these special cases.
How is the counter-example constructed? Assume we observe that
in our economy only two goods are produced, steel and wheat, each
measured in their own physical units, tons and bushels, respectively.
We also observe the physical quantity flows in each industry, which
I am going to write in a somewhat cryptic manner. Define
d( 0 ) = a12 a01 + ( 1 - a11 ) a02 (12)
Suppose the physical quantity flows are as in the following table
on a per worker basis:
INPUTS STEEL INDUSTRY WHEAT INDUSTRY
Labor [a01 a12/d(0)] person-years [(1 - a11) a02/d(0)] person-years
Steel [a11 a12/d(0)] tons steel [(1 - a11) a12/d(0)] tons steel
OUTPUTS [a12/d(0)] tons steel [(1 - a11)/d(0)] bushels wheat
where all quantities are positive and
0 < a11 < 1 (13)
Notice that the sum of person-years in both industries is unity, as
promised. Also notice that the sum of the inputs of steel is equal to
the steel produced by the steel industry. As a further clarification,
we observe that these inputs are purchased at the beginning of the
year, and the outputs become available at the end of the year.
Furthermore, the steel input is totally used up in these production
processes. The output of the steel industry just replaces the
steel used up in the economy. The net output consists solely of
wheat, which we observe to be a consumption good.
Assume constant returns to scale. This means we can express the
observed quantity flows as follows:
a01 person years & a11 tons steel PRODUCE 1 ton steel
a02 person years & a12 tons steel PRODUCE 1 bushel wheat
This explains the puzzling notation above. The parameters reflect
unit (gross) outputs in both industries.
Notice nothing has been assumed about the available technology other
than constant returns to scale and that these proportions are possible.
Based on our observations of the quantity flows actually used in this
little model economy, we can draw no conclusions about what outputs
will be produced when the inputs of either industry are in different
proportions. It might even be the case that wheat can be used as a
capital good for some other technology, or that copper or some other
capital good might be used at a different set of prices. We certainly
haven't assumed a Leontief fixed-coefficients technology.
We have observed that [a12/d(0)] tons steel per worker are used in the
economy, but this is not the value of capital per worker, k, used in
Equation 5. The units are different. If aggregate output per head, y, is
measured in units of bushels wheat per capita, capital per head, k, must
also be measured in bushels wheat per capita in the aggregate production
function framework. We have to figure out how many bushels of wheat this
quantity of steel represents. But that's what prices are for.
Suppose we observe that prices are unchanged over the year in
which we are making our observations, and that the wage w is paid at
the end of the year. Suppose that we also observe that competition
has brought about the same rate of interest in both industries. Let
wheat be numeraire. Then the following price equations obtain:
a11 p (1 + r) + a01 w = p (14)
a12 p (1 + r) + a02 w = 1 (15)
I have not specified enough equations to fully define the price
system. Thus, we can solve for two of the price variables in terms
of the third, say the rate of interest. Define:
d( r ) = a12 a01 (1 + r) + [ 1 - a11 ( 1 + r ) ] a02 (16)
The price of a ton of steel as a function of the interest rate is
then given by Equation 17:
p = a01 / d( r ) (17)
Hence, the quantity of steel, when measured in bushels wheat, is:
k = [ a12 a01 ] / [ d( 0 ) d( r ) ] (18)
This value quantity of steel varies with the interest rate.
The wage can also be found as a function of the rate of interest:
w = [ 1 - a11 (1 + r) ] / d( r ) (19)
Equation 19 expresses the factor price curve [7] associated with the
observed technique. A different technique may be preferred at a
different rate of interest. All these curves can be graphed on the
same diagram with the wage as the ordinate and the interest rate
as the abscissa. The cost-minimizing technique(s) at any interest rate
will correspond to the technique(s) with the highest wage at that
interest rate [8]. The factor price frontier is thus formed from
the outer-envelope curve of the factor price curves corresponding to
each individual technique. Points on this frontier that lie on two or
more curves for individual techniques are known as "switch points."
