Laws of production and laws of algebra: Humbug IIgesd.free.fr/shaikh80.pdfChapter 5 of: Edward J. Nell (ed.), Growth, profits, and property. Essays in the Revival of Political Economy,

Laws of production and laws of algebra:Humbug II

Shaikh

The theoretical basis

Recent debates on capital theory have focusedon the notion of capital as a factor of production,which along with labor, to the in Though the intricate point and counterpoint ofthe controversy often obscure this simple fact, ithas become increasingly clear that what is atstake in the current debate is in essence thesame issue with which the classical economists,particularly Ricardo, grappled that of the divi-sion of income between wages and profits. Theargument thus rages around eco-nomic theory, whose aim it is to represent theworkings of a competitive capitalist economy. Ina sense this is a return to relevance, since muchof modern mathematical economics has stu-diously concerned itself, not with descriptive.but instead with normative theory, such as thestudy of optimal and efficient growth paths, etc..(Lancaster, 1968, pp. 9-10).

In neoclassical theory, the model of pure ex-change occupies a central position, for it illus-trates simply and elegantly the fundamentaltruths of the paradigm, truths which any morecomplex representations may modify but cer-tainly cannot undermine.’ Thus, in the model of

exchange. trading begins with selfish indi-viduals each having an arbitrarily determinedinitial endowment of goods, and proceeds to afinal state in which no one individual can im-prove his or her basket of commodities withoutmaking someone else worse off. Such a situation

modity other things being equal -the lower itsrelative price.

The next step in the analysis requires its to the case of production. Initial endow-

ments are now assumed to contain not just con-sumer goods but also means of production, suchas land, machines, raw materials, etc.; in addi-tion, since the game cannot continue unlessevery individual has at least some wealth, it isgenerally assumed that each and every initial en-dowment includes potentially saleable labor ser-vices. By assumption, the ultimate objective ofevery individual is consumption: means of pro-duction and labor services, however, are notdirectly consumable. At this point, therefore,production is introduced as a roundabout way ofconsumption, a process in which inputs aretransformed into outputs. In order to translateany given initial endowment into the productionpossibilities inherent in it. neoclassical econom-ics commonly relies on the assumption of a wellbehaved neoclassical production function, onefor each commodity produced.

Each individual then faces three basicmethods of arriving at some preferred final allo-cation, methods which he or she is free to use inany combination permitted by the initial endow-ment and consistent with the utility function.First, he can trade any of the means of production in his possession for othergoods he desires; second, he may rent out the

n fand/or rent out his labor power: and third, if his

is known as a pareto-optimal allocation, and itimplies a set of final exchange ratios betweencommodities that is, a set of This chapter is an expanded, revised version of a

tive prices. What is more, given paper entitled “Laws of Production and Laws ofof well-behaved neoclassical utility functions for

The Humbug which

each individual, the equilibrium prices of theappeared as a note in of’

model of exchange will Vol. LVI, No. 1 (February, pp.

the higher the relative availability of some along with comment by Robert The

postscript to this chapter assesses comment.

MichelZone de texte Chapter 5 of: Edward J. Nell (ed.), Growth, profits, and property. Essays in the Revival of Political Economy,Cambridge University Press, 1980

81

initial endowment so permits, he may choose tobecome a producer, renting and/or buyingmeans of production and labor-power and com-bining these with the elements of his initial en-dowment to turn out one or more commoditiesvia a well-behaved neoclassical production func-tion. Ruled only by his enlightened self-interest,which dictates that more is better, and con-strained only by his native abilities and initial en-dowment, he is assumed to eventually arrive atsome most “efficient” combination of thetrader-rentier-producer modes, thereby a t -taining his personal optimum in the form of somefinal allocation.

Because preferences (utility functions) andinitial endowments are of the analy-sis, the whole structure of equilibrium is ruledby them, so that once again, the forces of con-sumer sovereignty lead us ineluctably toPareto-optimality . Equilibrium relative pricesare once again prices, a term which nowcovers the prices of consumption goods, thewage rate for labor services, and the rentalsale prices of means of production (Hershleifer,1970).

Under carefully fashioned assumptions in-volving well-behaved utility and productionfunctions, these sorts of models are determinatein the sense that one or more possible equilibriacan be shown to exist. But the model, as out-lined here, contains no reference to the uniformrate of profit which is supposed to characterizecompetitive capitalism. The explanation of thisrate of profit is what (descriptive) neoclassicalcapital theory is all about. Moreover, given thatthe basic parables of the theory have alreadyidentified the equilibrium price of every good orservice as a scarcity price, one that reflects itsindividual and social scarcity, the task that con-fronts the theory is clear: somehow, the rate ofprofit too must be explained as the scarcity priceof some thing with both the price and quantity ofthis thing to be mutually determined in somemarket. This market, it turns out, is the capitalmarket, in which demand is determined by indi-vidual’s preferences for present versus futureconsumption their “taste for investment”(Dewey, 1965) and supply is determined by thetechnological structure. The price that suppos-edly emerges from this interaction is the of’interest, the scarcity index of the quantity of

and with the addition of a few more con-venient assumptions, the rate of profit is madeequal to this rate of interest. cun then, it is argued, the distri-bution in is sequence the in fact, within this wondrous construct, capital-ism itself represents the resolution of one of

ture’s most problematical gifts the “natural”selfishness of every individual!

