A Mathematical Formulation of the Dual Nature of Unproductive Labor

Abstract

In this paper, we present a novel model that explicitly formalizes the dual nature of unproductive labor and derives the optimal size of the unproductive sector. Two sets of results emerge regarding the relationship between the unproductive sector and the profitability and accumulation of capital. First, when the unproductive sector does not make any contribution to improving the efficiency of productive labor’s value creation process, the unproductive sector will cease to exist. In this case, when the labor supply is flexible, profit continues to be produced although it is smaller than otherwise, as long as the exploitation rate is positive; in contrast, if the labor supply is fixed, the absence of the unproductive sector necessarily squeezes the exploitation rate to zero, consequently yielding zero profit. Second, when the labor supply is flexible, an economy can always rely on improving the efficiency of the unproductive sector to prevent economic stagnation and to enhance growth performance; however, when the labor supply is fixed, an economy can do so only when the wage rate is sufficiently high relative to the labor productivity conditions in the productive sector.

Keywords

unproductive labor monetary expression of labor time value transfer

1. Introduction

One of the common observations that emerges from empirical studies on the productive versus unproductive labor distinction is that the unproductive sector has been surging (see, e.g., Moseley 1983, 1988, 1990, 1997; Mohun 2014; Paitaridis and Tsoulfidis 2012; Rieu and Park 2020). For instance, in the United States, according to Moseley (1997), the ratio between unproductive and productive labor rose from 0.64 in 1975 to 0.78 in 1994. More recently, Paitaridis and Tsoulfidis (2012) reported that the share of unproductive labor to total employment increased from 0.43 in 1963 to 0.49 in 2008. According to Rieu and Park (2020), in Korea, the share of unproductive labor employment increased from 0.43 in 1995 to 0.57 in 2015. Most of these studies also suggest that the expansion of unproductive employment harms the growth of the economy.¹

Despite the detrimental effect of the expansion of unproductive labor, some positive roles it plays are well recognized in Marxian theory. Consider, for example, the circuit of capital framework: an expansion of commercial capital provides wider outlets for commodities produced, and the development of financial capital allows easier access to credit for industrial capital, all of which contribute to increasing the turnover rate. Given such a dual nature of unproductive labor, some interesting questions can be raised. What is the optimal level of unproductive employment that balances the two opposing effects? What is the mechanism governing the way unproductive labor relates to the profitability and accumulation of capital? In this paper, we provide a novel mathematical model that addresses these questions.

One of the unique aspects of the model is that it uses as the central variable the concept of monetary expression of labor time (MELT) from the New Interpretation (NI) of the Marxian labor theory of value. Based on the idea that money value added results from expenditure of living labor, the NI proposes to adopt the equivalence between the two as an axiom, and the associated proportionality coefficient is the MELT, defined as the aggregate money value added divided by total living labor time.

Foley (1986) defined the MELT at the aggregate level only. Rieu (2008), however, suggested applying the MELT at the sectoral level, and the suggestion was accepted by Duménil, Foley, and Lévy (2009). In a similar vein, Rieu and Park (2020) extended the NI framework to estimate the MELTs at the industry level with an explicit distinction between productive versus unproductive industries. According to Marxian theory, values are produced only in productive industries, which are located in the productive phase of the circuit of capital, while thus-produced values are transferred to unproductive industries, which are located in the circulation or realization phase, where only redistribution of value takes place (see, e.g., Moseley 1985; Mohun 1996, 2014; Foley 2013).²

In this context, we distinguish the MELTs of productive and unproductive sectors. The MELT of the aggregate unproductive sector is defined as the ratio of total money value transferred to the unproductive sector against total unproductive labor time, and the MELT of the aggregate productive sector is defined as the ratio of total money value remaining and realized within the productive sector against total productive labor time.

In these definitions, we assume that all workers employed in the productive sector are productive. We know that this is not true in reality; there are laborers employed in productive industries who are unproductive, such as supervisory and managerial laborers who do not directly produce value and surplus value, or workers employed in sales or financial departments of productive firms.³ Our model presented below can be extended to incorporate supervisory versus non-supervisory labor within the productive sector.

Note that from the definition of unproductive labor as labor that does not produce value and surplus value but only redistributes them, it logically follows that an expansion of unproductive activities undermines capital accumulation and economic growth. On the other hand, although the unproductive sector does not produce any value but only spends the redistributed values, that does not mean that unproductive industries such as wholesale and retail trades, real estate, and finance are unnecessary. On the contrary, they are socially necessary for the maintenance and reproduction of the system, which justifies the value transfer to these unproductive industries. This contradictory nature of the unproductive sector leads us to ask what the optimal level of the size of the unproductive sector would be.

As a way to address these issues, our model derives the optimal levels of value transfer and unproductive labor time—optimal in the sense that they maximize the productive sector’s profit, which is assumed to be the sole source of capital accumulation and growth. The key behavioral assumption of the model is that unproductive industries are socially necessary and efficient in the sense that, although they do not directly create value and surplus value, they make indirect contributions to value creation by enhancing the efficiency of the value-producing process.

From our model, with the dual nature of the unproductive sector explicitly formalized, two sets of results emerge regarding the relationship between the unproductive sector and the profitability and accumulation of capital.

The first result is that the unproductive sector exists due to its indirect contribution to value creation by productive labor. To prove this, we consider a hypothetical case where the unproductive sector does not even make any indirect contribution to the value-creating process. In that case, the model suggests that the unproductive sector will cease to exist. What, then, is the consequence of the absence of the unproductive sector to capital profitability? On the one hand, if labor supply is elastic, profit continues to be produced as long as workers gain less than (the money equivalent of) the value they produce⁴—in other words, as long as the exploitation rate is positive; but profit, when there is no unproductive sector, is strictly less than otherwise. On the other hand, if the labor supply is perfectly inelastic, a complete absence of the unproductive sector makes it always the case that the exploitation rate is zero, thereby yielding zero profit.

In the second set of results, we examine the possibility of an economy relying on the unproductive sector for enhancing capital profitability and accumulation. For this exploration, we propose the concept of the unproductive sector-led phase, defined as a phase where the economy can prevent stagnation or achieve growth by strengthening the efficiency of the unproductive sector. Our analytical result suggests that when labor supply is elastic, the economy is unproductive sector-led, whereas when labor supply is perfectly inelastic, capital profitability and accumulation depend on the level of the productive sector’s wage rate. If the productive sector’s wage rate is sufficiently high relative to labor productivity conditions, the economy can rely on the efficiency of the unproductive sector for capital profitability and accumulation.

The rest of the paper is organized as follows. We discuss the related literature in section 2. Section 3 introduces the basic setup of the model. In section 4, an economy with unconstrained labor supply is examined as the benchmark. We then move on to an economy constrained by labor supply in section 5. As a robustness check, in section 6 we consider maximizing aggregate profit, instead of the productive sector’s profit. Section 7 presents the conclusion.

2. Related Literature

Our paper is related to various strands in the literature on the productive/unproductive distinction. First of all, empirical evidence that depicts the increase of unproductive labor’s share in total employment as the cause of stagnation in the US economy is forcefully presented by Moseley (1983, 1985, 1988, 1990, 1997) and Mohun (1996, 2006, 2014). Here, the central mechanism is the growth of unproductive labor leading to a reduction in the profit share and profit rate, which ultimately causes the economy to stagnate. More recently, Paitaridis and Tsoulfidis (2012) and Cogliano (2018) arrive at a similar conclusion on the deleterious impact of the expansion of unproductive activities on economic growth. In particular, in comparison to our paper where the main concern is value transfer between the aggregate productive and unproductive sectors, Cogliano (2018) provides an empirical account of value transfer at the industry level, with an emphasis on competition between productive and unproductive industries. However, all of these papers focus only on the depressing effect of unproductive activities on profitability and accumulation of capital.

In contrast, there are papers that shed light on the dual character of unproductive labor, a perspective similar to ours. Smith (1993) discusses circulation labor—that is, unproductive labor employed by unproductive capital such as commercial capital or financial capital, a government employee, and supervisory labor; all of these labors facilitate and indirectly contribute to the valorization and accumulation of capital, despite not directly contributing to the augmentation of social surplus value. According to Smith (1993: 268), these are “the socially necessary forms of unproductive labor that serve the reproduction of the capitalist socioeconomic order.”

Based on the same idea, Duménil and Lévy (2011) call these unproductive labors “profit rate-maximizing labor” since, although their compensations reduce profits, they enhance the realization of value, thereby improving capital profitability. Corresponding to these views, our model assumes that an increase in either unproductive labor time or value transfer to the unproductive sector enables each productive labor time to produce more value.

