Toward a Marxian Model of Economic Growth

David Laibman

The American Economic Review, Vol. 67, No. 1, Papers and Proceedings of the Eighty-ninthAnnual Meeting of the American Economic Assocation. (Feb., 1977), pp. 387-392.

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RADICAL ECONOhfICS

Toward a Marxian Model of Economic Growth

In 1941. Joan Robinson urged us to the frontiers of our economic understanding by comparing Marxian and orthodox economics in these terms: "if there is any hope of progress in economics at all. it must be in using academic methods to solve the problems posed by Marx" (p . 95). In the spirit of that injunction. the present essay pursues the comparison on the limited terrain of models of economic grou th . Its major finding is that an adequate conception of technical change as it occurs within capitalist production relations makes it possible to ad-vance on both the Marxian and the growth- model fronts. With regard to the fo r~ner . it is the key to overcoming seemingly insurmount- able indctcr~ninacies in the dynamics of the crucial variables: with regard to the latter, it brings into focus a capitalist-investor who is neither the neoclassical plaything of the sovereign consumer . nor the al l -powerful del is e.r rllnc,hincz of the pos t -Keyncs ians . The model suggested here will also transcend the limitation\ of the steady state. \\ithout losing coherence. although like the main proto- types of the neoclassical and post-Keynesian models i t will skirt the complexities arising from a rigorous treatment of heterogeneous outputs and fixed capital.

The hlarxian ambiguities stem from Xlarx's unsubstantiated law of rising tendency of the "organic composition of capitalv-the ratio of the value of the stock of physical capital to

"4ss1,tant Protes5or. Uroohl)n College C I ~ ) Cn!\er\rt) of Yew Yorh I u l \h to ~chnowledge the indthenia t~c~l assi5tance of Ercurnent Ozizrnir of Richmond College. and the cornputationnl a\sistance of Carl Luni of the Hrooh-

College Computer Center, paper is a report a iargcr \tudy in progress.

the flow of current labor cooperating with that stock in production. For this ratio to rise. thc accun~ulation of physical capital per unit of labor must exceed the rate at \\hich rising pro- ductivity cheapens the elements of phqsical capital; in one-co~nrnodity macro terms, the physical capital-labor ratio. k , must grow more rapidlq than the net productivity of labor, J .

Marx never supplied a rationale for this asser- tion. although he repeatedly affirms that the rising organic composition of capital is "but another expres\ion for the rising productivity of labor." (On this see Maurice Dobb, Paul Sweezy and Laibman 1976.) The "rate of exploitation." or ratio of the unpaid to the paid portions of the working day. is a second crucial variable which, despite much discussion, does not have clear determinants. althoilgh Marx's conception of the t-plorive impoverishment of the ~vorking class implies that this variable should have a rising tendency. The well-known upshot is that the famous "law of the falling tendency of the rate of profit" is deprived of a cons i s ten t f o u n d a t i o n . Never the less , the Marxian vision has unique synthetic power: choice of technique is inseparable from tech- nical change, and both are aspects of the invest- ment process, which in turn is a moment of the wider concept of a~~c~li/nrrlntion. notentailing only quantitative growth but also the "ex-tended reproduction" of antagonistic social relations in production. i . e . . the accumula-tion of c~lpitcllin the full Marxian sense of thdt

B, contrast. the grouth models of academic economics miss this " i s ion , by or rigidly separating its several aspects. In neo- classical growth. to illustrate. technical change

is generall!, seen as autonomous and exterior to the economic process; choice of technique simply adapts the capital intensity of production to the ratc of savings coming from sovereign consumers. Positive net investment is the passive result of an optimal allocation deci- sion. like supplying consumers with anything else that has positive demand price. The post- Keynesian models go to the opposite extreme, postulating full investor sovcreignt! and banish- ing the production function from their world altogether.' A Marxian growth model should avoid these opposite reductions of the capitalist, who will appear as a powerful subjective force in pursuit of identifiable goals but within a system that is a process without a subject. i . e . . Fvithout totall) unconstrained actors.