Scarcity pricing parables and the aggregateproduction function

Traditionally, several models have been used toextend scarcity pricing to the theory of distribu-tion. The simplest, and by far the most widelyused in both the theoretical and empirical litera-ture, is the aggregate production function model.

we is an aggregated ver-sion of the general equilibrium model outlinedabove, constructed as an empirically usefulapproximation, supported the

Even the sophisticates, the so-calledhigh-brows of neoclassical theory, at one time,took this and similar parables seriously:

. . . In various places I have subjected to de-tailed analysis certain simplified models in-volving only a few factors of production . . .[These] simple models or parables do. I think.have considerable heuristic value in giving in-sights into the fundamentals of interest theoryin all of its complexities. (Samuelson, 1962,

The originators of the “production func-tion” theory of distribution (in the staticsense, where I still think it should be takenfairly seriously) were Wicksteed, Edgeworth,and Pigou. (Hicks, 1965, p. 293, footnote 1)Though aggregate or surrogate production

function models occupy the bulk of the theoreti-cal and empirical literature on the distribution ofincome in a capitalist society, the essential char-acteristic of this and all other parables of neo-classical theory concerns their attempt to ex-plain the wage rate and the rate of profit as scar-city prices of labor and capital, respectively,determined in the final analysis by efficiencyconsiderations. It was precisely this techno-cratic apologia for capitalism which became thetarget of the neo-Keynesian counterattack ofthe during the so-called Cambridge cap-ital controversies.

One of the most striking, and for neoclassicaleconomics most devastating, results of theabove capital controversies was the proof that

version of the neoclassical parable, in whichthe rate of profit varied inversely with the quan-tity of capital and the wage rate inversely withthe quantity of labor (so that each at least be-haved like a scarcity price) was valid in staticconditions prices in possiblecompetitive equilibria were proportional to laborvalues.” I apply,

to that particular version of the parableknown as the aggregate (or surrogate)

8 2 S h a i k h

tion function, in which the wage rate and the rateof profit not only move inversely to the quan-tities of labor and capital, respectively, but arealso equal to and determined by their respectivemarginal products. Considering that the neoclas-sical in a counterrevolution the classical school,against Ricardo and Marx in particular” (Dobb,1970, p. and above all, against the labortheory of value in form, it is gratifying to dis-cover in the end these parables themselvesdepend on the simple labor theory of value. Theirony is inescapable.

These and other inimical results were not loston the faithful. As awareness of the internal in-consistencies of neoclassical theory began togrow, many were led to abandon it. But forothers, hope died hard; and hope, it seems, layin the data. “As a neoclassical theorist, canonly that the relevant question is what isrelevant: should we make our predictions on thebasis of what Mrs. Robinson has called perversetechnical behavior thethat been repeatedly

1971, p. 254, emphasis added)What has been “repeatedly observed,” it is

argued, is the empirical efficacy of aggregateproduction functions. spite of the verystrongest theoretical requirements for theirexistence, the use of such functions flourishes the current justification being that their empiri-cal basis appears strong. In study after study,empirically derived functions appear to stronglysupport both the constancy of returns to scaleand the equality of marginal products with

rewards”: in for hnthseries and cross-section studies (within any onecountry), the Cobb-Douglas function appears todominate the field,

For the neoclassical faithful, these results rep-resent their salvation; no matter what those crit-ics from Cambridge say, the “real” world, itwould is Or is i t ‘? Theanswer is simple: no.strength production is illusion,due not to some mystical laws of production, butinstead, to some rather prosaic laws of algebra.To see why, however, we must first examinehow production functions are estimated.

The empiricalfunctions

basis of aggregate production

The most popular methods of estimating ag-gregate production functions have been thesingle equation least squares method and thefactor shares method (Walters, 1963). The

former can be most generally described as fittinga function of the Q(t) = toobserved data while the latter consists of

that aggregate marginal products of capitaland labor are equal to their respective unit

ify structural coefficients. In general, for bothtime series and cross-section data, theDouglas function wins out: “the sum of coeffi-cients usually approximate closely to unity”(thus implying constant returns to scale), withthe additional bonus of a close “agreementbetween the labor exponent and the share ofwages in the value of output” (thus support-ing aggregate marginal productivity theory)(Walters, 1963, p. 27).

In a recent paper, Franklin Fisher concedesthat the requirements “under which the produc-tion possibilities of a technically diverseeconomy can be represented by an aggregateproduction function are far too stringent to bebelievable” (Fisher, 1971, p. 306). He proposestherefore to investigate the puzzling uniformityof the empirical results by means of a simulationexperiment: each of industries in this simu-lated economy is assumed to be characterizedby a microeconomic Cobb-Douglas productionfunction relating its homogeneous output to itshomogeneous labor input and its ownmachine stock. The conditions for theoreticalaggregation are studiously violated, and thequestion is, how well, and under what circum-stances, does an aggregate Cobb-Douglas func-tion represent the data generated? In such aneconomy, the aggregate wage share is often vari-able over time, so that in general an aggregateCobb-Douglas would not be expected to give agood fit. What seems to surprise Fisher, how-ever, is that when the wage share happens coin-cidentally to be roughly constant, a Douglas production function will not only fit thedata well but provide a good explanation ofwages,

m aggregate Cobb that “the the

constancy is presenceof p r o d u c t i o n

is runs the the success

is due tothe constancy shut-e.(Emphasis added.) (Fisher, 1971, p. 306).

It is obvious that so long as aggregate sharesare roughly constant, the appropriate economet-ric test of aggregate neoclassical production anddistribution theory requires a Cobb-Douglasfunction. Such a test would then apparently castsome light on the degree of returns to scale(through the sum of the coefficients), and the

applicability of aggregate marginal productivitytheory (through the comparison of the labor andcapital exponents with the wage and profitshares, respectively). What is not obvious, how-ever, is that so long as aggregate shares are con-stant, an aggregate Cobb-Douglas functionhaving apparently “constant returns to scale”will always provide an exact fit, for any datawhatsoever.

to

T h e s epropositions, it will be shown, are consequences of constant shares, and it will beargued that the puzzling uniformity of the empir-ical results is due in fact to this law of algebraand not to some mysterious law of production.In fact, in order to emphasize the independenceof any laws of anillustration is provided in the form of the ratherimplausible data of the Humbug economy, foreven data such as this is perfectly consistentwith a Cobb-Douglas function having “constantreturns to scale,” neutral technical change,”and satisfying “marginal productivity rules,” solong as shares are constant.