The optimal cost-minimizing technique is unique at interest rates
for non-switching points.
Assume the observed technique is a non-switching point in an
interval in which that technique is selected. Then the factor price
curve for the selected technique will be tangent to the factor
price frontier at this rate of interest. The desired derivative, dw/dr,
in Equation 9 is the slope of the factor price frontier at the observed
rate of interest. From this tangency relationship one can see that the
slope can be found by differentiating Equation 19, the factor price
curve for the observed technique.
It seems useful to provide an aside on a correct understanding of
marginal productivity relationships before proceeding with this
differentiation. The analysis of the choice of technique in long
run equilibrium through the construction of the factor price frontier
is completely general in circulating capital models. It applies to a
Leontief technique, a choice among several Leontief techniques, or
continuously differentiable micro-economic production functions in
which all inputs are specified in physical units (e.g. tons, bushels,
person-years). In the last case, all points along the frontier will
be non-switching points, although the chosen technique will vary
continuously with the interest rate [9]. Price Wicksell effects, as
explained in this essay, result in the difficulty in defining a unit
of "capital" in any case. Marginal productivity is another method of
analyzing the choice of technique in the continuously differentiable
case. When correctly applied, marginal productivity will not determine
the distribution of income, and no equation analogous to Equation 7 will
arise [10]. If my example is supplemented by the needed assumptions, one
can show that the price of a ton of steel is equal to the value of the
marginal product of (ton) steel in both sectors. Since time discounting
is used in this relationship, the interest rate will appear in the
mathematical statement of these equalities. But these equalities clearly
differ from Equation 7, for capital is measured in the same units as
output in Equation 7. The two good model has other properties that can
differ from the simple one good model.
Now we can return to our problem of examining Equation 9 for this
simple two-good model. From Equation 19, the slope of the factor price
frontier is given by Equation 20:
dw/dr = - a12 a01 / [ d( r ) d( r ) ] (20)
We can now compare the value of capital with the additive
inverse of the tangent to the factor price frontier.
The right hand sides of Equations 18 and 20 do not look like additive
inverses of one another. As a matter of fact, assuming a positive
interest rate, the interest rate is equal to the marginal
product of capital at any given interest rate if and only if
Equation 21 holds:
a01 / a11 = a02 / a12 (21)
So if neoclassical theory is compatible with a steady state in
which Equation 21 does not hold, macroeconomic models in which the
interest rate is equated to the marginal product of capital are
not the general case.
Equation 21 is quite interesting. It implies that equilibrium
prices are proportional to labor values, as defined in classical
economics. As a matter of history, the reliance of the labor theory
of value on this sort of extremely special case was thought to be a major
weakness. If neoclassicals find this condition too extreme for the
labor theory of value, they can hardly find it general enough as a
defense of neoclassical macroeconomics [11].
Perhaps the solution lies in adopting another method of evaluating
the physical quantity of capital in the same units as net output.
Champernowne has proposed a "chain index" measure of capital that will
restore the macroeconomic equality [12]. However, this measure only works
under special cases, too. Burmeister has shown that the macroeconomic
equality can be established with this chain-index if and only if
real Wicksell effects are always negative. However, as was shown
by the Cambridge Capital Controversy, this assumption of "negative
real Wicksell effects" is not a general case either. In fact, nobody
has determined what are necessary assumptions on technology to
ensure the desired conclusion will follow [13]. Finally, if this index
is used to express an aggregate production function in per capita
terms, the wage is no longer equal to the marginal product of
labor as that marginal product is typically expressed in such
functions [14].
But, some may object, aggregate production functions work
empirically. So if economists cannot even state their assumptions, they
may say, this empirical success justifies the continual use of aggregate
production functions. This is an extremely weak defense. Income
distribution has been stable over much of the period in which
macroeconomists have been using aggregate production functions. Franklin
Fisher has shown through simulation that the supposed empirical success
of aggregate production functions can arise under these conditions even
in cases where the needed assumptions do not hold. Thus, this supposed
empirical success of aggregate production functions fails to test the
models with the unstated assumptions of aggregate neoclassical theory or
to test among alternative theories. In fact, economists who rely on this
defense seem to be confusing their empirical results with another
accounting relationship [15].