In addition, there are papers that explicitly formalize the dual character of unproductive labor and examine its macroeconomic consequences. In response to Moseley (1990), who emphasizes the dampening effect of the growth of unproductive labor on capital profitability and growth, Cullenberg (1994) provides a model that assumes that supervisory labors contribute to raising the rate of surplus value; it is shown that as long as this positive effect is sufficiently strong, an expansion of unproductive labor can possibly increase the profit rate. Olsen’s (2015) model produces a similar result based on various types of unproductive labor; some have a profit-squeezing effect, while others raise the relative surplus value by increasing work intensity and promoting technical innovation. The simulation results demonstrate that, depending on the relative strength of these contradictory forces, unproductive labor can reduce or promote profitability and growth. In relation to these views, our paper contributes to explicitly deriving the optimal level of the size of the unproductive sector in terms of labor time and value transfer. The solution of the model shows that the optimal level depends on the efficiency of the unproductive sector in enhancing the value-creating capacity of productive labor.

In terms of modeling, Cuyvers’s (1978) study is worth mentioning. He attempts to replicate existing Marxian results on two aggregate equalities and the rate of profit along the balanced growth path within the Leontief linear production model with unproductive industries explicitly incorporated. One of the relevant results is that in the economy without unproductive industries, the equilibrium rate of profit is greater than in the economy with the unproductive sector. This is because the unproductive sector in Cuyvers’s (1978) model is simply a drag on the system.

On the other hand, our model better compares to standard two-sector growth models driven by a labor productivity differential. Baumol’s (1967) seminal model consists of a goods sector with exponentially growing labor productivity and a service sector with constant labor productivity. The wage rate, therein, is equalized across sectors due to perfect competition in the labor market, and grows in step with labor productivity growth of the goods sector. In this setting, when the output ratio between the two sectors is held constant due to, for instance, government subsidy of the service sector, the growing share of total labor supply is devoted to the service sector, thereby raising the ratio of service sector employment to goods sector employment. In addition, due to the productivity differential and equalized wage rate, the unit cost ratio between the service sector and the goods sector rises without limit permanently, a phenomenon known as Baumol’s disease. As more scarce resources are continuously allocated to the lower-productivity, cost-inefficient sector, the economy-wise growth rate eventually converges to zero in the limit.

Extending Baumol’s model, Wolff (1987) replaces the distinction between goods and service sectors with the distinction between productive and unproductive sectors, and derives similar results. Dutt’s (1991) model is also a Marxian growth model of productive and unproductive sectors; it adds the reserve army of labor and thus there is no full employment, in contrast to the two previous papers. Moreover, in line with Moseley (1988), stagnation is explained in relation to the fall in the profit rate. However, these Marxian models essentially follow the Baumol model and feature the labor productivity differential as the main cause of economic stagnation. In these Marxian models, the unproductive sector is unproductive because it has lower labor productivity. Moreover, Dutt (1991) distinguishes the unproductive sector from the productive sector in that the unproductive sector produces an intermediate good; however, it is not different from the productive sector in that it also generates value added and profit.

In contrast, the unproductive sector in our model is unproductive because it does not produce value and surplus value; the source of incomes in the unproductive sector is the value transferred from the productive sector. To formalize this, our model is built around the aggregate and sectoral MELTs, a measure of value creation and value transfer.

3. A Model: Basic Setup

In this section, we introduce the basic setup of a model to investigate the implications of the unproductive sector in relation to value and surplus value production, capital accumulation, and growth.

First of all, we use the following notation. On the one hand, MVA is the aggregate money value added, $MV A_{U}$ is the money value transferred to the unproductive industries, and $MV A_{P}$ is the money remaining in productive industries. Since values are produced only in productive industries while unproductive industries have values transferred from the productive sector, we obtain the following relation:

MV A_{P} = MVA - MV A_{U}

(1)

On the other hand, $L_{P}$ is total productive labor time employed in the productive sector, $L_{U}$ is total unproductive labor time employed in the unproductive sector, and $L = L_{P} + L_{U}$ is total labor time employed in the entire economy.

From these, three distinctive measures of the MELT can be defined as follows:

(i) $m = \frac{MVA}{L_{P}}$ : value added per productive labor time

(ii) $m_{P} = \frac{MV A_{P}}{L_{P}}$ : value realized, after value transfer, per productive labor time

(iii) $m_{U} = \frac{MV A_{U}}{L_{U}}$ : value transfer per unproductive labor time

where m, the aggregate MELT, is a measure of value added per productive labor time; $m_{P}$ , the productive sector MELT, is a measure of the value remaining and realized within the productive sector per productive labor time after the value transfer; and $m_{U}$ , the unproductive sector MELT, is a measure of the value transferred to the unproductive sector per unproductive labor time.

The total value added by productive labors for a given m is:

MVA = m L_{P}

(2)

On the other hand, profit realized in the productive sector is expressed as, by definition, $Π_{P} = MV A_{P} - w_{P} L_{P}$ , where $w_{P}$ is the average hourly wage rate prevailing in productive industries. Then, using equations (1), (2), and $MV A_{U} = m_{U} L_{U}$ , we get an expression for productive sector profits:

Π_{P} = m L_{P} - m_{U} L_{U} - w_{P} L_{P}

(3)

The main reason for considering the productive sector’s profit instead of total profit $Π = Π_{P} + Π_{U}$ is to focus on the consequence of total value transfer from the productive sector to the unproductive sector, which includes not only wages for unproductive labor but also profits for capital in the unproductive sector. For the sake of analytical convenience, suppose that the profit of the productive sector is used productively for capital investment, while the profit of the unproductive sector is used unproductively, in which case the productive sector’s profit is the sole source of accumulation and growth. Then the intra-capitalist class struggle surrounding the distribution of profit among the productive and unproductive sectors becomes crucial from the perspective of the economy as a whole.⁵

Note that the total value transfer to the unproductive sector reflects how big the unproductive sector is. Since it is measured by $MV A_{U} = m_{U} L_{U}$ , we have two measures for the size of the unproductive sector in our model—the unproductive labor time $L_{U}$ and the value transfer per unproductive labor time $m_{U}$ . These two measures can be schematically distinguished in the following way. On the one hand, an increase in $m_{U}$ for given $L_{U}$ can be viewed as a qualitative expansion of the unproductive sector, reflecting an enhancement of technological and market conditions in the unproductive sectors. On the other hand, an increase in $L_{U}$ for given $m_{U}$ can be viewed as a quantitative expansion of the unproductive sector without any technological development or improvement in market conditions.

Whether qualitative or quantitative, however, equation (3) demonstrates that an expansion of the unproductive sector is unambiguously a drain on capital profitability and economic growth; mathematically, $\frac{\partial Π_{P}}{\partial L_{U}} < 0$ and $\frac{\partial Π_{P}}{\partial m_{U}} < 0$ can be easily verified. This corresponds to studies discussed in section 2, such as Moseley (1983, 1985, 1988, 1990, 1997), Mohun (1996, 2006, 2014), Cogliano (2018), and Paitaridis and Tsoulfidis (2012), where the highlight is on the stagnating effect of the expansion of the unproductive sector.

On the other hand—in line with Smith (1993: 268), where unproductive labor is characterized as “socially necessary” for the maintenance and reproduction of capitalist system; Duménil and Lévy (2011: 218), who characterize unproductive labor as “profit-rate-maximizing” because its main goal is to guarantee the greatest profit for a given level of capital; and Cullenberg (1994) and Olsen (2015), who depict unproductive labor as contributing to raising the rate of surplus value—we propose to characterize the unproductive sector as making indirect contributions to capital accumulation and growth by enhancing the efficiency of the process of value production.⁶ Since the model has two measures for the size of the unproductive sector, that is, $L_{U}$ , unproductive labor time, and $m_{U}$ , value transfer, we can formalize the idea by formulating m as a function of these two variables in addition to other variables that positively affect m which we denote by a comprehensive variable $Ω$ :

m = m (L_{U}, m_{U}, Ω)

(4)

where $\frac{\partial m}{\partial Ω} > 0$ holds. $Ω$ captures technological and market conditions of the productive sector, which enhance the value-creating capacity of any given productive labor time.⁷

Once the relation of (4) is adopted and incorporated into the expression for the productive sector’s profit in (3), it can be verified that $\frac{\partial Π_{P}}{\partial L_{U}} < 0$ and $\frac{\partial Π_{P}}{\partial m_{U}} < 0$ do not hold any longer; the result becomes more complicated since it now depends on the impact of $L_{U}$ and $m_{U}$ on m. For this, we adopt the following assumptions.

Assumption 1:

(i) $\frac{\partial m}{\partial L_{U}} > 0$ : for given $m_{U}$ , an increase (decrease) in unproductive labor time makes the value creation per productive labor time more (less) efficient.

(ii) $\frac{\partial m}{\partial m_{U}} > 0$ : for given $L_{U}$ , an increase (decrease) in the value transferred to unproductive sectors makes the value creation by productive labors more (less) efficient.

Assumption 1 (i) concerns the effect on value creation of a simple quantitative expansion of the unproductive sector with the value transfer per unproductive labor time being constant. On the other hand, Assumption 1 (ii) concerns the effect of a qualitative expansion, where the total unproductive labor time remains the same but the value transfer for each unproductive labor time is greater. Under these assumptions, although the unproductive sector does not directly produce value, expanding it either qualitatively or quantitatively makes the value production by productive labor more efficient.

