The Falling Rate of Profit under Constant Rate of Exploitation: A Generalization

Abstract

In a two-sector model with circulating capital, Laibman (1982) shows that a capital-using and labor-saving technical change in the consumption goods sector lowers the rate of profit under the assumption of constant rate of exploitation. This paper generalizes his finding in a two-department multi-sector model that considers the capital advanced.

JEL Classification: B51, C67

Keywords

falling rate of profit technical change constant rate of exploitation

1. Introduction

Marx’s law of the tendency of the rate of profit to fall states that a rising organic composition of capital leads to a decrease in the general rate of profit when the rate of exploitation is constant, or even rising (Marx 1998: 210–11). Marx also discusses the counter-tendencies that could check the decline in the rate of profit, including “the cheapening of elements of constant capital” that is due to “the same development which increases the mass of the constant capital in relation to the variable” (234, my emphasis). Marx’s demonstration of the law by way of numerical examples hardly proves it, and leads to different interpretations of its meaning and validity. In contrast, the Okishio (1961) Theorem states that a profitable (cost-reducing) technical change increases the general rate of profit if the real wage rate is constant. Okisho’s assumption of constant real wage rate is related to but arguably different from Marx’s underlying assumptions. Laibman (1982), in a heuristic two-sector model with circulating capital, shows that under the assumption of constant rate of exploitation, a profitable, capital-using, and labor-saving technical change in the consumption goods sector necessarily lowers the general rate of profit; such a technical change in the capital good sector is also compatible with (though does not necessitate) a fall in the rate of profit. Laibman’s result can serve as a theoretical reference point that is different from Okishio’s in a potentially more general framework where wage and exploitation are allowed to be endogenously determined by technical change.¹ Is Laibman’s finding robust to generalization? This paper seeks to answer this question. In a two-department multi-sector input-output model that takes into account the stock-flow relation of the capital advanced, a profitable, capital-using, and labor-saving technical change in the consumption goods department lowers the general rate of profit under the assumption of constant rate of exploitation; as a corollary, even if the rate of exploitation rises, but stays within a certain threshold, the rate of profit also falls.

2. The Debate on Marx’s Law

The debate on Marx’s law can be broadly divided into two stages, with the Okishio Theorem being the watershed.² The first stage centers around whether Marx’s law is well-formulated, typically based on the following formula that concerns the aggregate value magnitudes when only circulating capital is considered:

\begin{matrix} r \equiv \frac{S}{C + V} \equiv \frac{S / V}{C / V + 1} \equiv \frac{ϵ}{C / V + 1} \end{matrix}

(1)

where C, V, and S are respectively constant capital, variable capital, and surplus value; r and $ϵ$ are the general rate of profit and the rate of exploitation. The organic composition of capital is often taken as C / V and assumed to be rising. Marx’s law is then interpreted as follows: holding $ϵ$ constant, a rising C / V lowers r (interpretation 1); or, even if $ϵ$ is rising, a rising C / V lowers r (interpretation 2, for example, see Meek 1967 and Heinrich 2013). These interpretations give rise to four serious doubts. The first concerns the tautology in interpretation 1: A rising C / V lowers r by definition, as long as $ϵ$ is constant; it “needs no ghost come from the grave to tell us this” (Robinson 1966: 42). Secondly, it is unclear why Marx designates “the cheapening of elements of constant capital” as one of the counter-tendencies, now that the change in C / V already incorporates that counteracting factor (Sweezy 1942: 104). The third and more fundamental doubt concerns the rationality of C / V to rise with the process of mechanization, since the devaluation of constant capital caused by the same technical change process tends to lower this ratio and the net effect is ambiguous. The fourth doubt concerns the movement of $ϵ$ in the context of relative surplus value production: With increasing mechanization, the value of workers’ consumption bundle tends to decrease, and so does the value of labor power, which raises $ϵ$ even if the working day or labor intensity stays unchanged (Sweezy 1942: 100–2). Marx does try to incorporate a rising $ϵ$ in his law but fails to arrive at any determinate answer according to Heinrich (2013). It is here where the functional aspect of Marx’s law is brought into picture, i.e., $ϵ$ is not assumed constant and is treated as endogenous to technical change, among other things. Proponents of interpretation 2 usually argue that because the rise in $ϵ$ has “certain insurmountable limits,” the rise of C / V must dominate the movement of r such that r tends to decline (Marx 1998: 246). This argument is not convincing in that the unit value of workers’ consumption goods, and thus the value of labor power can approach zero with the unbounded increase of labor productivity. As a result, $ϵ$ can approach infinity (Robinson 1966: 39–40).