4.0 CONCLUSION
This article has presented a simple explanation of the
nonequality of the interest rate and the marginal product of
capital, as that equality is understood in macroeconomic models.
Thus, neoclassical microeconomics does not imply that equality.
Various attempts to defend the macroeconomic models considered
here have been examined and have been found wanting. An interesting
aspect of this criticism is that it does not seem to be about index
number problems [16]. Nor has this argument depended on the
phenomena of reswitching at all, and it depended on capital reversing
only in criticizing Champernowne's chain index [17]. If the
demonstration of the theoretical possibility of these phenomena
are taken as central to the Cambridge Capital Controversy, then
that controversy should have been about something other than
aggregate production functions [18]. As far as I can see, this argument
is fairly well understood on the Cambridge, England side.
I conclude that the CCC was indeed about something else. I happen to
think the topic under debate was a confrontation of two rival paradigms
of value and distribution, namely the Classical theory and the so-called
Neoclassical theory [19]. In particular, it was shown, I think, that
Neoclassical economists cannot consistently maintain in their equilibrium
framework that owners of capital goods make any contribution to
production. Hence interest cannot be a payment for such a
contribution [20].
Finally, it is curious that economists continue to use aggregate
production functions despite the clear warnings of this traditional
argument. Many of those economists who follow Solow or Lucas do not
seem concerned about their inadequate concept of capital. Although these
researchers may be interested in technical improvements in their models,
capital-theoretic problems do not seem high on their agenda. Furthermore,
much of graduate education in economics seems to leave newly emerging
economists unaware of capital-theoretic problems with their models. These
young economists do not seem to possess the analytical tools that were
forged in the CCC, such as the correct analysis of the choice of
technique, a correct analysis of the relationship between the theory of
rent and income distribution, or even how to analyze depreciation and
the economic life of machines in a framework of joint production.
What explains this apparent continuation of the miseducation of
economists that Joan Robinson decried over forty years ago [21]? My
hypothesis is partly ideological. Any advanced treatment of capital
theory and the appropriate analytical tools [22] would expose the
student to the Cambridge Capital Controversy. The student would then
learn about some serious questioning of the internal consistency of
many claims of neoclassical economists. There are obviously normative
overtones to this controversy, for example, over the exploitative
nature of profits and the capitalist system as a whole. Neoclassical
economics might be claimed to currently fill the social role of "hired
prize fighters" for capital, what Marx characterized as "vulgar
economics" [23]. This social role is threatened by the CCC.
5.0 FOOTNOTES
[1] Elements of this argument can be found in Joan Robinson's article
"The Production Function and the Theory of Capital," _Review of Economic
Studies_, 1953-4 and Geoff Harcourt's _Some Cambridge Controversies in
the Theory of Capital_, 1972. The closest formulation to my argument is
in the following papers:
Amit Bhaduri, "The Concept of the Marginal Productivity of Capital
and the Wicksell Effect," Oxford Economic Papers, XVIII, 1966,
pp. 284-88
Amit Bhaduri, "On the Significance of Recent Controversies on Capital
Theory: A Marxian View," Economic Journal, LXXIX, 1969, pp. 532-9.
But the argument was also in Sraffa. See, for example:
Piero Sraffa, "Production of Commodities: A Comment," _Economic
Journal_, V. LXXII, June 1962, pp. 477-9.
[2] "Now a major problem existed because capital, unlike either labor or
land, is a produced means of production and cannot be measured
unambiguously in purely physical terms: the amount of capital can be
measured only in value terms. The problem was to establish the idea of
a market for capital, the quantity of which could be expressed
independently of the price of its service (i.e. the rate of profit)...