Laws of algebra

Let us begin by separating the aggregate data inany time period into output data the value ofoutput), distribution data (W, wages and prof-its, respectively), and input data (K, L, the indexnumbers for capital and labor, respectively).Then we can write the following aggregate iden-tity for

+ Given

always write:index numbers c a n

q(t) w(t) +

and are the andcapital -labor ratios, respectively, and w(t)

are the wage andprofit The istherefore the fundamental identity relating out-put, distribution, and input data. Defining theshare of profits in output as and the share ofwages as 1 s, we can differentiate identity 2 toarrive at identity 3 (time derivatives are denotedby dots, and the time index, t, is dropped to sim-plify notation):

Dividing through by

By definition, the profit and wage shares,respectively, are

so WC may write,

= + where B (1 + (3)

It is important to note that all relations givenso far are true for aggregate data atall, irrespective of production or distributionconditions.

Suppose now we are faced with particulardata which for some unspecified reasons exhibitconstant shares, so that Re-membering that the dotted variables are timederivatives etc.), we can immedi-ately integrate the identity (3):

dt In k +

where for convenience the constant of integra-tion is as In Rewriting, we have,

= =

where by definition

B = exp

Equation (4) is strikingly reminiscent of a con-stant returns to scale aggregate Cobb-Douglasproduction function with shift But in fact, it is not a function at all,but merely an algebraic relationship whichalways holds for output-input data L ,even data which could not conceivably comefrom any economy, so long as the distributiondata exhibits a constant ratio. Furthermore,since the term in identity is a weightedaverage of the rates of and respectively, it seems empirically reasonable toexpect of L would give capital-labor ratio which is weakly correlatedwith With measures for which the above is

so that B will also be solely afunction of time. Then we can write

= (5)

and since and we getQ = B(t)

The algebraic relationship just given has propcrtics. First, it is homoge-

neous to the first degree in and L. Second,since = the partial derivatives

are equal to w, respectively.And third, the effect of time is “neutral,” asincorporated in the shift parameter B(t). Whatwe have, actually, is mathematically identical toa constant returns to scale Cobb-Douglas pro-duction function having neutral technical changeand satisfying marginal productivity “rules.”And yet, as we have seen, production whatsoever presented being “gen -erated by such so long as shares areconstant and the measures of capital and laborsuch that k is uncorrelated with Therefore,precisely because is a mathematical rela-tionship, holding true for large classes of data as-sociated with constant shares, it cannot be inter-preted as a production function, or any produc-tion relation at all. If anything, it is a distributiverelation, and sheds little or no light on the under-lying production In fact, since theconstancy of shares has been taken as an empiri-cal datum throughout, equation (5a) does notshed much light on any theory of distributioneither.

I emphasized earlier that the theoretical basisof aggregate production function analysis wasextremely weak. It would seem now that itsapparent empirical strength is no strength at all,but merely a statistical reflection of an algebraicrelationship. For the neoclassical old guard, theretreat to data is really a rout.

Applications

It is obvious that one can apply Equation (5a) inmany ways. The section that follows will reex-amine famous paper on measuring tech-nical change. The “humbug production func-tion” section will present a numerical exampleto illustrate the generality of Equation Thesection on Fisher’s simulation experiments willextend the preceding analysis; and the final sec-tion will touch briefly on cross-section produc-tion function studies.

Technical change and the aggregate productionfunction: In what is considered a “semi-nal paper” Robert intro-duced in 1957 a novel method for measuring thecontribution of technical change to economicgrowth. Since that several refinements of

original calculations have been estab-lished, all aimed at providing better measures oflabor and capital by taking account of education,

vintages of machines, etc., but has remained unchanged.

the

approach is by now a familiar one.Equation (6) expresses the assumption of a con-stant returns to scale aggregate production func-tion, with the parameter A(t) expressing the as-sumption of neutral technical change.

For such a function, the marginal product ofcapital is = A(t) = since A(t) = By assumption, this marginalproduct is equal to the rate of profit

and by rewriting, wethe profit share s:

can express this in terms of

k rk share of profit in output

expressed purpose was to distinguishbetween shifts of the assumed production func-tion (due to “technical change”) and move-ments along it (due to changes in thelabor ratio,

Figure 5.1 illustrates the geometric assump-tion implicit in paper. Points A, andare observed points, at times and respec-tively, while represents the “adjusted” pointafter “neutral technical change” has been re-moved. Thus points and lie on the “under-lying production function.”

Algebraically, in terms of Equation theaim of his procedure is to partition output perworker into A, the technical change shiftparameter, the “underlying productionfunction” to which 1 just referred. In order to dothis, first differentiates Equation (6):

(Value of

Figure 5.1

Rearranging,

= + 1Since from = and from

k rk

being the share of profit in gross output, we canwrite

Equation (8) is derived from the assumptions ofa constant returns to scale aggregate productionfunction, with distribution determined by mar-ginal productivity rules. Equation de-rived earlier from an identity and thereforealways true any production and distributionbehavior, is mathematically identical to (8).It follows therefore that =

t h a t i s , m e a s u r e technical change is merely a weighted average ofthe growth rates of the wage rate, and the rateof profit,

data provide him with a series forgross output per worker capital per worker and profit share for the United States from1909-1949. From this data, he calculates therates of change and and using theserates along with the data for the profit share hederives a series for k/A =

To the series for represents therate of change of technology; since a scatter dia-gram of on k shows no apparent correla-tion, he concludes that technical change is es-sentially neutral. By setting A(0) = 1, he is ableto translate the rate of technical change intoa series for A(t), the shift parameter.’ Finally,since by definition = he is able tocombine his derived series for A(t) with hisgiven series on to derive the underlying pro-duction function q/A(t).