The aforementioned doubts are partly resolved if we consider Marx’s distinction of the three concepts of composition of capital, i.e., the technical (TCC), value (VCC), and organic (OCC) composition of capital (Fine and Harris 1976). In this case, equation (1) could be re-formulated as follows Shaikh (1990).

Let $\tilde{C}$ and $\tilde{V}$ be the constant and variable capital measured with their base-year unit values, respectively. Let $I^{c}$ , $I^{b}$ , and $I^{w}$ be the indexes of the unit value of constant capital, the unit value of a consumption bundle, and the real wage rate respectively, all relative to the base year. Then $C = \tilde{C} I^{c}$ , $V = \tilde{V} I^{b} I^{w}$ , and equation (1) can be rewritten as:

\begin{matrix} r = \frac{ϵ}{1 + \frac{\tilde{C}}{\tilde{V}} \cdot \frac{I^{c}}{I^{b} \cdot I^{w}}} \end{matrix}

(2)

Henceforth, the VCC is understood as C / V, and the OCC as $\tilde{C} / \tilde{V}$ , while the TCC cannot be defined as a ratio due to the non-additive material units of different constant capital inputs. This formulation is consistent with Marx’s (1998) definition: “The value composition of capital, inasmuch as it is determined by, and reflects, its technical composition, is called the organic composition of capital” (145). More specifically: “The variable capital thus serves here (as is always the case when the wage is given) as an index of the amount of labor set in motion by a definite total capital” (144, my emphasis). Regarding constant capital, Marx states: “On the other hand, a difference in the magnitude of the constant capital may likewise be an index of a change in the mass of means of production set in motion by a definite quantity of labor-power” (145, my emphases). In addition, “the material growth of the constant capital implies also a growth—albeit not in the same proportion—in its value, and consequently in that of the total capital” (209–10). These quotes seem to suggest that what Marx has in mind when defining the OCC is the constant and variable capital measured with their constant unit values, and that Marx’s primary assumption is a rise in the TCC, with the OCC being its practical measure and the change in the VCC being its effect.³

With this new formulation of the problem, Robinson’s tautology critique becomes irrelevant, and the true problem lies in this: It is not clear as to how the OCC affects the VCC when the net effect of the three indexes remains unspecified. Besides, it is logical to assign the role of counter-tendency to the cheapening of constant capital ( $I^{c}$ ) once it is isolated from the VCC and considered as a secondary factor—its effect on the rate of profit is not conflated with that of the OCC. Therefore, the entire debate boils down to the question whether mechanization necessarily leads to a rising C / V, and, if so, whether the supposed increase in $ϵ$ is strong enough to offset the negative effect on the rate of profit (Mage 1963: 140). This question cannot be adequately answered by arguments or numerical examples, and “cries out for mathematical treatment” (Meek 1967: 141) because $ϵ$ and the three indexes in equation (2) are endogenous to the OCC and the net effect is ambiguous.

Okishio (1961) presents such a mathematical treatment in a multi-sector input-output model by imposing a special restriction on the rate of exploitation—a constant real wage rate. He also allows for endogenous change in relative price. His stark conclusion that the rate of profit rises after any profitable technical change brings the debate into a new stage. The Okishio Theorem has proved to be robust in various theoretical settings, including the case of fixed capital (Roemer 1979), joint production (Bidard 1988), product innovation (Nakatani and Hagiwara 1997; Fujimori 1998), and historical cost accounting (Laibman 2001).⁴ Nonetheless, the assumption of constant real wage rate is too restrictive (Okishio 2000), and the Okishio Theorem does not rule out the possibility that a falling rate of profit is compatible with a constant or even rising rate of exploitation when the real wage rate is allowed to vary. Laibman (1982) demonstrates exactly this. Moreover, the question on how a change in $\tilde{C} / \tilde{V}$ affects $ϵ$ cannot be adequately answered by only looking at technical factors, because the relationship between the two hinges not only on the devaluation of commodities (I^c and I^b ), but also on the change in the real wage rate (I^w ) that is subject to class struggle—how that plays out is supposedly historically contingent and can only be answered with the assistance of empirical study (Roemer 1981: 145).