I We begin by d e f i n i n ~ the ratc of profit, as

profits per head divided by the value of (phys- ical) capital per head, noting that this irnplics wages paid out of revenue, and Marx's "vari- able capital" not subject to accumulation:

'The neoclass~cal literature ma) be well represented by Robert Solow (195h), Tre\or Swan and James Meade. The leading post-Keynesian models ha\e been propounded by N~cholas Kaldor and Robinson (1956. 1972). A seminal sur\ey o f the field is F. H. Hahn and R. C. 0.Matthews.

u.here E is the ra te of exp lo i ta t ion ; w =

1 / ( 1 + E ) = the wage share of value added; 11 and k = physical net output and capital per head, respectively. We may note here that 1 / y is the labor value of a unit of output; that the ~ ~ t r l u eof (physical) capital per unit of labor is thercfore k ( l /y) = Q, the organic coinposi- tion of capital.

We also posit a short-rut1 (in a sense to be defined below) tcchr~ology constraint in the form of the familiar Cobb-Douglas relation.'

The kc) to technical change in the (capitalist) short run is that capitalists maximize the trun-sitioncrl profit rate, i , defined as the rate of profit during the "transition period, when the use of machinery is a sort of monopoly. (in which) the profits are therefore exceptional . . .. 3 . accruing as they d o to an innovator for whom the "social value of the article produczd (is) above its individual value . . ." (Marx, pp. 443-44). If the individual capitalist pur- chases inputs from the outside, he will value thern at the appropriate unit value 1 / y , , where the zero subscript identifies the prevailing technique. The same will apply of course to the outputs. The maximand is therefore

'This function is dcfined for a short run. conceived as a period in wh~ch firm$ inake dcc~sions based on glven prices: in the longer run. changes in baluation cause erratic or "perver\e" iito\ements in v and k . so that (2) does not ha\e a "well-behaved" long-run counterpart: w e Piero Sraffa, Geoffrek Harcourt. As a micro function. it should not be interpreted as a "p\eudo" aggregate production function (Robinson 1956). but rather as a high11 siiil-plified cxprebsinn of stsicti) engineering relationshipi; only the rn~cro behavior \ *h~ch results need be aggregated Contrary to appearances. linear homogeneit! is nor being assumed here; under nunconstant returns to scale the parameters of (2) would be functtions of scale, but this uould not affect the argument.

389

1

L'OL.. 67 \O. I RADICAL ECONOMICS

from uhich

This is a straight line in \-k space; since is exogenous to the f i rm, the interccpt term y o / ( l + E ) is fixed. The firm ~vil l rotate (4) upivard as far as possible, i . e . , until it is tangent to the production function (see Figure 1). The tangency point thus identifies the target position o f the firm, y, and k l . At this point. ? clearly equals the marginal product of capital, d y l d k . The marginal productivity principle. thcn, ex-presses the behavioral rule for thc individual capitalist-~naxi~nize the trrrnsitionul profit ratc -and qualifies the neoclassical theory based on this rule as "bourgeois economics" in the pre- cise sense of an economics uhose limits art. those which the capitalist class "docs not transcend in practice."

Our task now is to find the parameters of y1 and k l , and then the growth rates of y, k and Q. First, the ? ,,,,, locus:

We will wish to assume that the prociuction function shifts during the transitional period, although the shift will be small. The production function, before and after the shift, is written

(2b) y1 = cOehtkY(1 + bdt)

Lastly, we need the condition that the maximum transitional profit rate equal the slope of the (shifted) production function. From (2b) we find dy, /dk l to get

( 5 ) f,,, = coae"'k?-'(I + bdt) = y l a / k ,

Substituting into (4a), and rearranging:

from which

(7) y* = y,/y,, - 1 = [(I - a ) ( l + E)]- ' -

a h c r e the a5tcrish denotes a grolith rate. Put- ting (2'1) and (2b) into (6) and solving for k , :

(8) k1 = kO [(1 + bdt ) ( l - a ) (1 + E)]-',"

from \+hich

Finall), an expression for the gro\bth ratc of the organic colnposit~on of cap~ta l f o l l o ~ s easily.