In Marxian theory, this dual nature of the unproductive sector can be explained within the circuit of capital framework, which describes the relation between industrial capital—corresponding to the productive sector—and commercial capital and financial capital—corresponding to the unproductive sector. An expansion of, or development within, commercial capital provides wider outlets for commodities that industrial capital produces or more efficient ways of selling them, and an expansion of, or development within, financial capital provides easier or more cost-efficient access to credit, which allows industrial capital to initiate a new circuit of capital without having to wait for the previous circuit to complete and the initial capital outlays to be recovered. All of these contribute to increasing the turnover rate so that industrial capital can make more efficient uses of given resources.

It is worth quoting from the chapter on commercial capital in Capital, volume 3:

Merchant’s capital, therefore, does not create either value or surplus-value, at least not directly. In so far as it contributes to shortening the time of circulation, it may help indirectly to increase the surplus-value produced by the industrial capitalists. In so far as it helps to expand the market and effects the division of labor between capitals, hence enabling capital to operate on a larger scale, its function promotes the productivity of industrial capital, and its accumulation. In so far as it shortens circulation time, it raises the ratio of surplus-value to advanced capital, hence the rate of profit. And to the extent that it confines a smaller portion of capital to the sphere of circulation in the form of money-capital, it increases that portion of capital which is engaged directly in production. (Marx 1981: 393)

Similarly, Duménil and Lévy (2011: 218), in characterizing unproductive labor as “profit-rate-maximizing” labor, offer the following explanation:

Unproductive labor is useful and necessary. The most synthetic expression of its function is the maximizing of the profit rate: Economizing on inputs, producing as efficiently as possible, selling as rapidly as possible and at the best price. The purpose of this activity is to guarantee the maximum profit to the capitalist, in relation to the amount of capital invested. Obviously, unproductive costs must be subtracted from profits and, in this sense, encroach on the profit rate, but their function is to diminish other costs or to increase the total output or sale that can be realized on the basis of the same amount of capital.

We are now ready to formally define what is meant by the statement that the unproductive sector is socially necessary.

Definition 1: The unproductive sector is socially necessary insofar as assumption 1 holds—that is, $\frac{\partial m}{\partial L_{U}} > 0$ and $\frac{\partial m}{\partial m_{U}} > 0$ . The levels of $\frac{\partial m}{\partial L_{U}}$ and $\frac{\partial m}{\partial m_{U}}$ are a measure of the degree by which an increase in unproductive labor time (quantitative expansion of unproductive sector) and a value transfer (qualitative expansion of unproductive sector), respectively, make the value production per productive labor time more efficient.

A word of caution is required to avoid any misunderstanding about Assumption 1. This assumption does not imply that the expansion of the unproductive sector necessarily improves profitability and accumulation of capital. Since our model is premised upon the unproductive sector having both positive and negative effects on capitalist profitability and growth, it depends on which of the opposing forces is dominant.⁸ The empirical findings in the literature on the slower growth of economies with large unproductive sectors, rather than contradicting our model, can be well explained by it. As becomes clearer in our discussion of the optimal size of the unproductive sector in the next section, those findings imply that the unproductive sector’s size is suboptimal, that is, too big compared to the optimal level.

In order to investigate the optimal size of the unproductive sector, we construct the following optimization problem. Suppose there exists what Engels (1970) called the ideal personification of total national capital, or ideeller Gesamtkapitalist, a hypothetical agent who decides the levels of $L_{U}$ and $m_{U}$ that maximize profit of the productive sector as the source of accumulation and growth. Of course, a decentralized market economy does not have such a social optimizer. We do not consider the ideeller Gesamtkapitalist aiming to maximize capital profitability as a description of social reality. Rather, we use optimization as a counterfactual method to examine what type of relationship between productive and unproductive sectors would be desirable from the bird’s eye view of total capital, and investigate the implications that ensue.

$L_{U}$ and $m_{U}$ are two control variables of hypothetical total capital, with $w_{P}$ and $L_{P}$ taken as given. Treating $w_{P}$ as constant instead of formalizing its determination in relation to m or $m_{P}$ as a measure for labor productivity reflects the possible and likely deviation between wage and labor productivity. That is, we consider the wage rate as determined outside of the model by factors such as class conflict.⁹ Once the optimal values of $L_{U}$ and $m_{U}$ are found, the optimal values of m and $Π_{P}$ are obtained from equations (4) and (3), and finally we are able to get the optimal value of $m_{P}$ from its definition:

m_{P} = \frac{m L_{P} - m_{U} L_{U}}{L_{P}}

(5)

While $L_{P}$ is constant, since $L_{U}$ varies it is useful to adopt the following assumption.

Assumption 2: $MV A_{U} = 0$ and $m_{U} = 0$ if $L_{U} = 0$ , whereas $MV A_{U} > 0$ and $m_{U} > 0$ if $L_{U} > 0$ .

Suppose $L_{U} = 0$ , which is equivalent to a complete absence of the unproductive sector unless the latter is characterized by full automation technology. Then $MV A_{U} = 0$ would be the case since value transfer to a sector that does not exist would be a non sequitur. In this case, although $m_{U} = \frac{0}{0}$ would be the case by definition, it would be reasonable to take it as $m_{U} = 0$ , which is the main point of Assumption 2.

In analyzing the optimization problem, we consider two cases in turn: an economy with an unconstrained labor supply and an economy constrained by labor supply. When there is a labor supply constraint, a quantitative expansion of the unproductive sector, that is, an increase in unproductive labor time, diminishes the pool of labor for productive employment, while this is not the case when there is no such constraint. Developing countries with huge non-capitalist sectors, which may be the source of an elastic labor supply to industrialized cities, are examples of economies with unconstrained labor supplies. On the other hand, developed countries where such non-capitalist sectors are very small are likely to be economies constrained by labor supply, although the constraint can possibly be relaxed by, for example, immigration, female participation in the labor force, and so on.

4 An Economy with Unconstrained Labor Supply

We first consider an economy with unconstrained labor supply.

4.1 The optimal value transfer with unconstrained labor supply

The optimization problem of the ideeller Gesamtkapitalist is as follows:

\begin{matrix} \max_{L_{U}, m_{U}} & Π_{P} = m (L_{U}, m_{U}, Ω) L_{P} - m_{U} L_{U} - w_{P} L_{P} \end{matrix}

(6)

An implicit constraint may be added to this optimization problem, which is that the total capital stock is given. In this case, the profit maximization problem (6) becomes effectively identical to a profit rate maximization, which makes our model more conforming to the Classical approach, where the rate of profit is the central driver. In addition, suppose the savings rate of capitalists, or recapitalization rate, to be constant. Then, overall, along with the assumption of fixed capital stock, maximizing profit becomes identical to maximizing not only the rate of profit but also the rate of growth.

The solution of the optimization problem is summarized in lemma 1.

Lemma 1: The optimal size of the unproductive sector of an economy with unconstrained labor supply is:

L_{U}^{*} = \frac{\partial m}{\partial m_{U}} L_{P}

(7)

m_{U}^{*} = \frac{\partial m}{\partial L_{U}} L_{P}

(8)

Proof. From the first-order conditions:

\begin{matrix} \frac{\partial Π_{P}}{\partial m_{U}} = \frac{\partial m}{\partial m_{U}} L_{P} - L_{U} = 0 \\ \frac{\partial Π_{P}}{\partial L_{U}} = \frac{\partial m}{\partial L_{U}} L_{P} - m_{U} = 0, \end{matrix}

we obtain $L_{U}^{*}$ and $m_{U}^{*}$ . Q.E.D.

The idea behind the solution of the two choice variables is clear and intuitive. When the unproductive sector’s indirect contribution to value creation becomes stronger (weaker), unproductive labor time, $L_{U}$ , and value transfer to the unproductive sectors, $m_{U}$ , for given $L_{P}$ , will be greater (smaller). The levels of $L_{U}$ and $m_{U}$ beyond (7) and (8) are suboptimal, crowding out capital accumulation and growth, and will be considered as too big. Some findings, such as Cecchetti and Kharroubi (2015), that the financial sector crowds out real economic growth, may be understood in this context.