The basic deficiency with this approach is in its treatment of capital,
which cannot be measured independently of the rate of profit. As
observed above, the value of capital, like that of all produced
commodities, depends on the rate of profit, or interest."
-- J. E. Woods, _The Production of Commodities: An Introduction to
Sraffa_, Humanities Press International, 1990, pp. 306-307.
[3] [the divergence between the marginal product of capital and the rate
of interest] "is attributable to the fact that it is impossible to
find an invariant unit in which to measure the social quantity of
capital."
"To put the matter another way, we may say that a change in the supply
of capital - arising, for example, from new voluntary savings - alters
the units in which all the previously existing capital is measured;
and it is therefore incorrect to say that the supply of capital as a
whole has increased by the amount of the voluntary saving. It is
important to emphasize that this problem of measuring the quantity
of capital is not an index-number problem. There are, to be sure,
numerous index-number problems of the greatest complexity in the
theory of capital. But the problem to which I now refer would exist
even in the simplest economy in which all output consisted of a single
type of consumer's good and firms were exactly alike."
-- L. A. Metzler, "The Rate of Interest and the Marginal Product of
Capital," _Journal of Political Economy_, Vol. 53, 1950, pp.
284-306.
Metzler provides a brief literature review of awareness of this problem
going back to Wicksell. My analysis is closest to his comments on
Knight's capital theory, though I think my presentation is clearer.
[4] A more general statement of these relations, abstracting from
price Wicksell effects, is given by Equations 7' and 8':
df+/dk <= r <= df-/dk (7')
f( k ) - k df-/dk <= w <= f( k ) - k df+/dk (8')
where df-/dk is the left-hand dervative and df+/dk is the right hand
derivative. In the discrete case without price Wicksell effects, the
neoclassical aggregate production function is supposed to consist of
positively sloped line segments connecting "kinks" where the left-hand
and right-hand derivatives are not equal. The line segments correspond
to switch points (defined below), while the "kinks" are non-switching
points.
[5] A good explanation of price and real Wicksell effects can be found
in:
Edwin Burmeister, "Wicksell Effects," _The New Palgrave_.
Burmeister writes:
"The value of capital, however, is not an appropriate measure of the
'aggregate capital stock' as a factor of production except under
extremely restrictive assumptions. Wicksell (1893, 1934) originally
recognized this fact, which subsequently was emphasized by Robinson
(1956)."
[6] One can show that Equation 9 follows from 7' at a non-switching
point in the discrete case. Consider two sets of values for y, w,
r, and k:
y2 = w2 + r2 k2 (I)
y1 = w1 + r1 k1 (II)
The difference is given by Equation III:
y2 - y1 = w2 - w1 + r2 k2 - r1 k2 + r1 k2 - r1 k1 (III)
Or, in obvious notation:
dy = dw + k2 dr + r1 dk (IV)
Assume 7'. There are two cases.
Case 1: dk > 0 in Equation IV. Thus, dr < 0. Ignoring higher-order
terms:
y2 = y1 + (k2 - k1) (df+/dk)(k1) (V)
Equation 7' gives:
y2 <= y1 + (k2 - k1) r1 (VI)
Or:
dy <= r1 dk (VII)
Thus,
dw + k2 dr <= 0 (VIII)
Rearranging and taking the left-hand derivative gives:
- dw/dr- <= k (IX)
Case 2: dk < 0 and dr > 0 in Equation IV. Once again, ignoring
higher order terms:
y2 = y1 + (k2 - k1) (df-/dk)(k1) (XI)
The inequality in Display VI follows once again from Equation 7'.
Thus, Display VIII holds here, as well. Rearranging and taking the
right-hand derivative yields:
k <= - dw/dr+ (XII)
Since the left-hand and right-hand derivatives of the factor-price
frontier are equal at a non-switching point in the discrete case,
k = - dw/dr (XIII)
which was to be shown.
[7] See:
Heinz D. Kurz, "Factor Price Frontier," _The New Palgrave_.