Plotting f(k) versus k, gets a diagramwith noticeable curvature, and notes with obvi-ous satisfaction that the data “gives a distinctimpression of diminishing returns”1957, p. 380). fact, underlyingproduction function to be extremely well repre-sented by a Cobb-Douglas function:

= + = ( 9 )

Given our preceding analysis in the sectionon laws of algebra, it is not difficult to see why

results turn out so nicely. We know forinstance that his data exhibit roughly constantshares, and the residual term = iscorrelated with k. From purely algebraic consid-erations, therefore, one would expect the data to

be well represented by the functional form in = B(t) a form which is mathematically

identical to a constant returns to scaleDouglas function, with neutral technical changeand “marginal products equal to factor re-wards. In fact, the algebra indicates that

underlying production function shouldbe of the form:

= f(k) + k

is of course the (roughly) constant share and is a constant of integration which depends onlyon the initial points of the data.

1909-1942 in his for these years the average profit share s

Moreover, since in any period = from Equation in period we

may write = B(0) which gives us = (0) For

this residual B(t) represents the shift parameterA(t) (compare Equations (3) derived from anidentity, and Equation (8) derived fromlow’s assumptions), so that B(0) = A (0); asmentioned earlier, he takes A(0) = FromTable 1, p. 315 of his article, we get =

= 2.06, which when combined with B(0) =A(0) = gives 0.725.

Thus, on purely algebraic considerations onewould expect underlying productionfunction to be characterized by

= + kThis, of course, is virtually identical to

regression result, equation as itshould it is a law of not a law of

The humbug production function. The analysis ofthe laws of algebra led to the conclusion that production series k whatsoever, berepresented being by Douglas production neutraltechnical change and satisfying marginal produc-tivity “rules,” so long as shares are constantand the measures of capital and labor such that is uncorrelated is possible to illus-trate the generality of the above result by meansof a numerical example. Consider, for example,an economy with illus-trated in Figure 5.2 and having the same profitshare as in data for the United States.

The Humbug data set gives us a series for k,and from which we can calculate rates ofchange and k/k. From these, in turn, wederive = (The calculationsappear in Figure 5.5

Plotting B/B on gives us a scatter diagram

years, share is Moreover, since SO, = 2.00, and

for Humbug data, wethe constant term to be In In

B(O) Algebraic consid-erations therefore tell us that the constant termwill be -0.459 and theThe actual regression of on k, presentedbelow, gives virtually identical results.

- 0 . 4 5 3 =

The function is of course much more trou-blesome. A simple glance at Figure 5.3 tellsthat no linear or log-linear function will sufficefor a numerical approximation.even in this case a approximation is

corrected of

Figure 5.3

, I I I I I I2 . 0 2 . 4 2 . 8 3 . 2 3 . 6 4 . 0

Figure 5.4

Combining these two fitted functions, one ar-rives at a numerical specification for even theHumbug data (Table 5. I)!

Fisher’s simulation experiments. Earlier, I men-tioned Franklin Fisher’s extensive (and expen-sive) simulation experiments, in which he finds,to his surprise, that aggregate Cobb-Douglasfunctions seem to “work” for his simulatedeconomy even when the theoretical conditionsfor such an aggregate function are carefully vio-lated, so long as the particular simulation runhappens to have roughly constant wage (andhence profit) shares (Fisher, 1971, p. 306).

It is worth noting at this point that what Fishermeans by aggregate production functions work-ing, is not simply that they give a good fit togross output or gross output per worker

but also that the estimated marginalproducts of labor, and presumably of capital,closely approximate the actual wage and profitrates, respectively (Fisher, 1971).

1 have already demonstrated in section on thelaws of algebra why in general an aggregateCobb-Douglas may be expected to work, in thesense explained earlier, for data whichconstant wage shares. In this section, however,it will be shown that even Fisher’s massive com-puter simulation is in reality only an applicationof the laws of algebra.

The the Fisher’s simu-lated economy consists of N industries, eachproducing the same type of output Q, usinghomogeneous labor but its own typeof machine stock Thus and are bothquantities of the same good, produced by indus-tries i and j, respectively, whereas and arestocks of different types of machines.

by a microeconomicfunction:

Cobb -Douglas production

where = 1, . . . , N (14)(The are constant over time, but in generalA,(t), and K,(t) are not.)

At any instant of time, the total stock of laborL(t) in the economy is given. The basic proce-dure followed in the model is to allocate thisgiven supply among the existing industries so asto equalize the industry marginal products oflabor MPL): this of courseyields the maximum aggregate output Q(t)

In general, the marginal product of aDouglas function is =Since these are all equalized for the various in-dustries to a single ievel, we cancommon level by and write:

W(t)

denote this

represents the “imputed rental” (uniformwage rate) of a unit of labor, so that the wage billin the industry is:

Thus, the aggregate wage bill is:

= =

so that the wage share in total output Q(t)is:

wage share =

Finally, since Q,(t) is the gross output of the industry, and = Q,(t) its wage bill,the difference between the two, the gross Each industry is assumed to be characterized

Table 5. I. data

Year

Actualshare ofpropertyincome

“Humbug”output perworker

“Humbug”capital perworker

B/B

1 9 0 9 0.335 0.80 2.001 9 1 0 0.330 0.70 2.00

0.335 0.60 2.001 9 1 2 0.330 0.70 2.001913 0.334 0.70 2.101 9 1 4 0.325 0.70 2.201915 0.344 0.60 2.201 9 1 6 0.358 0.80 2.201917 0.370 0.80 2.301 9 1 8 0.342 0.60 2.301 9 1 9 0.354 0.60 2.401 9 2 0 0.319 0.60 2.501921 0.369 0.70 2.501 9 2 2 0.339 0.80 2.501 9 2 3 0.337 0.60 2.601 9 2 4 0.330 0.80 2.601925 0.336 0.75 2.651 9 2 6 0.327 0.70 2.701 9 2 7 0.323 0.75 2.751 9 2 8 0.338 0.80 2.801 9 2 9 0.332 0.60 2.801 9 3 0 0.347 0.60 2.901931 0.325 0.60 3.051 9 3 2 0.397 0.70 3.051933 0.362 0.70 2.901 9 3 4 0.355 0.80 2.901935 0.351 0.80 3.051 9 3 6 0.357 0.70 3.051 9 3 7 0.340 0.80 3.151 9 3 8 0.331 0.60 3.151939 0.347 0.60 3.251 9 4 0 0.357 0.60 3.351941 0.377 0.80 3.351 9 4 2 0.356 80 31943 0.342 0.80 3.451 9 4 4 0.332 0.60 3.451945 0.314 0.60 3.60