Now that the optimal level of unproductive labor time and optimal value transfer are determined, we examine below two sets of issues on the relationship between the unproductive sector and capital profitability. First, we compute the optimal profit of the productive sector using the definition of the productive sector’s profit in equation (3) and the optimal solution in Lemma 1:

Π_{P}^{*} = m^{*} L_{P} - \frac{\partial m}{\partial m_{U}} \frac{\partial m}{\partial L_{U}} L_{P}^{2} - w_{P} L_{P}

(9)

In addition, it is useful for the analysis below to have the function of $m$ in equation (4) linearly specified as follows:

m = α_{0} + α_{1} L_{U} + α_{2} m_{U} + α_{3} Ω α_{0}, α_{1}, α_{2}, α_{3} > 0

(10)

Note that $α_{1} \equiv \frac{\partial m}{\partial L_{U}}$ and $α_{2} \equiv \frac{\partial m}{\partial m_{U}}$ are two measures of the degree by which unproductive industries—measured by either $L_{U}$ or $m_{U}$ —make the production of value more efficient as discussed in Definition 1. Since an increase in $L_{U}$ is a quantitative expansion of the unproductive sector, and an increase in $m_{U}$ is a qualitative expansion of the unproductive sector, we call $α_{1}$ the quantitative effect of unproductive sector efficiency and $α_{2}$ the qualitative effect of unproductive sector efficiency. On the other hand, $α_{3}$ measures the degree by which the productive sector’s technological and market conditions make value production more efficient. These efficiency coefficients, $α_{1}$ , $α_{2}$ , and $α_{3}$ , are the deep parameters of the model that measure more structural aspects of the contributions $L_{U}$ , $m_{U}$ , and $Ω$ make to enhancing m.

Under this linear specification of m, the optimal profit of the productive sector in equation (9) is rewritten as follows:

Π_{P}^{*} = [α_{0} + α_{1} α_{2} L_{P} + α_{3} Ω - w_{P}] L_{P}

(11)

4.2. The existence of the unproductive sector without labor supply constraint

The first theoretical issue we deal with concerns a hypothetical case where the unproductive sector does not make any contribution to value creation. What would be the consequence if the unproductive sector is not socially necessary in the sense of Definition 1? Our model provides an answer to this interesting question, which is summarized in the following proposition.

Proposition 1: In an economy with unconstrained labor supply, if $\frac{\partial m}{\partial L_{U}} = 0$ and $\frac{\partial m}{\partial m_{U}} = 0$ , then

(i) $L_{U}^{*} = 0$ and $m_{U}^{*} = 0$ ,

(ii) $Π_{P}^{*} \frac{>}{<} 0$ if and only if $m^{*} \frac{>}{<} w_{P}$ ,

(iii) in the case of the linear specification of m as in equation (10), $Π_{P}^{*}$ with $\frac{\partial m}{\partial L_{U}} = 0$ and $\frac{\partial m}{\partial m_{U}} = 0$ is strictly less than $Π_{P}^{*}$ with $\frac{\partial m}{\partial L_{U}} > 0$ and $\frac{\partial m}{\partial m_{U}} > 0$ .

Proof. Suppose $\frac{\partial m}{\partial L_{U}} = 0$ and $\frac{\partial m}{\partial m_{U}} = 0$ . Then, part (i) is immediate from (7) and (8), whereas part (ii) is immediate from equation (9). On the other hand, in the case of a linear specification of m as in equation (10), since the optimal profit of productive sector is expressed as in equation (11), part (iii) immediately follows. Q.E.D.

Part (i) of Proposition 1 states that if the unproductive sector were not socially necessary in the sense of Definition 1, that is, if it did not contribute to value production even indirectly, unproductive labor would not exist and consequently there could be no value transfer to the unproductive sector. The contrapositive of this statement provides a more intuitive economic insight about the unproductive sector, namely, the reason why the unproductive sector exists is due to its contribution to making the process of value creation by productive labor more efficient.

On the other hand, according to part (ii), even when the unproductive sector does not exist because of not making any contribution to value creation, profit can still be produced as long as the wage rate is not so high as to eliminate profit, that is, as long as the exploitation rate is positive. Note that by using the definition of the exploitation rate, denoted by $e = \frac{m - w_{P}}{w_{P}}$ , the condition in part (ii), $m^{*} \frac{>}{<} w_{P}$ , can be expressed as $e^{*} \frac{>}{<} 0$ .

However, although profit can still be produced even when the unproductive sector ceases to exist as long as the exploitation rate is positive, part (iii) suggests that profit in the case of the absence of the unproductive sector is strictly less than profit in the case of the existence of the unproductive sector. Note that in the model, the negative consequence of the unproductive sector’s absence on profit works through lowering m according to Assumption 1.

In toto, Proposition 1 states that in an economy with an unconstrained labor supply, the unproductive sector exists since it contributes to making value creation process more efficient; otherwise, it would not exist, in which case productive labor’s value creation would be undermined, thereby reducing profit. However, positive profit can still be produced as long as workers’ compensation is less than the value they added. Interestingly, it is shown later in Proposition 3 that for an economy with a perfectly inelastic labor supply, $\frac{\partial m}{\partial L_{U}} = 0$ and $\frac{\partial m}{\partial m_{U}} = 0$ always lead to $m^{*} = w_{P}$ and hence $e^{*} = 0$ , thereby ultimately resulting in $Π_{P}^{*} = 0$ .

4.3. Unproductive sector and capital profitability without labor supply constraint

To further pursue the issue on the relation between the unproductive sector and capital profitability and accumulation, we examine the possibility of the economy relying on the unproductive sector to enhance capital profitability and accumulation. For this, we introduce the concept of an unproductive sector-led phase. In constructing this concept in Definition 2 below, we adopt the linear specification of m in (10) and focus on how the two efficiency coefficients of the unproductive sector, $α_{1}$ and $α_{2}$ , affect the productive sector’s profit.

Definition 2: Suppose a linear specification of m as in (10). An economy is said to be in an unproductive sector-led phase if both of the following two conditions hold simultaneously:

\frac{\partial Π_{P}^{*}}{\partial α_{1}} \geq 0 and \frac{\partial Π_{P}^{*}}{\partial α_{2}} \geq 0

with at least one of them holding with strong inequality.

According to Definition 2, an economy is said to be unproductive sector-led when an improvement in either the quantitative or the qualitative effect of unproductive sector efficiency raises the optimal profit of the productive sector while neither lowers it.

Now, let us apply the concept and examine whether the economy with unconstrained labor supply is unproductive sector-led or not. The result is summarized in the following proposition.

Proposition 2: According to Definition 2, the economy with unconstrained labor supply is in an unproductive sector-led phase since:

\begin{matrix} \frac{\partial Π_{P}^{*}}{\partial α_{1}} = α_{2} L_{P} > 0 \\ \frac{\partial Π_{P}^{*}}{\partial α_{2}} = α_{1} L_{P} > 0 \end{matrix}

Proof. The result immediately follows from equation (11) and Assumption 1. Q.E.D.

Proposition 2 states that the economy with an unconstrained labor supply is unproductive sector-led in the sense that improving the quantitative or the qualitative effect of unproductive sector efficiency necessarily enhances the profitability and hence accumulation of capital. That is, as long as the economy is not constrained by labor supply, it can always improve economic growth by relying on the unproductive sector’s socially necessary character of making the process of value production more.

This result is mathematically obvious. Consider the productive sector profit in (3), which is the objective function of the optimization problem. By using the envelope theorem, it can be readily confirmed that as long as $m$ has a linear function, as in (10), an increase in $α_{1}$ or in $α_{2}$ unambiguously raises profit; thus, the economy is in an unproductive sector-led phase.

This result is also consistent with the results of Baumol (1967) and Wolff (1987). Note that in those studies, the main reason for the labor productivity differential between two sectors leading to a permanent reduction in the growth rate is that the total labor supply is fixed. Due to the unproductive, or service, sector having a lower labor productivity, the share of fixed labor supply allocated to this sector rises continuously, and implies that more of the given resources are continuously being used less and less efficiently, which drags on the economy. Therefore, once the assumption of a fixed labor supply is relaxed, as we have done in this section, and the efficient and socially necessary character of the unproductive sector is introduced, as in our model, it becomes obvious that making the efficiency of the unproductive sector stronger, that is, raising $α_{1}$ or $α_{2}$ , increases profit and economic growth.

5. An Economy Constrained by Labor Supply

We now consider an economy constrained by labor supply. In contrast to the economy dealt with in the previous section, now the total labor supply is fixed, and as a result, an increase in unproductive labor time has an additional negative consequence, that is, a reduction in productive labor time, which is the sole source of value and surplus value.

5.1. The optimal value transfer with a constrained labor supply

In an economy constrained by labor supply, the ideeller Gesamtkapitalist solves the following constrained optimization problem:

\begin{matrix} \max_{L_{U}, m_{U}} & Π_{P} = m (L_{U}, m_{U}, Ω) L_{P} - m_{U} L_{U} - w_{P} L_{P} \\ s . t . & \bar{L} = L_{P} + L_{U} \end{matrix}

Using the same approach as the optimization problem for the economy with an unconstrained labor supply in section 4, if we suppose the total capital stock is constant, and the capitalists’ savings rate is also constant, then the profit maximization problem becomes effectively identical to the problem of maximizing the rate of profit and the rate of growth.

The solution is summarized in Lemma 2.