Therefore, I treat the Okishio Theorem and Marx’s law as two propositions that are not contradictory to one another, but rather based on different, though related, assumptions. I follow Laibman’s (1982) suggestion to examine the relationship between technical change and profitability in the context of what he calls class struggle neutrality: “It is the rate of exploitation—the ratio of unpaid to paid labor time—which expresses the balance of class forces at a given time. It is when wages keep in step with productivity that the balance of class forces does not change; this then, is the neutral framework in which to analyze technical change” (103). The assumption of constant rate of exploitation can be a useful reference case, though not necessarily true in a systematic manner.⁵ It is also compatible with interpretation 2 (see section 1) and the general law of accumulation in chapter XXV of volume 1 of Capital (Marx 1996) which stresses the important role of the rising OCC in increasing the rate of exploitation. The reason is: if one can show a falling rate of profit under a fixed rate of exploitation, there certainly exists the case where the rate of exploitation rises (but not too much) and the rate of profit falls simultaneously (see section 4 for a formal proof).

It is worth noting that the hypothesis examined by Laibman (1982) and thus this paper differs from Marx’s formulation of his law in at least two respects. First, following Okishio (1961), we treat the effect of the cheapening of elements of constant capital and workers’ consumption goods as mixed with that of the OCC insofar as the former is induced by the change in the latter. The second difference has to do with the transformation problem: Marx formulates his law in value terms but the average value rate of profit generally deviates from the price rate of profit (Morishima 1973). To strictly follow Marx’s formulation, one has to decide how output proportions change with technical change since they are needed to calculate the average value rate of profit; I instead go with the price rate of profit which is more tangible to capitalists. Nonetheless, I maintain the definition of the rate of exploitation in value terms as Laibman does. This treatment is at least consistent with Marx’s (1998) attempt to derive the price rate of profit from a given rate of exploitation—the ratio between unpaid and paid labor time—that reflects social relations, regarding labor as the fundamental substance of value and the price system a phenomenal structure to be derived from the value system (Roemer 1981: 153–4).⁶

3. The Input-Output Model

Consider a closed capitalist economy where there are m different sectors, each producing a single type of goods. For i, j = 1, …, m, denote x as the m × 1 output vector whose generic element x_i $x_{i}$ is the output level of good i; K as the m × m constant capital stock matrix where K_ij is the amount of constant capital good i tied up in production, either as fixed capital or inventories of circulating constant capital, to produce x_j $x_{j}$ units of good j; D as the m × m turnover matrix of capital stock where D_ij ≥ 1 is the number of rounds of production that could be sustained by the constant capital stock of good i used in the production of good j; A as the m × m constant capital flow matrix where A_ij is the amount of good i consumed (as depreciation of fixed capital or depletion of inventories of circulating constant capital) in producing x_j units of good j (note that K_ij = D_ij A_ij $K_{i j} = D_{i j} A_{i j}$ by definition); l as the 1 × m productive labor power vector (flow); b as the m × 1 reference consumption basket where b_i denotes the amount of good i that appears in this basket; $ω$ as the real wage rate (a scalar); p as the 1 × m price of production vector; $v$ as the 1 × m labor value vector; $ϵ$ as the uniform rate of exploitation; π as the uniform price rate of profit.

Assume all wages are paid after production;⁷ when the real wage ( $ω b$ ) is changing, only the size ( $ω$ ) of workers consumption bundle is changing while the proportions (b) among different consumption goods remain the same. Assume production is continuous, i.e., a production cycle restarts when it ends, and assume the constant capital stock is replenished to its previous level at the end of the production period.⁸ Assume constant returns to scale in production and normalize output to unity, i.e., x_i = 1, such that all the input variables, stock or flow, is reinterpreted as per unit of output. Throughout this paper, I limit my analysis to the case $π > 0$ .