0.3121 9 4 7 0.327 0.70 3.55

-0.125 0.000 -0 .125-0.143 0.000 -0 .143+O. 167 0.000 167

0.000 -0.0170.000 0.048 -0.016

-0.143 0.000 -0.1430.000

0.000 0.045 -0.016-0.250 0.000 -0.250

0.000 0.044 -0.0150.000 0.042 -0.015

+ O . 1 6 7 0.000 +O. 167+ o . 1 4 3 0.000 + o . 1 4 3-0 .250 0.040 -0.264

0.000-0.063 0.019 -0.0690.067 0.019 -0.073

0.0190.018

-0.250 0.000 -0.2500.000 0.036 -0.0120.000 0.052 -0.018

+ O . 1 6 7 0.000 + O . 1 6 70.000 -0.049 0.019

+ o . 1 4 3 0.000 + o . 1 4 30.000 0.052 -0.018

-0.125 0.000 -0.1250.143 0.033 + O . 1 3 20.250 0.000 -0.2500.000 0.032 -0.0110.000 0.031 -0.011

0.0000.000 0.070 -0.026 000

-0.250 0.000 -0 .2500.000 0.044 -0.015

+O. 0.000 + O . 1 6 7-0.014

1 .ooo 0.8000.875 0.8000.750 0.8000.875 0.8000.860 0.8140.846 0.8260.725 0.8280.965 0.8300.948 0.8430.710 0.8450.700 0.8570.690 0.8700.805 0.8700.921 0.8690.678 0.8850.902 0.8870.810 0.8930.780 0.8970.830 0.9030.880 0.9080.660 0.9080.652 0.9200.641 0.9350.74% 0.3350.764 0.9160.874 0.9160.860 0.9300.752 0.9300.852 0.9400.638 0.9400.633 0.9480.626 0.9600.843 0.9500.820 0.9750.832 0.9640.624 0.9640.614 0.9780.717 0.9750.721 0.970

in the industry, is treated as the “imputed chine being a different type. An index ofrental” of its unique machine stock K,(t). De- capital has therefore to be constructed,fining this gross profit and i t is known that in gcncrnl any such as (t), we have: wil l violate the s t r ic t condit ions under which

= the microeconomic Cobb-Douglas production= gross profits in industry functions can bc theoretically aggregated into

a macroeconomic Cobb -Douglas productionSince output and labor L,(t) are homoge- function (Fisher, 1971, pp. 307-08). On theneous across industries, their respective basis of aggregation theory, therefore, one wouldgregates are derived by simple addition. But not expect the macroeconomic variables in thissince each industry has a type of simulated economy to behave as if they werechine, an capital h y e v e nderived by adding machines together, each if aggregate shares happen to remain roughly

constant over time. That, of course, is the rea-son for Fisher’s surprise at his results.

Fisher chooses to construct an aggregateindex in two steps. First, he runs the modeleconomy over its 20-year period, from which hegets the gross profits of any given industry,for each of 20 years. Similarly, over each of the20 years he knows the machine stock K,(t) in thesame industry: the ratio of the sums ofthese two is the average rate of return in the industry:

= 20 year average rate of returnin industry (19)

The units of each average return are outputper machine type Thus Fisher can use these in any one period t to aggregate the individual in-dustry machine stocks into an aggregate index of

J(t) = =

It is useful to note that in the above expressionthe are nut functions of time, since they repre-sent average rates of r e t u r n t h e w h o l e20-year period.

The From Equation the wage share is

wage share =

Now, as Fisher notes, since the parametersare independent of time, the wage share will beroughly constant over time only if the relativeoutputs are roughly constant overtime (Fisher, 1971, p. 321, footnote 21). Let usdenote these roughly constant relative outputsby Pi, and the constant wage share by (1 s),the lack of time subscript denoting their con-stancy:

In each industry, the wage bill, as derived inEquation is = From the aggregate wage bill is = (1

and dividing one by the other, we get:

1 I

Finally, to prepare us for the last step, weneed to note that the rough constancy of relative

relative employment

implies that each firm’s output andemployment grow at roughly the same rate. Thatis, dropping time subscripts and denoting timederivatives by dots:

It is the central resultof this paper that constant shares, any ag-gregate data Q , L whatsoever can bedescribed by a function of the form Q(t) =

providing the residual is a function of time. What we must there-

fore do for Fisher’s experiments. in order to seewhy aggregate Cobb-Douglas functions workfor them, is to examine this residual

By definition, from Equation (3)

= = Fisher’sindex of capital is denoted by J. Thus =Q/Q and K/K = and:

L- - - L J L

Since and are profitand wage shares, respectively, we need onlyexamine the rates of change of

The first is easy. In all of his simulations,Fisher specifies that “labor grows at an averagerate of 3% trend” with small random deviationsfrom the trend (Fisher, 1971, p. 309). Ignoring

small random deviations

L

The growth rate of the aggregate capital indexJ(t) is a bit more complicated. In Equation (20)we defined

J,(t) =

where the are constant overtiating this with respect to time,

Dividing through we get: 1

time.