Lemma 2: The optimal size of the unproductive sector of the economy constrained by labor supply is:

L_{U}^{*} = \frac{\frac{\partial m}{\partial m_{U}}}{1 + \frac{\partial m}{\partial m_{U}}} \bar{L}

(12)

m_{U}^{*} = \frac{\frac{\partial m}{\partial L_{U}}}{1 + \frac{\partial m}{\partial m_{U}}} \bar{L} + w_{P} - m^{*} (L_{U}^{*}, m_{U}^{*}, Ω)

(13)

Proof. From the first-order conditions,

\begin{matrix} \frac{\partial Π_{P}}{\partial m_{U}} = \frac{\partial m}{\partial m_{U}} (\bar{L} - L_{U}) - L_{U} = 0 \\ \frac{\partial Π_{P}}{\partial L_{U}} = \frac{\partial m}{\partial L_{U}} (\bar{L} - L_{U}) - m - m_{U} + w_{P} = 0 \end{matrix}

we obtain $L_{U}^{*}$ and $m_{U}^{*}$ . Q.E.D.

The result for $L_{U}^{*}$ in equation (12), on the one hand, is identical to the one in (7) for the unconstrained optimization problem: a higher $\frac{\partial m}{\partial m_{U}}$ leads to a higher $L_{U}$ for given $L_{P}$ . According to this result, an increase in the share of unproductive labor, commonly observed in the literature cited in sections 1 and 2, may suggest that there could have possibly been a strengthening of the qualitative effect of unproductive sector efficiency. On the other hand, $m_{U}^{*}$ in equation (13) is not in a closed form. Therefore, the existence and uniqueness of the solution $m_{U}^{*}$ needs to be verified, which is done in Lemma 3.

Lemma 3: In the case of an economy constrained labor supply, the unique solution $m_{U}^{*}$ exists.

Proof. Let $g (x; \dots) = \frac{\frac{\partial m}{\partial L_{U}}}{1 + \frac{\partial m}{\partial m_{U}}} \bar{L} + w_{P} - m^{*} (L_{U}^{*}, x, Ω)$ , which is the right-hand side of equation (13) with the $m_{U}^{*}$ term removed and $m_{U}^{*}$ in $m^{*}$ replaced by some variable x. Since g is continuous and monotonic, according to a fixed-point theorem, there exists a fixed point $\bar{x}$ such that $g (\bar{x}) = \bar{x}$ , and we have $\bar{x} = m_{U}^{*}$ . Q.E.D.

We may now conduct similar exercises to those in section 4 to examine the relationship between the unproductive sector and capital profitability in an economy constrained by labor supply. For this, we first compute the optimal profit of productive sector:

Π_{P}^{*} = (m^{*} - w_{P}) \bar{L} - \frac{\frac{\partial m}{\partial m_{U}} \frac{\partial m}{\partial L_{U}}}{{(1 + \frac{\partial m}{\partial m_{U}})}^{2}} {\bar{L}}^{2}

(14)

5.2. The existence of an unproductive sector with a constrained labor supply

Recall that when the unproductive sector is not socially necessary in the sense of Definition 1 in an economy with an unconstrained labor supply, the unproductive sector ceases to exist, but profit continues to be produced as long as workers gain less than what they produce, which yields a positive rate of exploitation. What if the labor supply is fixed so that an increase in unproductive labor has an additional negative effect by reducing the availability of productive labor, the sole source of value and surplus value? The answer is summarized in the following proposition.

Proposition 3: In an economy constrained by labor supply, if $\frac{\partial m}{\partial L_{U}} = 0$ and $\frac{\partial m}{\partial m_{U}} = 0$ , then

(i) $L_{U}^{*} = 0$ and $m_{U}^{*} = 0$ ,

(ii) $Π_{P}^{*} = 0$ .

Proof. Suppose $\frac{\partial m}{\partial L_{U}} = 0$ and $\frac{\partial m}{\partial m_{U}} = 0$ . Then, $L_{U}^{*} = 0$ is immediate from equation (12). Due to assumption 2, $L_{U}^{*} = 0$ implies $m_{U}^{*} = 0$ , which proves part (i). Furthermore, since equation (13) simplifies to

m_{U}^{*} = w_{P} - m^{*}

(15)

which, due to $m_{U}^{*} = 0$ , implies $w_{P} = m^{*}$ . The latter, along with $\frac{\partial m}{\partial L_{U}} = 0$ and $\frac{\partial m}{\partial m_{U}} = 0$ , simplifies equation (14) into $Π_{P}^{*} = 0$ , which proves part (ii). Q.E.D.

The result of part (i), on the one hand, is the same as part (i) of Proposition 1. In other words, when the unproductive sector does not even make an indirect contribution to value production and therefore is not socially necessary in the sense of Definition 1, the unproductive sector ceases to exist. The contrapositive of this result is that the unproductive sector exists because it contributes to making the value-creating process more efficient. But will profit continue to be produced if the unproductive sector does not exist due to not making any contribution? Part (ii) of Proposition 3 says no, a stronger result than that found in Proposition 1.

Recall that according to part (ii) of Proposition 1, in an economy with an unconstrained labor supply, even when there is no unproductive sector because it is not socially necessary, profit will continue to be produced if and only if workers’ wages are less than the value they produce. In comparison, according to part (ii) of Proposition 3, in an economy constrained by labor supply, the consequence is always that the worker’s wage exactly equals the value they added and therefore profit is driven to zero. As was the case with Proposition 1 for an economy with an unconstrained labor supply, the key mechanism lies in the behavioral specification of m. That is, the absence of the unproductive sector damages productive labor’s value-creating capacity and thus lowers value added.

Formally, as equation (15) evidently demonstrates, $L_{U}^{*} = 0$ and $m_{U}^{*} = 0$ makes $m^{*} = w_{P}$ a logical consequence.¹⁰ This analytical relation can be interpreted as stating that the absence of the unproductive sector reduces the value created by productive labor all the way to their wage, pressing the rate of exploitation down to zero.

Overall, Proposition 3 suggests that in an economy constrained by labor supply, when the unproductive sector is not socially necessary in the sense of Definition 1, not only do the unproductive sectors cease to exist, but value creation by productive labor becomes so weak that profit is driven to zero.

To summarize, as a way to highlight the role played by the unproductive sector for capital profitability, Propositions 1 and 3 examine the consequence of a hypothetical case where the unproductive sector does not make any contribution to productive labor’s value-creating process. Both propositions suggest that in that case, the unproductive sector will cease to exist. By implication, according to the contrapositive of this result, the reason why the unproductive sector exists is because it contributes to making value creation by productive labor more efficient. Once again, it should be emphasized that the existence of a social optimizer is only a working hypothesis for a comparative analysis of a real capitalist economy.¹¹

Propositions 1 and 3 also state that the absence of the unproductive sector would undermine productive labor’s value-creating capacity and would reduce profit; that is, profit in the case of the absence of unproductive labor is strictly less than profit in the case of the presence of unproductive labor. However, if labor supply is elastic, positive profit can still be produced as long as workers’ compensation is less than the value they added, making the exploitation rate positive (Proposition 1), whereas if labor supply is perfectly inelastic, then the degree by which productive labor’s value-creating capacity is undermined is so severe that the value creation by productive labor becomes as low as their wage compensation, hence the exploitation rate being zero, which makes zero profit a necessary consequence (Proposition 3).

What insights do these results provide for the Marxian theory of exploitation? Note that, in particular, the Fundamental Marxian Theorem states that a positive rate of exploitation is the necessary and sufficient condition for a positive rate of profit. A related theoretical question that is worth pursuing is under what conditions would the exploitation rate be positive? A contribution of Propositions 1 and 3 lies in providing one possible answer to this question, focusing on two elements as conditions for a positive rate of profit: (a) the unproductive sector, and (b) the labor supply constraint, which relates to the reserve army of labor.

More specifically, on the one hand, in cases where labor supply is flexible and there is a huge reserve army of labor, even when there is no unproductive sector, a positive exploitation rate can be achieved through other various means related to class struggle; on the other hand, in cases where labor supply is fixed, which implies a scarce reserve army of labor, if the unproductive sector does not exist there is no way for capitalists to exploit workers. That is, the presence of either an elastic labor supply or unproductive sector is a necessary condition for a positive rate of exploitation; if neither exists, the exploitation rate is inevitably pressed down to zero, eventually yielding zero profit.

Our zero profit result can be compared to Okishio’s (2001) model, wherein a model without an unproductive sector generates a result such that when labor supply is fixed, competition among capitalists forces the real wage rate to exactly correspond to labor productivity, thereby making the exploitation rate and the profit rate converge to zero. In Okishio’s (2001) model, it is the absence of a flexible labor supply—a reserve army of labor—that squeezes the exploitation rate and profit rate to zero. In contrast, by incorporating the unproductive sector, we are able to show that zero profit is yielded only when both the flexible labor supply and the unproductive sector are absent.

To highlight this contrast to Okishio (2001), it can be easily shown by using the optimal profit in equation (14) that despite the absence of a flexible labor supply, a positive rate of exploitation and $Π_{P}^{*} > 0$ are still possible as long as the unproductive sector’s impact on $m^{*}$ is strong enough.