[8] Textbook treatments of the connection between cost minimization and
the factor price frontier can be found in (Woods 1990) or
Heinz D. Kurz and Neri Salvadori, _Theory of Production: A Long
Period Analysis_, Cambridge University Press, 1995.
[9] These properties of the factor price frontier in the "continuous
substitution" case were brought out by Luigi Pasinetti in correcting
a technical mistake by Robert Solow. See:
Luigi Pasinetti, "Switches of Technique and the 'Rate of Return' in
Capital Theory," _Economic Journal_, 1969, pp. 508-513.
It seems worth pointing out, since many may be confused on this point,
that reswitching and capital reversing are possible when the optimal
technique varies continuously with the interest rate. See:
P. Garegnani, "Heterogeneous capital, the Production Function and the
Theory of Distribution," _Review of Economic Studies_, v 37, June 1970,
pp. 407-36.
[10] See:
Frank Hahn, "The neo-Ricardians," _Cambridge Journal of Economics_,
V. 6, 1982, pp. 353-374.
[11] Hence, Paul Samuelson's defense of aggregate production functions
is inadequate. This defense can be found in
Paul A. Samuelson, "Parable and Realism in Capital Theory: The
Surrogate Production Function," _Review of Economic Studies_, 1962,
pp. 193-206.
[12] D. G. Champernowne, "The Production Function and the Theory of
Capital: A Comment," _Review of Economic Studies_, V. 21, 1953-4, pp.
112-35.
[13] See the reference in footnote 5.
[14] Salvatore Baldone, "From Surrogate to Pseudo Production Functions,"
_Cambridge Journal of Economics_, V. 8, 1984, pp. 271-288. Baldone also
shows Burmeister's claims are problematic when used to compare
quasi-stationary economies with a positive rate of growth, instead of
just stationary economies.
[15] Anwar Shaikh, "Humbug Production Function," _The New Palgrave:
Capital Theory_, Macmillan, 1990. See also John S. L. McCombie,
"The Solow Residual, Technical Change, and Aggregate Production
Functions," _Journal of Post Keynesian Economics, V. 23, Winter
2000-2001, pp. 267-298.
[16] See footnote 3.
[17] "The construction of the production function does not even require
this refutation via the phenonomenon of returning techniques
('reswitching'), because a production function for which the marginal
product equals the factor price already becomes impossible if the
wage curves of single techniques are not straight lines (except for
a few unimportant cases; see Garegnani, 1970, Hunt and Schwartz,
1972). Contrary to the usual interpretations today, the debate about
the possibility of returning techniques is important not only because
it proves that the production function with its marginal products is
nonsensical, but because, on a more general level, it can be shown
that a demand function for capital...cannot be defined."
-- Bertram Schefold, _Mr. Sraffa on Joint Production and Other
Essays_, Unwin Hyman, 1989, p. 292.
[18] "[O]ne should emphasize the distinction between two types of
measurement. First, there was the one in which the statisticians were
mainly interested. Second, there was measurement in theory. The
statisticians' measures were only approximate and provided a suitable
field for work in solving index number problems. The theoretical
measures required absolute precision. Any imperfections in these
theoretical measures were not merely upsetting, but knocked down the
whole theoretical basis. One could measure capital in pounds or dollars
and introduce this into a production function. The definition in this
case must be absolutely water-tight, for with a given quantity of
capital one had a certain rate of interest so that the quantity of
capital was an essential part of the mechanism. One therefore had to
keep the definition of capital separate from the needs of statistical
measurement, which were quite diffent. The work of J. B. Clark,
Bohm-Bawerk and others was intended to produce pure definitions of
capital, as required by their theories, not as a guide to actual
measurement. If we found contradictions, then these pointed to
defects in the theory, and an inability to define measures of capital
accurately. It was on this - the chief failing of capital theory -
that we should concentrate rather than on problems of measurement."
-- Piero Sraffa, Interventions in the debate at the Corfu Conference
on the "Theory of Capital", 4-11 September 1958.