Figurc 2 shows the shape of the Q*-E rela-t ion , with the "shift factor" d t = 0 , for various values of a . The important feature here is the inverse relationship, with Q * bc- coming zero at some (relatively high) level of E . A "vertical integration" case. in which the

390 Z\IFKIC1\ E(O\O\II(

firm produces and values its inputs according to the novel technique, has also been analyzed; vertical integration shifts the curve outward, raising the critical value of c at which growth of the organic composition of capital comes to a halt. In a Marxian world. a must be con-sidered endogenous to the model , and the search for engineering practice which makes it possible to realize high transitional profits would lead this parameter to be high and pcr- haps rising. A high a , a small shift factor in- dicating intense competition. and progressing vertical integration all suggest that the critical E is far above historically observed levels. and this amounts to a rather qualified "law of the rising organic composition of capital.' '

I1 The Q*-E relation (10) is the-hitherto miss-ing-link that will enable us to sketch the con- tours of a Marxian growth path for the capitalist economy. The other ingredients, savings and the growth of the labor force, present no new problems. As in all postclassical models, la-bor force growth is assumed constant and exogenous:

(11) I I = constant

The appropriate savings function is the well- known classical savings function, with zero sav- ings out of wages and savings out of profits occurring at a high and constant rate u , detzr-mined by the interaction of "animal spirits" and the historically conditioned level of con-sutnption out of profits. We therefore have

where C is the vahre of the physical capital stock. From Q = C / L , where L is the labor force, we have

And finally, thc basic identity of the Marxian system:

15\0(14TIO\ FEBKC 4 R ) I977

(14) r = -E

Qcl + 6 , '

Combining (12) and (13), we find the general form of what I will call the (.011.~istetztp ~ t hof the economy:

( 1 3 r = Q*/LI + n / a .

When Q* = 0 , this reduces to the Cambridge growth ciluation, n = ar. In general, Q* > 0 , and growth is off the steady state; hence the terrn "consistent path" in place of the usual "stcad!.-state path."

By combining ( I 5 ) with (10). we fully specify the consistent path (its derivation is diagrammed in Figure 3):

1(16 ) r =

a [ ( l + bdt) (1 - a)( 1 + E)]""

In the lower left quadrant of Figure 3 , the growth factor Q* + n is plotted against E , the

391 1'01.. 67 \0.I RADICAL ECONOMICS

horizontal summation of (10 and ( 1 1 ) . In the upper left, we have (12) and, linking the two left quadrants, ( 13). which ensures that the level of unemployment will be constant over time, since the growth rate of the demand for labor, C * - Q*. is equal to the growth rate of the labor supply n . The consistent-path relationship between r a n d E is then derived in the upper right quadrant; it is the downward-sloping curve, whose curvature is determined by (10). The upward sloping curve in the same quadrant is (14). and it approaches the asymptote l / Q . The intersection of the two curves identifies a consistent point T, corresponding to some historically given level of Q . At T, Q * > 0, and (14) is shifting downward. The initial impact of this is to lower r ; with a fixed in the short period, C * will fall, the growth rate of demand for labor C * - Q* will be less than the growth rate of the labor s u p p l y , and unemployment will rise. For this lower. r and C * to be consistent with constant average unemployment u t~dwith the degree of capital- deepening that capitalists wish to carry out in search of maximum transitional profits, the rate of exploitation will have to rise. in the degree indicated by the slope of the consistent path locus in the upper right quadrant. If there is a positive relationship between the unemployment rate and the rate of exploitation, the rise in the former will have the desired effect; E will rise until Q * is choked off and C * increased until the difference C * - Q * = n. The opposite will occur if the economy is at any point to the northeast of the consistent path locus. The consistent path is , therefore, to the extent that this mechanism operates, a stable path.

Assuming the economy stays on the con-sistent path. it will move toward the point S at which the growth of Q comes to a halt. S rep-resents a stable steady state in which Q * = 0, and, as can be shown under certain strong con- ditions, the marginal productivity rule holds. In this version of Marxian accumulation. there- fore, the steady state represents a final resting place toward which the system moves. In this sense the system is stable "in the large," and

the question arises: How obtainable or unob-tainable is the point S.? The "dual" to the Marxian accumulation theorem would place the economy at S and reduce the consistent path to a historical or irrelevant exercise. It must suf- fice here to state that study of the coordinates of E,,, and r,,, indicates that the point S is probably out of reach for any existing capitalist economy, and likely to remain so. It should be noted that the consistent path embodies the two laws of motion that emerge from Marx's dis- cussion of accumulation: the falling tendency of the rate of profit and the rising tendency of the rate of exploitation.