5.3. Unproductive sector and capital profitability with a constrained labor supply

As we did in Section 4.3 for an economy with a constrained labor supply, we now apply the concept of unproductive sector-led suggested in Definition 2 to an economy constrained by labor supply. Since the concept requires a linear specification of m, we adopt the one suggested in equation (10), in which case the optimal profit of the productive sector in (14) is rewritten as, using equations (12) and (13):

Π_{P}^{*} = [α_{0} + \frac{α_{1} α_{2}}{1 + α_{2}} \bar{L} + α_{3} Ω - w_{P}] \frac{\bar{L}}{1 + α_{2}}

(16)

Recall that an economy with an unconstrained labor supply is always unproductive sector-led, as demonstrated in Proposition 2. On the other hand, the case of an economy constrained by labor supply is more complex. The result is summarized in the following proposition.

Proposition 4: According to Definition 2, the economy constrained by the labor supply is unproductive sector-led when $w_{P}$ is at least as high as a certain threshold, since

\frac{\partial Π_{P}^{*}}{\partial α_{1}} = \frac{α_{2}}{{(1 + α_{2})}^{2}} {\bar{L}}^{2} > 0

(17)

\frac{\partial Π_{P}^{*}}{\partial α_{2}} \frac{>}{<} 0 \Leftrightarrow w_{P} \frac{>}{<} α_{0} + α_{3} Ω - \frac{α_{1} (1 - α_{2})}{1 + α_{2}} \bar{L}

(18)

Proof. The results immediately follow from (16). Q.E.D.

Similar to the case of an economy with an unconstrained labor supply, an improvement in the quantitative effect of the unproductive sector efficiency, that is, an increase in $α_{1}$ , always has a positive impact on optimal profit, as shown in equation (17). Equation (18) shows that an improvement in the qualitative effect of the unproductive sector efficiency, or an increase in $α_{2}$ , has an indeterminate impact; it depends on the level of the productive sector’s wage rate. If it is greater (less) than a certain threshold, the increase in the qualitative effect of the unproductive sector efficiency has a positive (negative) impact on the capital profitability.

Note that the wage rate threshold $α_{0} + α_{3} Ω - \frac{α_{1} (1 - α_{2})}{1 + α_{2}} \bar{L}$ is shaped by two groups of parameters: the productive sector-related parameters, $α_{3}$ and $Ω$ , and the unproductive sector-related parameters, $α_{1}$ and $α_{2}$ . Recall the linear specification of m in (10): $α_{0}$ is the autonomous constant; $Ω$ reflects the productive sector’s technological and market conditions that positively affect m; and, $α_{3}$ captures the degree by which $Ω$ affects m. On the other hand, $α_{1}$ and $α_{2}$ are the efficiency coefficients of the unproductive sector. Equation (18) can be understood as stating the following: when the conditions that affect productive labor’s value-creating capacity are so disadvantageous that the given wage rate in that sector is considered too high,¹² the economy can rely on the qualitative effect of unproductive sector efficiency to enhance profitability and accumulation of capital.

Overall, Proposition 4 states that when an economy constrained by labor supply is in a phase with too high a wage rate in the productive sector compared to labor productivity conditions, it can rely on improving the efficiency of the unproductive sector to avoid stagnation or to achieve growth. The reason why the productive sector’s wage rate matters here in relation to the impact of $α_{2}$ (the qualitative effect of unproductive sector efficiency) is due to the existence of the labor supply constraint. When the economy is constrained by labor supply, an expansion of the unproductive sector in terms of labor hours generates an effect that was absent in the case of an economy with an unconstrained labor supply—that is, a reduction in the labor hours available for productive employment. In this case, too high a wage rate exerts an additional pressure on profit. This is the key element to understanding the formal result of Proposition 4.

To be more specific, regarding equation (18) of Proposition 4, consider an improvement in $α_{2}$ ; from Lemma 2, we verify that it leads to an increase in $L_{U}^{*}$ , which was also true in the case of an economy unconstrained by labor supply; see Lemma 1, remembering $\frac{\partial m}{\partial m_{U}} \equiv α_{2}$ . When there is the labor supply constraint, the result of the increase in $L_{U}^{*}$ is a reduction in $L_{P}^{*} \equiv \bar{L} - L_{U}^{*}$ . Since productive labor is the only genuine source of value production, the reduction in $L_{P}^{*}$ lowers the money value added and essentially has a negative impact on capital accumulation. But when the productive sector’s wage rate is so high as to significantly undermine that sector’s profitability, the fall in $L_{P}^{*}$ makes consequent savings in wage cost strong enough to outweigh its negative effect of reducing the money value added; therefore, the net effect is an improvement in profitability. This explains why an economy constrained by labor supply is unproductive sector-led when the wage rate is sufficiently high.

To further clarify the rationale behind Proposition 4, suppose $α_{2} < 1$ for the right-hand side of the second inequality of equation (18).¹³ Further suppose that the productive sector-related parameters, $α_{3}$ and $Ω$ , are sufficiently low—implying that technology and market conditions in that sector are disadvantageous for value creation by productive labor—so that overall the right-hand side of the second inequality of equation (18) is negative. Then, since $w_{P}$ is positive, the condition always holds as >. That is, when the conditions for productive labor’s value-creating capacity are sufficiently weak, the wage is always too high; consequently, relying on the unproductive sector necessarily helps boost capital profitability and accumulation.

In sum, Propositions 2 and 4, in regard to the condition of the unproductive sector-led phase (especially the first one where the economy with flexible labor supply is always in an unproductive sector-led phase), should not be taken as predicting that an economy with a sufficiently efficient unproductive sector will exhibit strong growth performance. Remember that the objective function of the model’s optimization problem is capital profitability, and that its connection to accumulation and growth is made by relying on the simplifying assumption of saving rate and investment rate being constant. In addition, although the results suggest that an economy is in an unproductive sector-led phase under certain conditions, the consequent economic growth can be weak depending on the parameter values of the model.

These words of caution are particularly important in relating our results to existing empirical studies that document weak growth performance of advanced countries characterized by an expansion of the unproductive sector. Our results do not contradict this finding but rather explain it. First, note that the financial industry started to expand during a period when real wage growth, on average, exceeded labor productivity growth. This observation is well explained by the result in Proposition 4, according to which an economy can rely on the quantitative and qualitative expansion of the unproductive sector when the wage is high relative to the productivity condition in the productive sector.

Second, the observed fact that growth was weak and sluggish in the economy during an expansion of the unproductive sector can also be easily explained by the results of the model. According to Propositions 2 and 4, it was simply due to the value of the parameters that define the effect of improving the unproductive sector’s efficiency as being insignificant and thus the effect as being very weak. For instance, in Proposition 2, these parameters are $α_{1}$ and $α_{2}$ . When their values are significantly small, the improvement in profitability and growth can be unimpressive.

6. Optimal Unproductive Labor Time When Aggregate Profits Are Maximized

We have thus far considered the productive sector’s profit as the objective function of the ideeller Gesamtkapitalist. The assumption was that the profit of the productive sector is productively used for capital investment, while the profit of the unproductive sector is unproductively used. In such a case, the distribution of aggregate profit between the productive and unproductive sectors matters for capital accumulation and economic growth.

In this section, we drop this assumption and consider the ideeller Gesamtkapitalist aiming to maximize the aggregate profit of total capital. That is, we assume that intra-capitalist class conflict surrounding value transfer and the distribution of profit is neutral to capital profitability and accumulation. In terms of the model, we assume that $m_{U}$ does not have any impact on m, i.e., $\frac{\partial m}{\partial m_{U}} = 0$ . In this case, the choice variable of the ideeller Gesamtkapitalist is only $L_{U}$ .

Accordingly, there are a couple of formulations in the model that require some modification. First, to obtain a definite analytical solution, it is necessary to endogenize wages by formulating them as a positive function of the demand for labor. Considering that a greater demand for labor diminishes the labor market slack and pushes up the wage, we assume $\frac{\partial w_{P}}{\partial L_{P}} > 0$ and $\frac{\partial w_{U}}{\partial L_{U}} > 0$ . In this regard, we adopt the following linear specifications:

w_{P} = β_{0} + β_{1} L_{P} β_{0} > 0, β_{1} > 0

(19)

w_{U} = γ_{0} + γ_{1} L_{U} γ_{0} > 0, γ_{1} > 0

(20)

Second, also to obtain a definite analytical solution, we need to adopt the linear specification of $m$ suggested in equation (10), but with $α_{2} = 0$ , since value transfer is now assumed to be neutral to productive labor’s value-creating capacity.

Below, we derive the optimal solution for $L_{U}$ in both cases with and without labor supply constraint. The exercise in this section can be considered as a robustness check of the model.

6.1. An economy with unconstrained labor supply

First, consider an economy with an unconstrained labor supply. The optimization problem faced by the ideeller Gesamtkapitalist is as follows:

\begin{matrix} max_{L_{U}} Π = m L_{P} - w_{U} L_{U} - w_{P} L_{P} \\ s . t . m = α_{0} + α_{1} L_{U} + α_{3} Ω \\ w_{U} = γ_{0} + γ_{1} L_{U} \end{matrix}

Since $L_{P}$ is considered to be constant in an economy with an unconstrained labor supply, we can take $w_{P}$ as constant as well. For this reason, the function of $w_{P}$ is omitted in the constraints of the optimization problem.