During all his simulations, Fisher assumes thateach capital K,(t) grows at an essentially

rate one which in general differs fromindustry to industry. Thus,

and this in turn implies

(27)

Therefore is a weighted average of the with weights which sum to one, since J(t)

(This type of weighted average is as a convex combination, and implies

that will always be between the largest andsmallest .)

Finally, we come to the growth rate of ag-gregate output From Equation

we know so

Q(t) Q;(t)

From this, we can derive

Of the terms in expression weknow that L/L

from and from (27). Tothis, we need only add the fact that in general,ignor ing smal l random Fisher as-sumes that the shift parameter grows at an es-sentially constant rate, which differs from

to industry.

(30)

All of this gives

But = = constant wage share,from (22). So

(1

Combining the expressions for j/J, and we return to the all important residual

of equation (25);

Given that the constant wage share I = we can write the profit share = 1 But by definition SO that

1

Thus,

= = (1 a,

where = (1 From this, we at long last

+ 11

in important to note that the and when summed over each

sum to I.

In theexpression (32) for the basic structuralparameters are and Of these, repre-sents the rate of growth of the machine stockover any given simulation run, whereas rep-resents the rate of technical change in the in-dustry. (Since the are constant over any givenrun, changes in the shift parameter repre-sent the only possible technical change in any in-dustry.)

Fisher partitions his simulations into twobasic groups. In the first of these, which he calls“Hicks experiments,” he sets all = 0. Thus,in each of these experiments, there is technicalchange 0) but no growth in the size of themachine stock = 0). Under these condi-tions, reduces to a constant over time.

constant over time)

Thus, for Hicks can expectfrom purely algebraic considerations that

In +

where is a constant of integration.From the laws of algebra (Equation we

know that in general if is solely a function oftime, any data associated with constant shares

can be represented by the functional form (since Fisher uses J as an index nf capital,

what we previously called = K/L is now

= =

Taking natural logs,

In = In B(t) + In + j = In j and combining the= In In terms into a single constant

and combining the constants into. a single con-stant we get

In = + What we have shown therefore is that for

purely (as opposedto econometric) lead us to theconclusion that whenever shares are (roughly)constant Fisher’s aggregate data can be gen-erated by what to be a Cobb-Douglas“production’ function with a constant rate oftechnical change and a marginal product of laborequal to the actual wage.

This is precisely Fisher gets hisHicks experiments: for this set of experiments,the functional form which repeatedly works thebest (in the sense that the estimated marginalproduct of labor most closely approximates theactual wage) is one which assumes constant re-turns to scale and a constant rate of technicalchange .

We now turn to the second set of experiments,what Fisher calls his “Capital experiments,” inwhich all = 0. In this set of experiments,therefore, there is positive or negative growth ofthe machine stock 0) but NO technicalchange = 0). Equation the genera1

the now becomes:

In Equation (36) each term in the brackets is aconvex combination (a weighted average whoseweights sum to one) of the so that each termlies between the largest and the smallest Onewould therefore expect the of theseterms to be close to zero; in addition, since theconstant wage share I = is itself aconvex combination of the parameters it it-self will be within range of

since the unweighted average of the is0.75, the profit share will be roughly around0.25. is to be small, multiplying it by will yielda number even closer to zero. In capital experi-ments algebraic considerations would thereforelead us to expect:

so that B where is a constant

In setting this result into the general functionalform of Equation (5) = and nat-ural logs of both sides, + +

For the capital experiments, therefore, purelyalgebraic considerations lead us to expect thatFisher’s data can be represented by what

be a Cobb-Douglas production func-tion with a constant level of technology and amarginal product of labor equal to the actualwage. Once this is precisely the result

gets his experiments. It is important to note that Fisher himself

never presents the exact regression results in-volved (an understandable omission consideringthat there were a total of 1010 runs of this simu-lated economy, each run covering a 20-yearperiod). Instead, he tells us only that the best fitsto the aggregate data were derived from an equa-tion of the form = + forHicks experiments, and one of the form =

j for experiments. To Fisherthis result comes as a surprise. But it should not,

have Fisher’s complicatedand expensive experiments have merely redis-covered the laws of algebra.

Cross-section productionThe direct analogy to constant shares in timeseries is the case of uniform profit marginsits per dollar sales) in cross-section data. Usingthe subscript i for the industry (or firm), anddefining as the uniform profitmargin, we can rewrite Equation (3) as

Then, so long as the term in brackets is related with the above equation is alge-braically similar to a simple linear regressionmodel = + with the term in bracketsplaying the part of the disturbance term

any data the bracketedterm is small and uncorrelated with the depen-dent variable the “best” fit will be across-section Cobb-Douglas production func-tion with constant returns and factors paid theirmarginal products.

There are still other ways in which one mayexplain the apparent success of a Cobb-Douglasin cross-section studies, the best single refer-ence being Phelps Brown’s (1957) critique. In subsequent note, Simon and Levy (1963) showthat any data having uniform wage and profitrates across the cross section can be closelyapproximated by the ubiquitous Cobb-Douglasfunction having “correct” coefficients, eventhough the data reflect only mobility of labor andcapital, not any specific production conditions.

Once again, it would seem that the apparentempirical success of the Cobb-Douglas functionhaving “correct” coefficients is perfectly con-sistent with wide varieties of data, and cannot beinterpreted as supporting aggregate neoclassicalproduction and distribution theory.

Summary and conclusions

It is characteristic of theoretical parables thatthey illustrate truth para-digm, truths which more developed theoreticalstructures may modify and elaborate, but cannotundermine. In the neoclassical progression ofparables from simple exchange to capitalism asthe final solution to Man’s “natural” greed, onecentral theme which emerges right in the begin-ning is the conception of equilibrium prices as“scarcity prices:” relative prices which reflectthe relative scarcity of commodities.