In summary: capitalists in competitive con-ditions pursue maximum transitional, or quasi- monopoly, profit rates. In the short run. in which this pursuit alone takes place, they are constrained by production functions whose parameters are determined by an engineering practice historically adapted to capitalist produc- tion re la t ions , and therefore emphas iz ing deepening over fundamental research and shift- ing ou tward . T h e model thus portrays an intimate connection between accumulation and technical change (in which the former tends to outrun the latter for reasons specific to capital- ism), rather than accumulation it1 ~ ,ucuo (as in the Cambridge post-Keynesian models) or dis- embodied technical change and choice without an investment function (as in the neoclassics).

The resulting relation between the growth rate of the organic composition of capital and the rate of exploitation is the capstone of the model. The derivative consistent path identifies combinations of c and r that imply balance between the growth rates of the labor force and of the capital stock. The model then sets the stage for study of deviations from the con-sistent path and their resolution; it confirms some major Marxian predictions. such as the long-term rising rate of exploitation and slowing down of the growth rates of output and the capital stock.

Models invariably d o violence to the full

texture of reality, as our best thinkers-Marx. Marshall, Keynes-all pointed out. Marx's vision included growing potential for "real-ization" crisis, the concentration and centraliza- tion of capital, the impoverishment of work, foreign economic-military expansion, increas- ingly authoritarian relations within production units, increasing reliance on the machinery of the state to reinforce reproduction at adequate levels of profit, and ultimately the cannibaliza- tion of civil society itself-and all this in a context of resistance on the part of the working class and allied strata, which pick up, one by one. the banners thrown down by the pillars of society and formulate ever-more thorough-going solutions. These, and more, are aspects of a Marxian theory of capitalist de~~eloprnent; any model of capitalist gro\r,th, l ike good seasoning. must accent the vision from which it derives. rather than turning one's attention away from that vision.

REFERENCES Raford Boddy and James Crotty, "Class

Conflict. Keynesian Policies. and the Busi- ness Cycle." Morlthlx Re l , . , Oct . 1974, 2 6 , 1-17.

Maurice Dobb, Political Econotily and Cupitul- isrx, New York 1945.

F. H. Hahn and R . C. 0. Matthews, "The Theory of Economic Growth: A Survey," in Sur\lc~ys in E~~otzornic Theory , Vol. I , New York and London 1967.

Geoffrey C. Harcourt, "Some Cambridge Controversies in the Theory of Capital," J . Econ. Lit. , Sept. '1969, 7 , 369-405.

Nicholas Kaldor, Essays otz Vuluc~ and Dis- tributiotz, London 1960.

David Laibman, "The Marxian Labor-Saving Bias: A Formalization," Quart. Rc~v. Ecotz. Bus. , Autumn 1976, 16, 25-44.

, "The Marxian Profit Cycle: A Macro- model," unpublished.

H. Lewin and J . Morris, "Marx's Concept of Fetishism," Scietlce & Society, forth-coming.

Karl Marx, Capital, Vol. I , Chicago 1906. James Meade, A Neoc~lussical Theory of

Econowzic Gro\r,th, London 196 1 . Joan Robinson, The Accumulatiot~ of Capital,

London 1956. , Atz Essay on Mar-sian Economics,

London 1942. . Ecotzomic Herc~sic~s, Cambridge 1972.

Robert Solow, "A Contribution to the Theory of Economic Growth," Quart J. E c o n . , Feb. 1956, 70.

. "Technical Change and the Aggre-gate Production Funct ion," Rel , . E c o t ~ . Statist., Aug. 1957, 39, 3 12-20.

Piero Sraffa, Production of Cornrnodities by Means of Commodities, Cambridge 1960.

Trevor Swan, "Economic Growth and Capital Accumulation," Econ. Record, Nov. 1956, 32.

Paul Sweezy, T h e T h e o r y of Capi ta l i s t Development, New York 1956.

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Toward a Marxian Model of Economic GrowthDavid LaibmanThe American Economic Review, Vol. 67, No. 1, Papers and Proceedings of the Eighty-ninthAnnual Meeting of the American Economic Assocation. (Feb., 1977), pp. 387-392.Stable URL:

http://links.jstor.org/sici?sici=0002-8282%28197702%2967%3A1%3C387%3ATAMMOE%3E2.0.CO%3B2-6

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References

Some Cambridge Controversies in the Theory of CapitalG. C. HarcourtJournal of Economic Literature, Vol. 7, No. 2. (Jun., 1969), pp. 369-405.Stable URL:

http://links.jstor.org/sici?sici=0022-0515%28196906%297%3A2%3C369%3ASCCITT%3E2.0.CO%3B2-W

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