aa r X i v : . [ q -f i n . E C ] M a y version 2.07 An equation for a time-dependent profit rate ⋆ Rafael D. SorkinPerimeter Institute, 31 Caroline Street North, Waterloo ON, N2L 2Y5 CanadaandDepartment of Physics, Syracuse University, Syracuse, NY 13244-1130, U.S.A.address for email: [email protected]
Abstract
Taking as a hypothesis a form of the labour theory of value, and withoutassuming equilibrium , we derive an equation that yields the profit-rate π as a function of time. For a mature economy, π ( t ) reduces to the productof two factors: ( i ) a certain retarded average of the sum of the growth-rates of productivity and of the size of the labour-force measured by hoursworked, and ( ii ) the ratio of the current rate of surplus value to its ownretarded average. We also suggest an empirical test of the equation. Keywords and phrases : labour-value, profit-rate, time-dependence, non-equilibrium economics
I. Introduction
Any model of capitalist dynamics that rests on an assumption of economic equilibrium canhave at best a limited explanatory value. Even without having witnessed the upheavalsof the past ten years, I doubt that any serious observer would want to ignore the factthat disequilibrium appears more characteristic of capitalist economies than its opposite. ⋆ The present article is a revised version of a 1982 manuscript that was submitted unsuc-cessfully to various economics journals. The main equations and results are the same asbefore, with some improvements and changes of notation. The surrounding discussion hasbeen revised significantly, especially the concluding section.1he history of capitalism presents us not with smooth development, but with a series ofbooms and subsequent “crises” interspersed with even more disruptive episodes like wars(including wars conducted within the capitalist “core” and wars inflicted by the core nationson the “halo” regions of the capitalist world). Faced with this chaotic panorama, we mustask whether there exist economic models robust enough to survive under conditions where“competitive equilibrium” (if it ever existed) has been lost.As soon as one decides to seek such a model one comes face to face with a problem.How are we to attribute exchange-values to commodities? Neo-Ricardian approaches assignvalues-qua-prices by reference to an input-output matrix or its generalizations. Otherapproaches, even farther removed from reality, appeal to the hypothetical “utilities” and“preferences” of atomized economic actors. In both cases, an appeal to uniform growth orto some other type of equilibrium is essential. One might attempt to rescue such modelsby adding in small fluctuations away from equilibrium, but that could be done only if theequilibrium could be claimed to be stable, or if it could be claimed to become stable viasome sort of coarse-graining. As far as I know, such claims have not been put forward withany seriousness, and no such generalization has been attempted.In the face of such difficulties, it is tempting to pursue the long-standing idea thatexchange-value derives from labour-value. According to this hypothesis, known usuallyas the “labour theory of value”, prices are merely a phenomenal form taken by valueswhich are more fundamentally defined by asking how much accumulated human labourwent into producing a given item. One can of course raise many questions about how welldefined this labour-value really is, how it relates to prices, etc. This is not a discussionI want to enter into here. Instead I simply want to demonstrate that if one uses labour-time (denominated in hours, for example) as one’s measure of value, and if one definesprofit in terms of labour-values, then one can derive a general equation for profit-ratethat expresses it as the “historical resultant” of variables that are relatively close to being2irectly observable, namely the rates of growth of productivity and of the labour-force,together with the rate of surplus value (or equivalently exploitation) † If one accepts the labour-value hypothesis, then one would expect that profit definedthereby would offer a conceptual tool useful for the analysis of capitalist economies. Onemight also go further and expect that “labour-value profit” would equate roughly to profitas measured by prices. If so then the underlying assumptions would be opened up to anempirical test.