The first-order condition is obtained as:

\frac{\partial Π}{\partial L_{U}} = α_{1} L_{P} - γ_{0} - γ_{1} L_{U} - γ_{1} L_{U} = 0

(21)

The first term, $α_{1} L_{P}$ , is the marginal benefit of one unit increase in $L_{U}$ , while the rest of the terms with negative signs are the marginal cost. The reason the term $γ_{1} L_{U}$ appears twice in marginal cost is because an increase in $L_{U}$ raises the total wage for unproductive workers not only directly but also indirectly through sending upward pressure on wage, $w_{U}$ .

From the first-order conditions, the optimal level of nonnegative $L_{U}$ is obtained as follows:

L_{U}^{*} = {\begin{matrix} \frac{α_{1} L_{P} - γ_{0}}{2 γ_{1}} if α_{1} > \frac{γ_{0}}{L_{P}} \\ 0 if α_{1} \leq \frac{γ_{0}}{L_{P}} \end{matrix}

(22)

It is interesting to note that if unproductive labor’s contribution to productive labor’s value creation is not sufficiently strong, in other words, if $α_{1} \leq \frac{γ_{0}}{L_{P}}$ , it is optimal to have $L_{U}^{*} = 0$ , which implies that the unproductive sector ceases to exist unless the unproductive sector is adopting a full automation technology. In addition, the impact that each of the parameters has on $L_{U}^{*}$ is intuitive; $\frac{\partial L_{U}^{*}}{\partial α_{1}} > 0$ , $\frac{\partial L_{U}^{*}}{\partial γ_{0}} < 0$ , $\frac{\partial L_{U}^{*}}{\partial γ_{1}} < 0$ , and $\frac{\partial L_{U}^{*}}{\partial L_{P}} > 0$ are all as expected.

6.2. An economy constrained by labor supply

When labor supply is perfectly inelastic, $L_{P}$ is no longer fixed but is inversely related to $L_{U}$ . Accordingly, $w_{P}$ cannot be considered constant, as in the case of the economy with unconstrained labor supply. Thus, in an economy constrained by labor supply, the optimization problem of the ideeller Gesamtkapitalist is as follows:

\begin{matrix} max_{L_{U}} Π = m L_{P} - w_{U} L_{U} - w_{P} L_{P} \\ s . t . m = α_{0} + α_{1} L_{U} + α_{3} Ω \\ w_{P} = β_{0} + β_{1} L_{P} \\ w_{U} = γ_{0} + γ_{1} L_{U} \end{matrix}

From the first-order conditions, we get the optimal level of nonnegative $L_{U}$ as:

L_{U}^{*} = {\begin{matrix} \frac{(α_{1} + 2 β_{1}) \bar{L} - α_{0} - α_{3} Ω - γ_{0} + β_{0}}{2 (α_{1} + β_{1} + γ_{1})} if α_{1} + 2 β_{1} > \frac{α_{0} + α_{3} Ω + γ_{0} - β_{0}}{\bar{L}} \\ 0 if α_{1} + 2 β_{1} \leq \frac{α_{0} + α_{3} Ω + γ_{0} - β_{0}}{\bar{L}} \end{matrix}

(23)

The expressions in equation (23) are more complicated than in equation (22), since now a variation in $L_{U}$ affects the labor market in the productive sector as well. Nonetheless, the idea is the same: the optimal level of $L_{U}$ is determined in such a way that marginal benefit and marginal cost of a unit increase in $L_{U}$ are equal to each other. Let us first identify negative channels that reduce profit. On the one hand, whereas in an economy with an unconstrained labor supply as examined earlier an increase in $L_{U}$ has a negative impact on profit by raising the wage expenditure for unproductive labor, in an economy constrained by labor supply there is an extra channel, i.e., an increase in $L_{U}$ lowers $L_{P}$ , which is the sole source of value creation and thereby reduces total value added.

Next, positive channels that raise profit are as follows. Whereas in an economy with an unconstrained labor supply as examined earlier an increase in $L_{U}$ has a positive impact on profit through raising m, in an economy constrained by the labor supply, there is an extra channel; in other words, an increase in $L_{U}$ lowers $w_{P}$ , thereby lowering the wage expenditure for productive labor. The two profitability-enhancing channels of $L_{U}$ are reflected in $α_{1}$ and $β_{1}$ , respectively. The result in equation (23) states that when the combined effect of the two positive channels, reflected in $α_{1} + 2 β_{1}$ , is sufficiently strong, some of the given labor pool is allocated to the unproductive sector, and the specific amount depends on the parameters appearing on the right-hand side of equation (23); the comparative static results of $L_{U}^{*}$ with respect to the parameters are all as expected. In contrast, if the combined effect is not sufficiently strong, it is optimal to have $L_{U}^{*} = 0$ , and the unproductive sector ceases to exist.

The results in this section, where the ideeller Gesamtkapitalist aims to maximize the aggregate profit, generally conform to the analysis in earlier sections, where the ideeller Gesamtkapitalist aims to maximize the productive sector’s profit. In essence, once the dual nature of unproductive labor is explicitly formulated, the optimal level of unproductive labor can be derived.

7. Conclusion

In this paper, we have presented a model to study the relationship between the unproductive sector and the profitability and accumulation of capital. Our model is unique in two senses. First, it uses the concept of MELT as the central variable and distinguishes the MELT of the productive sector and that of the unproductive sector as a way to measure the value transfer between the two sectors. Second, we explicitly formulate the dual nature of the unproductive sector. In the model, it is only productive labor that directly produces value and surplus value, but the unproductive sector makes an indirect contribution by enhancing the value-creating capacity of productive labor. Accordingly, we are able to derive the optimal size of the unproductive sector. Using the optimal solutions of the model, two sets of results are derived.

The first set of results shows that the unproductive sector exists due to its indirect contribution to value creation. If no contribution is made, the unproductive sector will cease to exist. However, profit will continue to be produced as long as workers’ compensation is less than the value they add, which is possible if the labor supply is flexible; otherwise, the absence of unproductive labor will necessarily squeeze the value added by productive labor so that it equals the wage, eventually driving profit to zero.

These results contribute to the literature on Marxian theory on the relation between exploitation and profit by elaborating on the condition for the positive rate of exploitation. On the other hand, they primarily rely on the assumption that wage rates for productive and unproductive workers are constant. This assumption is relaxed in section 6, but only for the purpose of obtaining definite analytical solutions of a slightly modified model where the intra-capitalist class struggle surrounding the distribution of profit is assumed to be neutral to productive labor’s value creation, in which case $L_{U}$ is the only choice variable. Future study should investigate how the results might change if the assumption of constant wages is relaxed in the original model so that both $L_{U}$ and $m_{U}$ can be used as choice variables.

For the second set of results of the model, we introduce the concept of the unproductive sector-led phase, defined as a phase in which an economy can rely on improving the efficiency of the unproductive sector to boost the profitability and accumulation of capital. According to the analytical result of the model, an economy with an unconstrained labor supply is in an unproductive sector-led phase, whereas an economy constrained by labor supply is in an unproductive sector-led phase only if productive labor’s wage is sufficiently high. That is, when labor supply is perfectly inelastic and the productive sector wage is too high relative to weak technology and market conditions in that sector, the economy can rely on the unproductive sector for accumulation and growth.

These results correspond well to the existing findings that the expansion of the unproductive sector started to take off during the period when real wage growth on average exceeded labor productivity growth. On the other hand, the fact that the expansion of the unproductive sector was followed by economic stagnation and sluggish growth suggests that the excess of real wage growth over the labor productivity growth was not strong enough to warrant the unproductive sector growth.

Lastly, since all the results reported in this paper are theoretical, their validity should be examined against empirical data, which is the current topic of our ongoing research.

Footnotes

Acknowledgements

The authors are grateful to Myungheon Lee for valuable comments on various stages of the development of the paper. We also thank Sohrab Behdad, Ted Burczak, Pedro Cadenas, Ann Davis, Quentin Duroy, Benan Eres, Sergio Cámara Izquierdo, Xiao Jiang, David Laibman, Fred Moseley, Jong-seok Oh, Jiwoong Park, Daniel Saros, Johan Uribe, and participants at the 2018 Summer Conference of the Korean Association for Political Economy in Seoul, the URPE 50th Anniversary Conference in Amherst, and Denison University Economics Research Seminar in Granville for useful comments and suggestions. The usual caveat applies.

Declaration of Conflicting Interests

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

Funding

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

ORCID iDs

Hyun Woong Park

Dong-Min Rieu

1

is an exception who criticizes these existing empirical studies of unproductive labor, putting forth a theoretical argument that the share of unproductive labor would not have increased and that the alleged economic stagnation was not due to the unproductive labor. See note 2.