In their most developed form, neoclassicalparables have sought to present the notion ofscarcity pricing as an explanation of the distribu-tion of income between workers and capitalists.Here, the task is to portray a capitalist economyin such a way that the wage and profit rates maybe seen to be scarcity prices of labor andcapital, respectively. But for this to be even alogical possibility, it is at the very least neces-sary that the wage and profit rates behave as they were scarcity prices i.e., that the profitrate fall as the capital-labor ratio rises, and thewage rate fall as the labor-capital ratio rises.This correlation is minimally necessary for theinternal consistency of the parable (though ofcourse its existence would hardly justify the im-plied causation).

Alas, the grand neoclassical parables havefallen on hard times, and after repeated demon-strations of their logical inconsistencies, theyhave been abandoned by the high-brows of thetheory; not without regret, though, for asuelson so insightfully notes, within the parable“the apologist for capital and for thrift has a lessdifficult case to argue” (Samuelson, 1966).

“If all this for those nos-talgic for the old time parables of neoclassicalwriting, we must remind ourselves that scholarsare not born to live an easy existence We respect, and appraise, the facts of life”uelson, 1966).

Not everyone was ready to give up the oldtime parables though, and those who chose to ig-nore the previously mentioned facts of lifesought succor where else? in the “facts.”The “real world,” whose vulgar intrusions neo-classical theory had in the past so carefullyavoided, became its last refuge. Facts, after all,are always better than facts-of-life.

And what are these facts? Simply, that againand again, aggregate Cobb-Douglas productionfunctions work that is, they not only give agood fit to aggregate output, but they also gener-ally yield marginal products which closelyapproximate factor rewards. Since the aggregateproduction function is the simplest form ofthe grand neoclassical parable, its apparentlystrong empirical basis has often been taken asproviding a good measure of support for the oldtime religion, regardless of what the theory says,

The of this has toshow that these empirical results do not, in fact,have much to do with production conditions atall. Instead, it is demonstrated that when the dis-

data (wages and profits) exhibit con-stant shares, there exist broad classes tion data (output, capital, and labor) that canalways be related to each other through a func-tional form which is mathematically identical to Cobb -Douglas “production with

constant “returns to scale,” “neutral technicalchange, and “marginal products equal to

tor Since this result is a mathematical conse-

quence of any (unexplained) constancy ofshares, it is true even for very implausible data.For instance, data that spell out word“HUMBUG” were used as an illustration, andit was shown that even the humbug economycan bc by Cobb-Douglastion function having all the previously men-tioned properties.

Similarly, we have examined paper on measuring technical change; and heretoo it is shown that the underlying productionfunction which he isolates, by removing the ef-fects of technical change, can be algebraicallyanticipated, even down to the fitted coefficientsof his regression.

Next, Franklin Fisher’s mammoth simulationexperiments are examined and once again it be-comes clear that the laws of algebra can antic-ipate the laws of simulation from the structure ofthe experiments alone.

Lastly, in the final part of this chapter, theanalysis is to provide a tion for cross-section aggregate production func-tions. The overall impact of these discussions, itis hoped, will be to demonstrate that the to which the neoclassical hangers-on clutch sodesperately is as empty as their own abstrac-tions.

Postscript

The point of this chapter is to demonstrate thatas long as distributive shares are constant, it isan algebraic law that the Cobb-Douglas

tion “fits” almost any data. Hence,paper and the Humbug data stand on the samefooting.

has recently claimed that all along theintention of his 1957 paper was to “yield anexact Cobb-Douglas and tuck everything elseinto the shift factor” 1974, p. 121). Buthis own printed words give quite a differentimpression: in the original paper, after he hasderived the so-called shift factor A(t), ex-pressly states his intention to “discuss the shapeof I) and the (underlying) ag-gregate production function” 1957, p.317). To this end, he constructs a graph of f(k)versus noting with obvious satisfaction thatin spite of “the amount of a priori doctoringwhich the raw figures have undergone, the fit isremarkably 1957. p. 317). givingrise to “an inescapable impression of curvature,of persistent but not violent diminishing re-turns” 19.57, p. 318).

If, as now claims, he knew all alongthat the underlying production function wouldbe a Cobb-Douglas, then why bother “recon-structing” it? Why the surprise at the tightnessof fit and the “inescapable impression of curva-ture”? Why does need regression analy-sis to “confirm the visual of dimin-ishing returns . . 1957, 319). If

had indeed understood his own method,he should have known that regardless of theamount of a priori doctoring of the data, the lawsof algebra dictate that the fit of f(k) versuswould be very tight as well as being inescapablycurved. But it is hardly necessary to rediscoverthese algebraic artifacts by means of graphs and

Having just said that his method and his edu-cation lead him to conclude that even theHumbug economy is neoclassical, nextasserts the very opposite. With the help of Sam-uel L. Myers, he runs a regression of the formIn = a, + + b on the Humbug data,and finds to his obvious that this not only to a very poor fit but also gives rise to anegative coefficient for The moral seemsclear: production functions do not “work” forthe Humbug data, whereas they do for real data

1974, p. 121).But once again, his method and education be-

tray him, The laws of algebra show that almostany production data associated with a constantprofit share could be cast in the form Q =

The Humbug data was an illustration ofthis, and it was sufficient for my purpose in theoriginal paper to show that even in this case the“underlying” function was extremely wellfitted by the Cobb-Douglas = =

and that the so-called shift factor was a function of Hcncc, Humbug

. . . the core of the theory of a private owner-ship economy is provided by the theory of ex-change” (Walsh, 1970, p. 159).Garegnani in fact does not state it this way. Heshows that the necessary and sufficient condition isthat the wage-curves all be straight lines, andshows that this in turn is true when all industrieshave the same capital-labor ratios, i.e., whenprices are proportional to labor values (Garegnani,1970, 421)Q(t) value of output; K(t) value of the utilizedstock of capital; L(t) employed stock of labor;t time.I thank Professor Luigi Pasinetti for having pointedthis out in his comments on an earlier version ofthis paper.R. R. Nelson a summary of subsequentrefinements (Nelson, 1964).“In order to isolate shifts of the aggregate produc-tion function from movements along it” 1957, p. 314).The discrete equivalent for is AA/A, whereAA = A(t + 1) A(t). Thus A(t + 1) = A(t) +AA/A]; in 1909, = 0, and by setting A(0) =

derives a series for A(l), A(2) . . . , fromthe data on Since calculations contained an arithmeti-c a l rcprcscnting ycnrs

data would be consistent with a neoclassical pro-duction function having “neutral technicalchange” and “marginal products equal to factorrewards”.