A production technique is then characterized by the tuple ${K, A, l}$ , and the value system is:

\begin{matrix} v = v A + l \end{matrix}

(3)

The value vector can be solved as $v = l {(I - A)}^{- 1}$ with some standard assumptions imposed on A. Then, the uniform rate of exploitation per unit of labor power is given by:

\begin{matrix} ϵ = \frac{1 - v ω b}{v ω b} = \frac{1}{v ω b} - 1 \end{matrix}

(4)

where $v ω b$ is the value of one unit of labor power. The rate of profit of, say industry j, is $π = (p_{j} - p A_{j} - p ω b l_{j}) / (p K_{j})$ , where subscript j of matrices A and K denotes their j th columns. Based on this we can form the price system in matrix terms as:

\begin{matrix} {\begin{array}{l} p = π p K + p A + p ω b l \\ p b = 1 \end{array} \end{matrix}

(5)

in which the second equation is a normalization condition imposed on p. With some other standard mathematical assumptions, it can be proved that there exists a unique and positive solution for π and p in the above price system (Brody 1970: 42). Furthermore, π is decreasing in $ω$ by the Perron-Frobenius Theorem.⁹

Next, I define two important types of technical change. A technical change in sector j, $(K_{j}, A_{j}, l_{j}) \mapsto (K_{j}^{'}, A_{j}^{'}, l_{j}^{'})$ , is said to be profitable when it creates room for extra profit under the old price and wage rate, i.e., $p_{j} = (π + Δ π) p K_{j}^{'} + p A_{j}^{'} + p ω b l_{j}^{'}$ where $Δ π > 0$ , or considering technical change in multiple industries at the same time:

\begin{matrix} p \geq π p K^{'} + p A^{'} + p ω b l^{'} \end{matrix}

(6)

in which an industry undergoes a profitable technical change when the inequality sign is strict for that industry, and does not when it is binding. Assume the turnover matrix D to be constant,¹⁰ then a technical change is said to be capital-using and labor saving (CU-LS) when:

\begin{matrix} A^{'} \geq A a n d l^{'} \leq l \end{matrix}

(7)

The elements of capital stock K change proportionally with the elements of A because K_ij = A_ijD_ij and D_ij remains constant. The CU-LS type of technical change captures the essence of rising OCC or mechanization, that is, the substitution of labor power by constant capital, although it does not allow for substitution of one type of constant capital by another. In what follows, I only consider profitable CU-LS technical change.

4. Generalization of Laibman (1982)

As stated previously, my research question is how a profitable CU-LS technical change affects the rate of profit if the rate of exploitation is constant. My main strategy to answer this question is to devise two threshold conditions regarding the growth of real wage such that the constancy of the rate of profit and that of the rate of exploitation are maintained, respectively. Comparing these two thresholds enable us to use the negative relationship between the real wage rate and the rate of profit to see whether the rate of profit rises or falls when the rate of exploitation is constant.

Threshold Condition 1: For a profitable technical change ${K, A, l} \mapsto {K^{'}, A^{'}, l^{'}}$ , the equilibrium rate of profit π falls, rises, or remains the same depending on whether the real wage rate $ω$ grows at a rate greater than, smaller than, or equal to the threshold growth rate $\hat{ω^{o}}$ as derived below.¹¹

From the price system (5), the price vector can be explicitly solved as:

\begin{matrix} p = ω l {(I - π K - A)}^{- 1} \end{matrix}

(8)

Let $ω^{o}$ be the threshold real wage rate such that the equilibrium rate of profit remains at its previous level π after the profitable technical change; let $p^{o}$ be the associated price vector. Then, we have a price system formed by $p^{o} = π p^{o} K' + p^{o} A' + p^{o} ω^{o} b l'$ and $p^{o} b = 1$ , which results in:

\begin{matrix} p^{o} = ω^{o} l^{'} {(I - π K^{'} - A^{'})}^{- 1} \end{matrix}

(9)

Apply $p b = 1$ and $p^{o} b = 1$ to equation (8) and (9); after some rearrangement, we have:

\begin{matrix} \hat{ω^{o}} \equiv \frac{ω^{o} - ω}{ω} = \frac{ω l {(I - π K - A)}^{- 1} b - ω l^{'} {(I - π K^{'} - A^{'})}^{- 1} b}{ω l^{'} {(I - π K^{'} - A^{'})}^{- 1} b} \end{matrix}

(10)

By the Okishio Theorem we know $ω^{o} > ω$ and thus $\hat{ω^{o}} > 0$ . Further because of the inverse relationship between the real wage rate and the rate of profit, the rate of profit falls, rises, or remains unchanged if the real wage rate grows at a rate greater than, smaller than, or equal to $\hat{ω^{o}}$ , by construction.¹²

Similarly, let the real wage rate be $ω^{*}$ in the new equilibrium such that the rate of exploitation $ϵ$ remains the same after a profitable technical change, and the corresponding rate of profit be π*. We have the second threshold condition:

Threshold Condition 2: For a profitable technical change ${K, A, l} \mapsto {K^{'}, A^{'}, l^{'}}$ , the rate of exploitation falls, rises, or remains the same depending on whether the real wage rate grows at a rate greater than, smaller than, or equal to the rate of reduction in the value of the reference consumption basket.