2

Even in Marxian literature, some, such as Laibman (1992, 1999), are skeptical about the theoretical consistency and empirical usefulness of the "productive/unproductive" distinction. However, as can be seen from the papers cited below, the distinction has played an important role in many theoretical and empirical Marxian analyses. Since dealing with the related controversial issues is not a topic of this paper, we refer the readers to Mohun’s (1996) response to Laibman (1992) and the textual evidence presented in Moseley (1988). provides an alternative, “value-theoretic” approach to the distinction, rejecting the existing empirical studies of unproductive labor that rely on the national income account.

3

See, for example, Moseley (1990, 1997) and Mohun (1996, 2006, for empirical estimations of unproductive labor with a “supervisory-nonsupervisory” labor distinction.

4

From now on, we use simply “value” when we actually refer to “money equivalent of value.”

5

We drop this assumption in section 6.

6

We are using “efficient” in the usual sense of the word as a nuanced expression.

7

Remember that the technological and market conditions of the unproductive sector are reflected in the determination of $m_{U}$ .

8

See , from which it can be easily verified that the sign values of $\frac{\partial Π_{P}}{\partial L_{U}}$ and $\frac{\partial Π_{P}}{\partial m_{U}}$ are indeterminate.

9

In Section 6, we endogenize the wage rate with a slight modification of the model.

10

This mechanism does not exist in the case of the economy with an unconstrained labor supply. In Proposition 1, while m decreases as $m_{U}$ falls, it is not always the case that m decreases all the way to $w_{P}$ ; $m^{*} > w_{P}$ is still possible even when $m_{U}^{*}$ is at the lower bound. One possible way to explain this difference is to pay attention to the mechanism behind $m_{U}^{*} = 0$ , since it is what guarantees $m^{*} = w_{P}$ in Proposition 3. To compare, first see that in the economy with an unconstrained labor supply, $m_{U}^{*}$ is a homogeneous function of the efficiency coefficient of the unproductive sector as shown in equation (7) in Lemma 1. Consequently, the efficiency coefficient being zero directly causes $m_{U}^{*}$ to be zero. In contrast, in an economy constrained by the labor supply, $m_{U}^{*}$ is not a homogeneous function of the efficiency coefficient of the unproductive sector—see equation (13)—and therefore $\frac{\partial m}{\partial L_{U}} = 0$ and $\frac{\partial m}{\partial m_{U}} = 0$ do not necessarily yield $m_{U}^{*} = 0$ . Rather, we obtain the latter by invoking Assumption 2. Once $m_{U}^{*} = 0$ is obtained, $m^{*} = w_{P}$ follows due to .

11

Without doubt, a microfoundation should be pursued in order to connect a social optimizer model to a real capitalist economy that is characterized by a decentralized market. Understanding how the results of our social optimizer model are reproduced by capital competition requires further study.

12

This is the case of, in more general terms, labor productivity being too low compared to wages, or, in Marxian terms, the exploitation rate being too low.

13

While we did not report it here, it can easily be shown that the optimal ratio between unproductive and productive labor time is equal to $α_{2}$ , i.e., $\frac{L_{U}^{*}}{L_{P}} = α_{2}$ , for the economy with an unconstrained labor supply, and $\frac{L_{U}^{*}}{L_{P}^{*}} = α_{2}$ for the economy constrained by labor supply. Moreover, in , it is shown that in the United States and Korea the ratio between unproductive and productive labor time has been less than one for the last couple of decades; thus, $α_{2} < 1$ .

Author Biographies

Hyun Woong Park is Assistant Professor of economics at Denison University in Granville, Ohio. His papers have been published in journals such as North American Journal of Economics and Finance, Structural Change and Economic Dynamics, etc.

Dong-Min Rieu is Professor of economics at Chungnam National University in Korea, where he teaches Marxian economics and history of economic thought. His papers have been published in various journals including Journal of the History of Economic Thought, Metroeconomica, Review of Political Economy and Review of Radical Political Economics.

References

Baumol

William J.

1967. Macroeconomics of unbalanced growth: The anatomy of urban crisis. The American Economic Review 57 (3): 415–26.

Cámara

Sergio

. 2006. A value-oriented distinction between productive and unproductive labour. Capital & Class 90 (3): 37–63.

Cecchetti

Stephen G

Kharroubi

Enisse

. 2015. Why does financial sector growth crowd out real economic growth? BIS Working Paper, No. 490. Basel. Switzerland: Bank for International Settlements. Accessed at: https://www.bis.org/publ/work490.pdf

Cogliano

Jonathan F.

2018. Surplus value production and realization in Marxian theory: Applications to the US, 1987–2015. Review of Political Economy 30 (4): 505–33.

Cullenberg

Stephen

. 1994. Unproductive labor and the contradictory movement of the rate of proﬁt: A comment on Moseley. Review of Radical Political Economics 26 (2): 111–28.

Cuyvers

Ludo

. 1978. A mathematical interpretation of Marxian unproductive labour. Economica 45 (177): 71–81.

Duménil

Gérard

Foley

Duncan K.

Lévy

Dominique

. 2009. A note on the formal treatment of exploitation in a model with heterogeneous labor. Metroeconomica 60 (3): 560–67.

Duménil

Gérard

Lévy

Dominique

. 2011. Unproductive labor as proﬁt-rate-maximizing labor. Rethinking Marxism 23 (2): 216–25.

Dutt

Amitava

. 1991. Unproductive activity and stagnation: A neo-Marxian model. Review of Radical Political Economics 23 (1–2): 95–103.

10.

Engels

Frederick

. 1970. Socialism: Utopian and Scientiﬁc. Moscow: Progress Publishers.

11.

Foley

Duncan K.

1986. Understanding Capital. Cambridge, MA: Harvard University Press.

12.

Foley

Duncan K.

2013. Rethinking financial capitalism and the “information” economy. Review of Radical Political Economics 45 (3): 257–68.

13.

Laibman

David

. 1992. Value, Technical Change and Crisis. New York: M.E. Sharpe.

14.

Laibman

David

. 1999. Productive and unproductive labor: A comment. Review of Radical Political Economics 31 (2): 61–73.

15.

Marx

Karl

. 1981. Capital Volume 3. London: Penguin Classics.

16.

Mohun

Simon

. 1996. Productive and unproductive labor in the labor theory of value. Review of Radical Political Economics 28 (4): 30–54.

17.

Mohun

Simon

. 2006. Distributive shares in the US economy, 1964–2001. Cambridge Journal of Economics 30 (3): 347–70.

18.

Mohun

Simon

. 2014. Unproductive labor in the US economy 1964–2010. Review of Radical Political Economics 46 (3): 355–79.

19.

Moseley

Fred

. 1983. Marx’s concepts of productive labor and unproductive labor: An application to the postwar US economy. Eastern Economic Journal 9 (3): 180–89.

20.

Moseley

Fred

. 1985. The rate of surplus value in the postwar US economy: A critique of Weisskop’s estimates. Cambridge Journal of Economics 9 (1): 57–79.

21.

Moseley

Fred

. 1988. The increase of unproductive labor in the postwar US economy. Review of Radical Political Economics 20 (2–3): 100–6.

22.

Moseley

Fred

. 1990. The decline of the rate of proﬁt in the postwar US economy: An alternative Marxian explanation. Review of Radical Political Economics 22 (2–3): 17–37.

23.

Moseley

Fred

. 1997. The rate of proﬁt and the future of capitalism. Review of Radical Political Economics 29 (4): 23–41.

24.

Okishio

Nobuo

. 2001. Competition and production prices. Cambridge Journal of Economics 25 (4): 493–501.

25.

Olsen

Erik K.

2015. Unproductive activity and endogenous technological change in a Marxian model of economic reproduction and growth. Review of Radical Political Economics 47 (1): 34–55.

26.

Paitaridis

Dimitris

Tsoulﬁdis

Lefteris

. 2012. The growth of unproductive activities, the rate of proﬁt, and the phase-change of the US economy. Review of Radical Political Economics 44 (2): 213–33.

27.

Park

Hyun Woong

Rieu

Dong-Min

. 2018. A puzzle about the monetary expression of labor time: An equilibrating mechanism or just a coincidence? Economics Department Working Paper Series 2018-15. University of Massachusetts Amherst. Accessed at: https://www.umass.edu/economics/publications/2018-15.pdf

28.

Rieu

Dong-Min

. 2008. Estimating sectoral rates of surplus value: Methodological issues. Metroeconomica 59 (4): 557–73.

29.

Rieu

Dong-Min

Park

Hyun Woong

. 2020. Unproductive activities and the rate of surplus value at the industry level in Korea, 1995–2015. Journal of Contemporary Asia 50 (2).

30.

Smith

Murray E. G.

1993. Productivity, valorization and crisis: Socially necessary unproductive labor in contemporary capitalism. Science and Society 57 (3): 262–93.

31.

Wolff

Edward N.

1987. Growth, Accumulation, and Unproductive Activity. Cambridge: Cambridge University Press.