[19] "Capital theory has been one of the most contentious areas of debate
in economic analysis. One reason for this is that it is the point at
which the classical theory of value and distribution, and the
neoclassical theory of price meet, so to speak, on the same ground.
Classical theory was devoted to explaining the determination of the
rate of profit and associated 'natural prices' in an economy using
reproducible means of production. In so far as neoclassical theory
attempts to determine the rate of profit and associated 'long-run
prices' it is offering an alternative explanation of exactly the
same things."
-- John Eatwell, Murray Milgate, and Peter Newman, "Preface,"
_The New Palgrave: Capital Theory_, Macmillan, 1990.
[20] "...while wages are paid for work, and one can (and in some
circumstances should) think of the wage bill...as reproducing the
power to work, *profits are not paid for anything at all.* The flow of
profit income is not an exchange in any sense. The Samuelson diagram is
fundamentally misleading; there is no 'flow' from 'household supply' to
the factor market for capital. The *only* flow is the flow of profit
income in the other direction. And this, of course, leads straight to
that hoary but substantial claim that the payment of wages is not an
exchange either, or at any rate, not a fair one. For Wages plus Profits
adds up to the Net Income Product; yet Profits are not paid for
anything, while wages are paid for work. Hence the work of labor
(using the tools, equipment, etc., replacement and depreciation of
which is already counted in) has produced the entire product. Is
labor not therefore exploited? Does it not deserve the whole product?"
-- Edward Nell, "The Revival of Political Economy," in _Economic
Relevance: A Second Look_, edited by Robert L. Heilbroner and Arthur
Ford, 1971, 1976.
[21] This miseducation is asserted on slightly different grounds in the
following quote:
"Observe that even in neoclassical theory full employment alone is
not enough to transform marginal-productivity relationships into a
long-run theory of distribution. In long-run neoclassical theory, the
capital:labor ratio is endogeneously determined, so that the wage rate
cannot be determined solely by marginal productivity of labor at full
employment - not even in Chicago. Instead, distribution must reflect
household preferences with respect to present and future consumption."
"Thus, it is fair to conclude that there are two marginal-
productivity theories. The first is a relatively innocuous, general
theory that involves nothing more controversial than competitive
profit maximization - and provides correspondingly little
contribution to the theory of growth and distribution under
capitalism. The second is more powerful, and very special, providing
by itself a theory of distribution, for the short run at least,
whose 'only' defects are (1) that it assumes full employment
and (2) that it begs the question of accumulation. The wonder is that
it is precisely this theory that so many students come away with from
their study of economics. Only slightly more wondrous is that by and
large they believe it!"
-- Stephen A. Marglin, _Growth, Distribution, and Prices_, Harvard
University Press, 1984, pp. 330-331.
Neither version of marginal productivity theory need include the equality
of the marginal product of capital and the interest rate in an aggregate
production function framework.
[22] Syed Ahmad, _Capital in Economic Theory: Neo-classical, Cambridge,
and Chaos_, Edward-Elgar, 1991.
[23] "In summary I believe that Marx's sociology of economic knowledge
was quite an impressive achievement, in spite of being flawed by its
reliance on functional explanations and the labor theory of value...
The recent 'capital controversy' shows that these are not dead issues.
Surely some cognitive confusion lay at the origin of the idea that
'capital' can be treated as a homogeneous 'factor of production,' for
instance an inference from the fact that *capitalists* form a fairly
homogeneous class. And conceivably the tenacity with which the
neoclassical economists stuck to the notion of aggregate capital has
something to do with non-cognitive interests. This, admittedly, is
sheer speculation, and I may be quite wrong. Vested intellectual
interests may suffice to explain the resistance. Be this as it may,
the sociology of economic conceptions and economic theory is a field
worth cultivating, if proper attention is paid to the many
methodological pitfalls in this domain."
-- Jon Elster, _Making Sense of Marx_, Cambridge University Press,
1985, p. 504.
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