We know from equation (4) that the rate of exploitation is determined by the value of labor power $v ω b$ , and the new value vector is $v^{'} = l^{'} {(I - A^{'})}^{- 1}$ according to $v^{'} = v^{'} A^{'} + l^{'}$ . For the rate of exploitation to be constant, the value of labor power must remain the same after the profitable technical change, i.e., $v ω b = v^{'} ω^{*} b$ , and then:

\begin{matrix} \hat{ω^{*}} \equiv \frac{ω^{*} - ω}{ω} = \frac{l {(I - A)}^{- 1} b - l^{'} {(I - A^{'})}^{- 1} b}{l^{'} {(I - A^{'})}^{- 1} b} \end{matrix}

(11)

This growth rate is obviously equal to the rate of reduction in the value of the reference consumption basket. Since the rate of exploitation is decreasing in the real wage rate by equation (4), it rises or falls depending on whether the real wage rate grows slower or faster than $\hat{ω^{*}}$ , which is positive when the technical change is also CU-LS, because in this case $v^{'} \leq v$ .¹³

After a profitable CU-LS technical change, does the new equilibrium rate of profit fall when the rate of exploitation is fixed? This question essentially boils down to whether $ω^{*} > ω^{o}$ holds, for if it does, the real wage rate has to grow faster than the threshold whereby the constancy of rate of profit is maintained. Hence, the rate of profit falls because of the inverse relationship between the real wage rate and the rate of profit. However, the answer is indeterminate at our current level of abstraction (Laibman 1982; Bidard 2004: 73–6).

At a more concrete level where the economy consists of one capital goods sector and one consumption goods sector, Laibman (1982) shows that a profitable CU-LS technical change in the consumption goods sector necessarily decreases the rate of profit. His result can be generalized in a two-department multi-sector framework as follows.

Let sectors 1 through n be the capital goods sectors, or Department I, and the rest, sectors n + 1 through m, the consumption goods sectors, or Department II. Accordingly, a_ij = 0 for $i = n + 1, \dots, m$ and $j = 1, \dots, m$ , and $b_{i} = 0$ for $i = 1, \dots, n$ . In matrix terms:

K = (\begin{matrix} K_{I} & K_{I I} \\ 0 & 0 \end{matrix}), A = (\begin{matrix} A_{I} & A_{I I} \\ 0 & 0 \end{matrix}), l = (l_{I}, l_{I I}), b = (\begin{matrix} 0 \\ b_{I I} \end{matrix})

with appropriate dimensions. The two-department price and value systems are:

\begin{matrix} {\begin{array}{l} p_{I} = π p_{I} K_{I} + p_{I} A_{I} + p_{I I} ω b_{I I} l_{I} \\ p_{I I} = π p_{I} K_{I I} + p_{I} A_{I I} + p_{I I} ω b_{I I} l_{I I} \\ p_{I I} b_{I I} = 1 \end{array}, {\begin{array}{l} v_{I} = v_{I} A_{I} + l_{I} \\ v_{I I} = v_{I} A_{I I} + l_{I I} \end{array} \end{matrix}

(12)

It can be proved that, if Department II undergoes a profitable CU-LS technical change, the two thresholds as in equation (10) and (11) are concretized as follows:¹⁴

\begin{matrix} \hat{ω^{o}} = \frac{[ω (l_{I I} - l_{I I}^{'}) - p_{I} (A_{I I}^{'} - A_{I I}) - π p_{I} (K_{I I}^{'} - K_{I I})] b_{I I}}{[π p_{I} K_{I I}^{'} + p_{I} A_{I I}^{'} + ω l_{I I}^{'}] b_{I I}} \end{matrix}

(13)

\begin{matrix} \hat{ω^{*}} = \frac{[l_{I I} - l_{I I}^{'} - v_{I} (A_{I I}^{'} - A_{I I})] b_{I I}}{(v_{I} A_{I I}^{'} + l_{I I}^{'}) b_{I I}} \end{matrix}

(14)