II. Assumptions
The analysis to be presented herein will be self-contained, but it is worth pointing out itsrelation to the discussion found in volume III of Das Kapital, in the section on the “ten-dentially falling rate of profit” [3] (which seems to be Engels’ working out of a preliminarydraft by Marx).The argument made there works entirely with labour-values and could perhaps besummarized as follows, if you will allow me to emphasize those aspects that relate bestto the present analysis. Profit, insofar as it is invested, implies an increase in the mass ofconstant capital, which in turn raises the ratio of total capital to surplus value. But, sinceonly surplus value ends up as profit, the rate of profit must fall secularly. In a word, presentinvestment causes future profit-rates to fall. However, this fall is only a tendency which hasto contend with a number of possible “counteracting causes” [Entgegenwirkende Ursachen],among them the lengthening of the working day, an increase in the rate of exploitation, † In this way, the formula offers a natural generalization of the equilibrium result thatprofit-rate is in essence just a measure of the rate of growth of the economy. See forexample the Sraffa input-output model in [1] or the von Neumann “balanced growth” rayin [2]. 3he cheapening of capital due to rising productivity, and the effects of foreign trade andimperialist super-profits.Apparently the falling rate of profit was not a prediction of the analysis but an “ob-served fact” that writers on economics were attempting to explain. And indeed, we willsee that the profit rate will fall for some time if it begins at a high value. However, wewill also see that if we more quantitatively take into account the first three of the above“counteracting causes”, as well as the related possibility of growth in the number of em-ployed workers, we will arrive at a floor below which the profit-rate need not fall. On theother hand, we will also arrive at an upper bound above which — on average and in thelong run — it cannot rise.Our analysis will treat the economy as a whole, that is we will treat everything inaggregate . Our main results will be equations (5) and (16). In deriving them, we willemploy for simplicity a continuous-time model of the economy, i.e. there will be flows of raw materials, of labour, etc. We will also work in the limit of zero “latency time”,whence the only capital that will enter into the profit-rate will be fixed capital. (Withnonzero latency time, other forms of capital like inventory and “advanced” wages wouldalso influence the profit-rate, but some of them could still be treated as special cases offixed capital.) Thus we will make (at least) the following assumptions and idealizations.(1) A closed capitalist economy. No (or only equal) trade, no destruction of capital (asby war).(2) The rate of profit is equal to total net profit divided by total fixed capital.(3) Profit and capital will be measured by their labour-values, as will all other items.(4) All profits are invested and there is no saving by workers.(5) All investment goes into capital formation.4
II. Some consequences
Let all values be measured in current labour time (replacement values at time t ) and let K = K ( t ) be the (value of the) total fixed capital at time t . (Recall that we are treatingthe economy in aggregate.) If π = π ( t ) is the rate of profit then in unit time the amount of profit will be by definition πK , whose reinvestment causes the value of fixed capitalto increase at a rate dK/dt = πK . At the same time however, the cheapening of capitalgoods reduces K at the rate dK/dt = − rK , where r is the rate of productivity growth inthe capital goods sector. Combining these competing effects yields dKdt = − rK + πK = ( π − r ) K (1)If only a fraction φ = φ ( t ) of profit ends up being invested then (1) becomes dKdt = − rK + φπK , (1 a )but for simplicity we have assumed that φ ≡ P as the number of workers, λ as the fraction of the dayeach spends working on average, and w as the wage rate (hours of wage paid to a workerper hour of his/her work). Recall here that wages are being measured directly in units oflabour-time. Accordingly, w and λ are both dimensionless numbers between 0 and 1. ⋆ Thetotal wages paid per unit time are thus wλP , which subtracted from the value produced,namely λP , leaves for the collective capitalist a surplus value s of s = λP − wλP (2) ⋆ We neglect those rare circumstances in which w might exceed unity or fall below zero.We similarly neglect the possibility (if it is one) that “speedup” and overtime become soextreme that λ effectively exceeds unity. 5t is convenient to combine λ and P into the product L = λP (which is the aggregatelabour-time expended per unit time) and to define σ = 1 − w , which will be called the rateof surplus value , and which is also a pure number between 0 and 1 representing the fractionof the workers’ production retained by the capitalists. † Combining these definitions yieldsfor the mass of surplus per unit time, s = (1 − w ) λP = σL Finally, since profit equals s by definition, we find for the rate of profit, π = s/K , simply π = rate of profit = σLK (3)Now how does this labour-value profit-rate change with time?To answer this question, we need only differentiate (3) and combine the result with(1) to obtain ˙ ππ = ˙ σσ + ˙ LL − ˙ KK = ˙ σσ + ˙ LL + ( r − π ) , an equation which will look slightly simpler if we define β = ˙ LL + r (4)so that it becomes ˙ ππ = ˙ σσ + β − π (5)Notice here that ˙ LL = ˙ λλ + ˙ PP is just the rate of growth of the number of workers plus therate of lengthening of the working day. Hence (as is anyway obvious) L = λP can rise overlong periods only if P does, since λ is bounded above by unity.One sees how almost all of the ingredients of the “falling profit-rate” discussion arerepresented in (5). If one were to keep only the third term in (5), it would reduce to † The commonly defined “rate of exploitation” e is related to σ by e = σ/ (1 − σ ).6 π = − π , and one would deduce that π must decrease forever, falling asymptotically tozero (or diverging to −∞ if it were negative). But this would be to neglect the effect of thetwo “counteracting” terms. The first of these, namely ˙ σ/σ , can check the fall to the extentthat the rate of exploitation increases, but its effect necessarily dies out as σ approachesits maximum value of unity. The other counteracting term, namely β = ˙ LL + r = ˙ λλ + ˙ PP + r ,receives three distinct contributions. The contribution ˙ λ/λ corresponds to lengthening theworking day, but its effect is limited in the same manner as is that of ˙ σ . Therefore, if β isto be effective for more than a limited time, it can only be because either P (the size of theworkforce) or the productivity of labour (corresponding to r ) grows indefinitely. The neteffect is that the long-term profit rate is governed by these two underlying growth rates.(Arguably they cannot act forever either, because no exponential growth can last forever.But the time-scales on which these limitations would assert themselves are evidently muchlonger than those belonging to σ and λ .)In the next section, we will draw some more quantitative consequences from (5). Infact (5) will let us compute the profit-rate at any time t , provided that we are given thevalues of σ and β at all earlier times. ♭ For now, we just note that, if we ignore the transienteffects of a changing σ , the profit-rate π necessarily moves toward β , decreasing when itexceeds β and increasing if it falls below β . In this sense β is the “reference value” that π always seeks.Before turning to the general analysis just referred to, let us mention a simple exampleof what can be expected. Suppose that both σ and β are independent of time and that at ♭ If the economy is insufficiently mature, then one needs also the initial value of π or itsequivalent. 7 = 0 there is some labour but no capital in existence (“new economy”). Integration of (5)(conveniently done by changing to the variable z = 1 /π ) yields in this situation π ( t ) = β − e − βt . (6)Thus the profit-rate under these conditions is infinite at first, but it falls to its equilibriumvalue of β on a time-scale whose duration β − is itself set by β , i.e. by the underlying rateof growth of productive capacity. The “tendency to fall” and the “counteracting causes”are then in balance. IV. The profit rate more generally
In general neither β nor σ will be constant. Nevertheless, it is clear from the fact that (5)is a first-order differential equation that given the initial value of π , we can, within thelimits of our simplified model, deduce π ( t ) for all t if we know the history of β togetherwith that of σ . By working out this dependence explicitly, we will arrive at the generalsolution for π ( t ) recorded in equation (16). In consequence of this equation, we will alsosee that the initial conditions tend to drop out in a “mature” economy, leading to theresult that the profit-rate at a given time is the product of two averages which summarizethe historically recent values of β and σ , respectively. We will also derive an upper boundon π that is entirely independent of initial conditions, and is relatively independent of σ as well.Our task, then, is to solve (5), presupposing that both β and σ are known functionsof time. As it stands, (5) contains a term quadratic in π , but we can render it linear byworking with 1 /π rather than π itself. In fact the new variable Z defined by Z = σ/π (7)8ill be a slightly more helpful choice. From equation (3) we see that Z = K/L measuresthe fixed capital per worker hour. It is thus closely related to what is sometimes called the“organic composition of capital”. Rewritten in terms of Z equation (5) becomes simply dZdt + β Z = σ (8)Thus β acts like a “restoring force” that (when positive) pulls Z toward 0, while σ actslike a “source” that pushes Z toward higher values. (More exploitation implies more rapidaccumulation of surplus value.) At any rate, we have in (8) a linear equation whose generalsolution is Z ( t ) = Z e − B ( t, + t Z e − B ( t,s ) σ ( s ) ds (9)where B ( t , t ) = t Z t β ( t ) dt (10)and Z = Z ( t = 0) is the initial value of Z . (One can verify this solution by directsubstitution into (8).) From Z ( t ) we can recover π ( t ) trivially, but first let us interpret theexpression B ( t , t ) and the integrals in which it occurs.To that end, notice first that since e − B is always positive the integral in (9) constitutesa weighted sum of σ with recent values counting most heavily (unless the economy isshrinking!) and very early values being exponentially damped. Appropriately normalizedthis integral therefore defines a certain kind of “moving average” of σ (or of any givenfunction of time) which I’ll denote by h · i : h σ ( t ) i = t Z e − B ( t,s ) σ ( s ) ds (cid:30) t Z e − B ( t,s ) ds (11)The denominator of (11) can also (though less obviously) be interpreted as an average,this time of β itself. To that end let us define ¯ β = ¯ β ( t ) through the equation1 − e − t ¯ β ( t ) ¯ β ( t ) = t Z e − B ( t,s ) ds . (12)9hen ¯ β is a kind of “self-weighted moving average” of β in the following sense (see Ap-pendix):(1) For each t , equation (12) defines ¯ β ( t ) uniquely.