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.

--

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