Because $D_{i j} \geq 1$ and is constant, $K_{i j}^{'} - K_{i j} = D_{i j} (A_{i j}^{'} - A_{i j}) \geq A_{i j}^{'} - A_{i j}$ , or $K_{I I}^{'} - K_{I I} \geq A_{I I}^{'} - A_{I I}$ . Then, we have $p_{I} (A_{I I}^{'} - A_{I I}) + π p_{I} (K_{I I}^{'} - K_{I I}) \geq p_{I} (A_{I I}^{'} - A_{I I})$ . Further because $p_{I} > ω v_{I}$ (see footnote 13):

\begin{matrix} \hat{ω^{o}} < \frac{[ω (l_{I I} - l_{I I}^{'}) - p_{I} (A_{I I}^{'} - A_{I I})] b_{I I}}{(p_{I} A_{I I}^{'} + ω l_{I I}^{'}) b_{I I}} < \frac{[l_{I I} - l_{I I}^{'} - v_{I} (A_{I I}^{'} - A_{I I})] b_{I I}}{(v_{I} A_{I I}^{'} + l_{I I}^{'}) b_{I I}} = \hat{ω^{*}} \end{matrix}

(15)

which implies $π^{*} < π$ as argued previously.

It is obvious that for any real wage growth rate $\hat{ω}$ such that $\hat{ω^{o}} < \hat{ω} < \hat{ω^{*}}$ , the rate of exploitation rises and the rate of profit falls at the same time. If $\hat{ω} \leq \hat{ω^{o}}$ , i.e., the exploitation rises too much, a falling rate of profit is impossible.

Laibman (1982) also demonstrates that if the above technical change takes place in Department I, the movement of the rate of profit is indeterminate.

5. Concluding Remarks

Laibman’s (1982) finding and its generalized version developed in this paper can be considered as a class-struggle-neutral benchmark for the debate on Marx’s law of falling rate of profit when exploitation is treated as endogenous to technical change. On the other hand, the Okishio Theorem can be treated as a real-wage-neutral benchmark for the same problem. Both cases are theoretical references—they tell nothing about the actual relationship between technical change and distribution, because there is no reason for the rate of exploitation or the real wage rate to stay constant systematically. However, they are potentially useful for studying the effect of technical change on profitability in a more general framework where the functional relationship between technical change and the real wage is considered. This issue is left for future study.

Footnotes

Acknowledgements

I would like to thank Deepankar Basu, Al Campbell, David Laibman, Peter Skott, Naoki Yoshihara, and the three reviewers for their helpful comments on earlier versions of this paper. The current version is substantially shortened from a previous one according to the reviewers’ suggestion. All errors are mine.

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Junshang Liang

1

It is beyond the scope of this paper to develop such a framework.

2

It is beyond the scope of this paper to provide a comprehensive review of the debate.

3

See also : 607–8) for a more detailed version of the tripartite distinction.

4

Shaikh (1978, ) challenges the empirical relevance of Okishio’s profitable criterion for the choice of technique. He argues that capitalists choose new production techniques to raise the profit margin rather than the temporary rate of profit, so that a new technique that increases the profit margin might lower the temporary rate of profit at the same time. Shaikh’s argument can be evaluated only by empirical investigation.

5

I thank the three reviewers for emphasizing this point; one of them also points out that a constant wage share, an alternative that also captures the unchanged balance of class forces, can be a market equilibrium outcome in a sequential bargaining model ().

6

As is pointed out by a reviewer, one might argue that the profit/wage ratio in price terms is more conceivable to the working class and thus has more behavioral foundation than the rate of exploitation defined in value terms. This is contentious because workers generally lack information on profits of the capitalist class as well as on the output (aggregate or sectoral) that is needed to calculate the profit/wage ratio. Moreover, using the assumption of constant profit/wage ratio comes at a modeling cost: To calculate the aggregate profits and wages, one has to impose some special restriction on how output changes with technical change. For example, Franke (1999) assumes a constant aggregate profit/wage ratio, but has to limit his analysis to the case of balanced growth. avoids the treatment of output by not using the aggregate profit/wage ratio, and instead assumes constant sectoral profit/wage ratios which are different across sectors. This implies that the labor market is non-competitive since the real wage rate is not necessarily uniform across sectors.

7

Allowing wages to be paid at any point between the start and the end of production does not alter the result of this paper. As suggested by a reviewer, I do not incorporate this factor in the model for the sake of simplicity, but the proof is available on request.