(2) If β is the constant function β then ¯ β = β .(3) If (for all t ) β ≤ β then ¯ β ≤ ¯ β ; in particular any (lower or upper) bound for β isalso one for ¯ β .Moreover this second average is closely related to the first one, since, as proven in theappendix, ¯ β − e − t ¯ β = h β i − e − B ( t, (13)In order to bring out further the intuitive meaning of the weighting factor e − B whichdefines the average h · i , let us observe that according to (10) and (4), B ( t , t ) is the sumof the integrated growth-rates of L and of productivity. If we temporarily introduce asymbol Π to represent productivity and define Q = L Π then β = ˙ L/L + ˙Π / Π is the rateof change of log Q and its integral B is just the logarithm of Q ( t ) /Q ( t ). Thus e B ( t,s ) = Q ( t ) /Q ( s ) (14)is a kind of measure of how much the productive capacity of the economy has grown between t and t , and the effect of our weighting factor e − B ( t,s ) in the above integrals is to discountthe contribution from a given historical moment in proportion as the productive capacitythen was smaller. In a growing economy, such an average damps out the contribution fromearly times.Thanks to these observations, we can also express the average (11) as h σ ( t ) i = t Z Q ( s ) σ ( s ) ds (cid:30) t Z Q ( s ) ds (11 a )10hile the corresponding recasting of ¯ β appears as1 − e − t ¯ β ( t ) ¯ β ( t ) = t Z Q ( s ) ds (cid:14) Q ( t ) . (12 a )With these definitions in mind, we can assemble equations (9), (14), (11a), and (12a)to obtain [where Z = Z ( t ), Q = Q ( t ), ¯ β = ¯ β ( t )] Z = Z Q Q + h σ i − e − ¯ βt ¯ β which with the aid of the identity (13) becomes Z = Z Q Q + h σ i − Q /Q h β i or after rearrangement, Z = h σ ih β i + Q Q (cid:20) Z − h σ ih β i (cid:21) (15)This is our main result expressed in terms of Z = σ/π . In a sufficiently mature economy(one which has already expanded by a large factor) Q /Q will be small compared to unityand we will be left with the approximate equality, Z = h σ i / h β i .Finally, we can substitute the definition (7) of Z to obtain an equation for the profit-rate at time t which, after a small rearrangement, furnishes our main equation for π : π = h β i σ/ h σ i Q Q h h β i π σ h σ i − i (16)Once again the term containing Q /Q can be dropped in a mature economy and we areleft in that case with the approximate equality π = h β i σ h σ i (17)Even without making any assumption about maturity, we deduce from (16) the inequality π ≤ h β i σ h σ i (1 − Q /Q ) − (18)11hich bounds the profit-rate above by an expression not much bigger than (17), and boundsit even more precisely by that value times (1 + Q /Q ), a factor close to unity which shouldprovide an adequate approximation in any but a very young economy.What the last three equations tell is that, apart from initial transients and temporaryfluctuations, the profit-rate coincides with the growth-rate β averaged over historicallyrecent times. They also tell us that the fluctuations about this average come from thefactor σ/ h σ i which will be appreciable only when σ (the rate of surplus value) rises sharplyabove, or falls sharply below, its historically recent average h σ i . And since σ is in anycase less than 1, a sharp increase is possible only if it was recently much less than unity. V. Comments and Extensions
Under conditions of equilibrium, one would expect the profit-rate to reflect directly theunderlying growth-rate, because the latter characterizes the entire process, and one wouldexpect to recognize this same rate no matter what variable one chose to monitor. It istherefore no surprise that equilibrium models (“neo-Ricardian” and others) produce theresult π = β . In disequilibrium conditions, however, some variables will be increasingwhile others decrease, fluctuations will be significant, if not dominant, and the reasoningbehind the equilibrium results will fail. It is thus interesting that, at least in the simplemodel of this paper, the profit-rate is still governed to a large extent by the growth rates, r and ˙ L/L ( β being their sum). The difference however is that the contemporaneous valueof π is no longer tied to the contemporaneous value of β , but to its historical average h β i taken in a precisely defined sense. Moreover, significant short-term fluctuations in π will in general take place, depending on the contemporaneous rate of surplus value σ inrelation to its historical average h σ i . In a relatively young economy the initial conditionswill be important as well. (One can of course set the initial time whenever one wants. Theequations will still hold.) 12 n illustration By way of illustrating our conclusions, let us imagine a closed economy whose initial time, t = 0, is chosen to be around the end of the last “world war”, so that at present t = 70 yr ,and suppose that its “productive capacity” Q has expanded since then by a factor often: Q/Q = 10. The historical averaging that enters into our equations would thenweight recent values of β and σ about 10 times more heavily than values from 70 yearsago. The latter could thus still make some difference, but not too much. Let us supposefurther that β recently has hovered around 2% per year, having been bigger previouslysuch that currently h β i = 0 . /yr , and that the average rate of exploitation has reached h σ i = 0 .