Obviously, given that the underlying func-tion f(k) was numerically specified by the laws ofalgebra (Equation (12) and note 9, in thischapter), all that would have been necessary fora complete numerical specification was a fittedfunction for B(t). However, since such a fittedfunction was not necessary to the logic of myargument, I was content withB(t) versus time, as in Figure 5.3.

A glance at Figure 5.3 is sufficient to indicatethat no simple linear or log-linear function will fitB(t). And yet this is precisely the that

uses in his regression, He naturally getsa very poor fit. How clever.

In this version of the paper, for the sake ofcompleteness, I do actually specify a fitted func-tion for B(t), with an = (Equation 13).But the logic of the argument does not requirethis step; it only requires that the so-called shiftfactor be a function solely of time: there isnothing in neoclassical theory, no law of pro-duction or of nature, which requires B(t) to belinear or log-linear. Struggling under the weight

Myers seem have forgotten that linearity is merely a conve-nient assumption whose applicability must at alltimes be not merely assumed.

Notes

9

11

12

13

1943-1949 clearly lay outside the range of any hy-pothesized curve. After expressing some

Suluw them out of his 1957, 318).

The deviation of the numerical value of the con-stant term is explained on pp. 20-21 of 1957 paper.I wish to thank Larry Heinruth and especially PeterBrooks, for the time and effort expended inderiving this function. Two in-volved in the fitting. First, a two-year movingaverage was constructed from the data forB(t), by means of the formula = [B(r)

in which the year 1909 represents =1, 1910 by = 2, etc. Second, the function ofEquation (13) was fitted to this moving averageB(t), a = Since the fitted function has = 16 parameters toit, and since there are T = 38 data points in themoving average the corrected for degreesof freedom is (Goldberger, 1964):

= =

By definition = Applyingthis to the expression for in Equation (14)yields = From Equation = where isconstant over time. Thus

= P i dt

and

Similarly for employment from (23). In = + + (small ran-

dom deviations). Ignoring the small deviations, anddifferentiating gives (Fisher, 1971, p. 309)

d t K , ( t )

=

Dropping the time subscript, and differentiating,

+

+ (1

1 6

17

18

19

20

21

so that

Fisher (1971, p. 309) assumes A, + sothatThe function form in Fisher’s equation, thebest form for Hick’s experiments, is log =

+ where his correspondsto our and his J/L to ourj. Fisher uses “log” fornatural logarithms (Fisher, 1971, p. 313).Fisher has two ranges of and

in both the unweighted average = 0.75(Fisher, 1971, p. 309).The functional form Fisher finds best for Capi-tal experiments is = which, allowing for notation differences, is iden-tical to equation 5.38 (Fisher, 1971, 313).Yet confronted with the humbug data, says:“If you ask any systematic method or any edu-cated mind to interpret those data produc-tion the

the answer will be that they are exactly whatwould be produced by technical regress with a pro-duction function that must be very close toCobb-Douglas” 1957, p. 121). What kindof “systematic method” or “educated that can interpret almost any data, even thehumbug data, as arising from a neoclassical pro-duction function?

uses the form = +since the general form under consideration is

so that A(t) + has obviously specified A(t) as log-linear: A(t)a ,

References

Brown, P. 1957. “The Meaning of FittedDouglas Functions”, of’ Eco-nomic&,

Dobb, M. H. 1970. “Some Reflections on the SraffaSystem and the Critique of the So-called Classical Theory of Value and Distribution,” un-published.

Dewey, D. 1965. York:Columbia University Press.

Ferguson, C. E. 1969. The Theory Production Distribution. Cambridge: Cam-bridge University Press.

1971. “Capital Theory Up to Date: A Comment onMrs. Robinson’s Article,” Journal Economics, Vol. IV, No. 2, 250-4.

Fisher, F. 197 1. ‘Aggregate Production Functions andthe Explanation of Wages: A Simulation Experi-ment,” of Economics Stat is t ics

+ (1 Garegnani, P. 1970. “Heterogeneous Capital, the Pro-

Function and the Theory of Distribu-tion,” Review Studies, 37, 407-36.

Goldberger, A. S. 1964. Econometric NewYork: John Wiley Sons.

1 1970. Investment. interest. and

Englewood Cliffs, N. J.: Prentice-Hall.

Hicks. J. R. 1965. New York:Oxford University Press.

1968. Economics. NewYork: Macmillan.

Nelson, R. R. 1964. “Aggregate Production Functionsand Medium-Range Growth Projections,”

Economic Samuelson, P. A. 1962. “Parable and Realism in

tal Theory: The Surrogate Production Function,” 29: 193-206.

1966. Summing Up,” QuarterlyLXXX,

Shaikh, A. 1974. “Laws of Production and Laws ofAlgebra. The Humbug Production Function: AComment,” Economics and Statistics,Vol. LVI, No. 1, 115-20.

Simon, H., and Levy, F. 1963. “A Note on theCobb-Douglas Function,” Economic

Vol. R. M. 1957. “Technical Change and the Ag-

gregate Production Function,”

1 9 7 4 . “Law of Production and Laws of Algebra:The Humbug Production Function: A Comment.”Review Economics and Statistics, Vol. LVI,No. 1,121.

Walsh, V. C. 1970. to Contemporary New York: McGraw-Hill.

Walters, A. A. 1963. “Production and Cost Functions:An Econometric Survey,” N o .l -2 , l -46 .