8

The replenishment of fixed capital is somewhat special at the firm level, because it only needs to be replaced at the end of its life, rather than that of a production period. Here for simplicity, I further assume that at the end of a production process, a depreciation fund is set aside and lies idle for future replenishment of fixed capital so that the fixed capital tied up in production remains constant over time.

9

Also note that the first m equations in system (5) can uniquely determine π and the relative price, i.e., p is unique up to multiplication of scalar. The last equation $p b = 1$ is only necessary to make p absolutely determined.

10

It is beyond the scope of this paper to treat the case in which D also changes. A special example in this regard is that allows K to increase while holding A constant, or the other way around.

11

This is a generalized version of the so-called “Marx-Okishio threshold condition” as in Foley (2009: 137–9) and .

12

The proofs of the Okishio Theorem and the inverse relationship between the real wage rate and the rate of profit within our framework are available on request.

13

From system (5) and $π > 0$ , we have $p > p A + ω l$ , and hence $p > ω l {(I - A)}^{- 1} = ω v$ . Based on the value system (3), it could be easily verified that $v^{'} - v = [v (A^{'} - A) + l^{'} - l] + (v^{'} - v) A^{'}$ or $v^{'} - v = [v (A^{'} - A) + l^{'} - l] {(I - A^{'})}^{- 1}$ . From inequality (6) we can form $p (A^{'} - A) + ω (l^{'} - l) \leq 0$ , which further implies $v (A^{'} - A) + l^{'} - l \leq 0$ because $A^{'} \geq A$ and $p > ω v$ . Moreover, ${(I - A^{'})}^{- 1} \geq 0$ according to the Perron-Frobenius Theorem. Therefore $v^{'} - v \leq 0$ , or $v^{'} \leq v$ .

14

The derivation of equation (14) is straightforward since the value of labor power in the two-department setting becomes $v_{I I} ω b_{I I}$ . To derive : From the price system as in (12) we have $p_{I} = ω l_{I} {(I - π K_{I} - A_{I})}^{- 1}$ . The price system after technical change, when π is held constant, becomes: $p_{I}^{o} = π p_{I}^{o} K_{I} + p_{I}^{o} A_{I} + p_{I I}^{o} ω^{o} b l_{I}$ or $p_{I}^{o} = ω^{o} l_{I} {(I - π K_{I} - A_{I})}^{- 1}$ , $p_{I I}^{o} = π p_{I}^{o} K_{I I}^{'} + p_{I}^{o} A_{I I}^{'} + p_{I I}^{o} ω^{o} b l_{I I}^{'}$ , and $p_{I I}^{o} b = 1$ . Changes in the prices thus are: $p_{I}^{o} - p_{I} = (ω^{o} - ω) l_{I} {(I - π K_{I} - A_{I})}^{- 1} = \frac{ω^{o} - ω}{ω} p_{I}$ , and $p_{I I}^{o} - p_{I I} = π p_{I}^{o} K_{I I}^{'} - π p_{I} K_{I I} + p_{I}^{o} A_{I I}^{'} - p_{I} A_{I I}$ $+ ω^{o} l_{I I}^{'} - ω l_{I I} = π (p_{I}^{o} - p_{I}) K_{I I}^{'} + π p_{I} (K_{I I}^{'} - K_{I I}) + (p_{I}^{o} - p_{I}) A_{I I}^{'} + p_{I} (A_{I I}^{'} - A_{I I}) + (ω^{o} - ω) l_{I I}^{'} + ω (l_{I I}^{'} - l_{I I})$ . Because $p_{I I} b = p_{I I}^{o} b = 1$ , postmultiply the price differences of Department II by $b_{I I}$ , we have $0 = (p_{I I}^{o} - p_{I I}) b_{I I} = [π (p_{I}^{o} - p_{I}) K_{I I}^{'} + π p_{I} (K_{I I}^{'} - K_{I I}) + (p_{I}^{o} - p_{I}) A_{I I}^{'} + p_{I} (A_{I I}^{'} - A_{I I}) + (ω^{o} - ω) l_{I I}^{'} + ω (l_{I I}^{'} - l_{I I})] b_{I I}$ . After plugging in the price changes of Department I and some rearrangement, we finally arrive at equation (13).

Author Biography

Junshang Liang is PhD candidate in the Department of Economics, University of Massachusetts-Amherst. His main research interest is Marxian economics, currently with a focus on the interplay of technical change, class struggle, and income distribution.

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