50. Let us also suppose that in the past couple of years the capitalists, in adesperate attempt to raise profits, have driven σ to its maximum possible value of unity: σ = 1 (which of course they could never really attain). The inequality (18) would thenallow at present a current labour-value profit-rate of at most0 . /yr × . . × − . . /yr In other words, the annual profit rate could not currently exceed 5.6% even if the workerswere made to “live on air”.Even this excess over h β i would only be temporary. In fact one can estimate ingeneral that π ( t ) can not exceed β ( t ) by an amount ε for a time much longer than 1 /ε , orsomewhat more precisely, this time multiplied by ln(1 /σ ) + ln(1 / ( β ′ + ε )) , β ′ being thevalue of β at the end of the period. Further remarks on equilibrium models
I have already expressed more than once herein, the view that equilibrium models arenot to be trusted. Perhaps the most devastating evidence for this is the observation that13quilibrium is simply not the historical rule, but one can also give more “technical” reasonswhy a hypothetical equilibrium profit rate is likely to be wrong. For this rate is essentiallythe maximum possible rate, a statement that is in some sense the content of the Frobenius-Perron theorem, which states in particular that for any set of prices there will always be atleast one industry in which profitability is at or below its equilibrium value. In itself this is afairly weak limitation, but presumably one could show that any growth trajectory far fromthe equilibrium one would soon encounter severe shortages and hence severe “realization”problems, resulting in markedly lower profits. A fuller analysis of this situation mightalso be able to elucidate the loss of profit inherent in any return to equilibrium afternew techniques or products have been introduced. In any case, if the equilibrium rateis an upper bound, it follows that slumps can lower profits below this hypothetical valuemore than booms can raise them. Thus, even if prices and product-mixes averaged outto their equilibrium values, the (nonlinear) relation between them and the (overall) profitrate would in general cause the latter to deviate systematically from its own equilibriumvalue. Similarly, any nonlinearity in the response of investors to fluctuating prices andmarkets would invalidate arguments of the Okishio-Theorem type, that employ a notionof capitalist rationality defined solely with respect to the current equilibrium price-vector(as in [4] for example).
Possible extensions
In deriving our main results, equations (5) and (16), we have relied on several assumptionsand idealizations. Some of them could easily be relaxed. For example one could accommo-date a situation where not all surplus value resulted in capital formation by using equation(1a) in place of (1), with φ chosen to take account of dividends and personal consumptionby capitalists. 14he assumption that capital is not destroyed could also be relaxed by incorporatinginto the variable r an additive contribution representing the rate of its destruction. Suchan r could no longer be interpreted simply as a growth-rate, but mathematically theabove analysis would go through as before. Equations like (11a) and (12a) would becomeincorrect, but re-expressed in terms of B ( t, s ) via (14), all our final results would remainthe same.Some of our other idealizations would be more difficult to do without, notably theassumption that the economy in question is a closed system. One might also wonderwhether (3) was a good approximation in a more service-based economy. Similarly, onemight question whether, or how, “fictitious” financial capital ought to be included in thevariable K , or how to take into account labour which in some sense is “unproductive”. Possible tests
Beyond these more “technical” questions are the conceptual questions raised by the use oflabour-values. In this paper, we are simply adopting labour-values as our starting pointand drawing out the consequences mathematically. The advantages of doing so are firstof all that we avoid the equilibrium fiction (if I can call it that), and secondly that onecan give at least a fairly clear definition of the crucial parameter r measuring productivitygrowth. One also has the feeling that labour values tap into something basic about theeconomy which empirical prices do not. ⋆ For this reason π might also have a certainanalytical interest in its own right. (A good question is why economists worry so muchabout “productivity” if labour values are irrelevant.) ⋆ The observation seems relevant here that labour-power is the one commodity which(ignoring slave markets and analogous exceptions) is not produced for sale by capitalists.15n this connection, one might wonder whether some other “universal” measure of valuecould have been used in place of labour-time. Indeed, the fact that β represents just thegrowth rate of productive capacity (more precisely of the hypothetical capacity for turningout capital goods if all available labour were devoted to them) suggests that some formof the equations relating π with β might be provable without recourse to any theory ofvalue at all. However any attempt to define β in this way (not to mention σ ) encountersthe ambiguity inherent in defining growth rates in the face of the continual emergenceof new commodities and techniques and the disappearance of old ones which economicgrowth entails. The present treatment, based on assigning to each commodity its labour-value, renders the crucial variable β unambiguous to a significant extent. Nevertheless, itseems clear that of the two factors entering into its definition, namely hours worked andproductivity, the latter is less well defined than the former.From an empirical standpoint the question would be whether, and to what extent, thephenomenal profit rate can be identified with the labour-value profit-rate studied in thispaper. Some economists seem to think that it can, many others think not. At this stageof economic theory, I believe one must treat either answer simply as a hypothesis that onecan adopt (or not) as the basis of further analysis, just as one does with hypotheses in thephysical sciences. Personally, I have very little idea how to address this issue theoretically,but our results above provide the beginnings of a way to address it empirically. (Beginningsbecause some of the simplifications we have made would need to be either corrected foror corroborated.) To the extent that r , π and σ can be observed ( L unquestionably can),one could compute (from (16) or (5) together with β and σ ) π as a function of time andcompare the resulting graph with the data. Or if this test were too difficult because σ ( t )was too hard to observe reliably given available statistics, one could as a second-best test,deduce σ ( t ) vice versa from (5) together with π and β , and see whether the resulting graphwas at least intuitively plausible. 16 ppendix. Some technical details In terms of the function F ( x ) ≡ − e − x x = Z e − xs ds , ( A β reads F ( ¯ β ) = t Z e − B ( t,s ) ds . ( A β is thereby well-defined and also to establish the further assertionsmade in the text.In the first place notice that, because the integrand in (A1) is a monotone decreasingfunction of x , F is also monotone decreasing. Moreover it is clear that F ( −∞ ) = + ∞ , F (+ ∞ ) = 0, so that F is a one-to-one map of the reals onto the positive reals. Hence (A2)has a unique solution ¯ β ; which was assertion (1) of the text.Now if β = β is constant then from (10), B ( t, s ) = ( t − s ) β , whence the integral in(A2) is t Z e − β ( t − s ) ds = 1 − e − β t β . Then the unique solution of (A2) is obviously ¯ β = β , which was assertion (2).Given this, the second part of assertion (3) follows from the first. To prove the firstsimply notice that if β increases pointwise then R e − B ( t,s ) ds decreases, so that ¯ β in (A2)must increase because F is monotone decreasing.Finally let us demonstrate the identity (13). To that end, let us first write out h β ( t ) i in the form (which follows immediately from (11) and (12)), h β i = ¯ β − e − t ¯ β t Z e − B ( t,s ) β ( s ) ds (19)17ext observe that ∂/∂s e − B ( t,s ) = β ( s ) e − B ( t,s ) , and substitute this into (19) to obtain h β i = ¯ β − e − t ¯ β t Z ds ∂∂s e − B ( t,s ) = ¯ β − e − t ¯ β (cid:0) − e − B ( t, (cid:1) , from which (13) follows immediately.This research was supported in part by NSERC through grant RGPIN-418709-2012. Thisresearch was supported in part by Perimeter Institute for Theoretical Physics. Research atPerimeter Institute is supported by the Government of Canada through Industry Canadaand by the Province of Ontario through the Ministry of Economic Development and Inno-vation. References [1] Piero Sraffa,
Production of Commodities by Means of Commodities: Prelude to a Cri-tique of Economic Theory (Cambridge University Press, 1960)[2] J. von Neumann, “ ¨Uber ein ¨okonomisches Gleichungssystem und ein Verallgemeinerungdes Brouwerschen Fixpunktsatzes”,
Ergebnisse eines Mathematischen Kolloquiums
Review of EconomicStudies
13 :
Das Kapital, Band 3 , 3. Teil: Gesetz des tendenziellenFalls der Profitrate,retrieved from http://ciml.250x.com/archive/marx engels/german/kapital3.pdf; trans-lated in K. Marx (ed. F. Engels),
Capital vol III (N.Y. International Publishers, 1967)Part III.[4] J.E. Roemer, “Continuing Controversy on the Falling Rate of Profit: Fixed CapitalAnd Other Issues”,
Cambridge J. Econ.3 :