Abstract
This article develops a two-sector model of surplus thermodynamic depth using the Sraffian approach. The effect on the model from ongoing advances in production technology is considered. The conclusion is that, absent countervailing tendencies, the per-worker-second production of thermodynamic depth under capitalism will tend to increase exponentially. The implications of this conclusion for political economists who conceive of “value” as the quantity of labor that may be treated as embodied in commodities by reference to average labor under given conditions of production are considered.
1. Introduction
This article discusses the relation between (a) the quantity of labor that may be treated as embodied in commodities by reference to average labor under given conditions of production (embodied labor); and (b) thermodynamic depth, which is a physical property of commodities having key features in common with embodied labor, notwithstanding that embodied labor is a social quantity of commodities as opposed to a physical one. This article builds on Quentin and Butler-Cole (2023), which introduced the concept of ergomochleusis (i.e., the quantity of thermodynamic depth yielded per period of average labor under prevailing conditions of production). That article considered the possibility of a single change in conditions of production (i.e., either an increase or a decrease in ergomochleusis) between the production of a commodity and such market exchange as may subsequently take place over it. This article—because it is concerned with the systemwide tendential increase in ergomochleusis that takes place in connection with tendential advances in production technology under capitalism—expands and generalizes that analysis.
The structure of the argument is as follows. First, the nature of thermodynamic depth and its unique features as a property of commodities (insofar as a surplus of commodities arises from production) are recalled. Next, a two-sector model of surplus thermodynamic depth using the Sraffian approach is developed, initially assuming no change in conditions of production. The effect upon the model of a single advance in production technology is then explored, followed by consideration of the effect of ongoing advances in production technology. The conclusion is that, absent countervailing tendencies, the per-worker-second production of thermodynamic depth under capitalism will tend to increase exponentially. The final section elaborates on the implications of this conclusion for political economists who conceive of value as the quantity of labor that may be treated as embodied in commodities by reference to average labor under given conditions of production.
Thermodynamic depth is defined in Lloyd and Pagels (1988), and an account of its utility in the context of objective measurement of surplus is provided in Quentin and Butler-Cole (2023); thus, the description of it below is brief. A core point to emphasize is that thermodynamic depth has unique features in common with embodied labor. Readers may be accustomed to dismissing physiocratic approaches to the conceptualizing of surplus on the grounds that such approaches are ahistorical, and with good reason (Burkett 2003), but thermodynamic depth is different. It is so close in nature to embodied labor that the difference between the two can be reduced to a simple conversion factor in which all that is required to accommodate the variance between an ahistorical measure (thermodynamic depth) and a socially determined one (embodied labor) is to recognize the behavior of that conversion factor as a historically contingent structural phenomenon under capitalism. The core contribution of this article is to anatomize that behavior.
2. Thermodynamic Depth
A foundational concept that underpins thermodynamic depth is entropy, although to be clear, the argument here by no means develops an entropy theory of value: entropy is simply a foundational concept that must be understood before thermodynamic depth can be understood.
A quantity of entropy may be thought of as a quantity of uncertainty: for instance, the greater the uncertainty as to the location and direction of travel of each of the particles comprised within an object on a microscopic level, the greater the entropy. A quantity of uncertainty may be characterized as a quantity of information in the sense that it constitutes a determinate quantity of information that is not known. For the purposes of the definition of thermodynamic depth, entropy is best understood as a quantity of information of that kind (i.e., information that is not necessarily useful because it includes vast amounts of chaotic and arbitrary information, such as the location and direction of travel of each of the particles comprised within an object at a microscopic level).
The thermodynamic depth of an object following a process is defined as its coarse-grained entropy at the end of that process minus the fine-grained entropy that it had at the beginning of the process (Lloyd and Pagels 1988). It may be characterized as a tally of the amount of useful information that went into producing a thing, as distinct from the useless, chaotic information that matter in a random state possesses (Lloyd 2007). Alternatively, it may be thought of as the degree of constraint that a process places on the amount of that useless, chaotic information that an object possesses (Quentin and Butler-Cole 2023), and to that extent, if entropy is broadly conceptualized as a measure of the extent to which a thing has arbitrary properties, thermodynamic depth may be broadly conceptualized as a measure of that thing’s non-arbitrary properties.
Quentin and Butler-Cole (2023) deploy the example of a brick, contrasting the fact that it is a rigid oblong with flat sides (a necessary feature in order for it to be a brick) with the precise position of all the particles that make it up (none of which matter). Broadly speaking, the thermodynamic depth of a brick is the total information contained by a lump of clay after it has been shaped into a brick and fired minus the precise position of all the particles that make it up, thereby leaving only the fact that it is a rigid oblong with flat sides. The deliberate, intentional process of turning the clay into a brick serves to constrain to a quantitatively determinate extent the useless, chaotic information it contains, in contrast to, for example, the negligible constraint that would be imposed by leaving the clay in the ground and permitting natural processes to have their effect upon it over geological slow time.
The physicists who first defined thermodynamic depth, Seth Lloyd and Heinz Pagels (1988), presented it as a measure of complexity, but that is not a particularly helpful way to think about it for the purposes of understanding its significance in this context. The point about thermodynamic depth that is crucial for present purposes—and which Lloyd and Pagels went to great pains to emphasize—is that thermodynamic depth satisfies their core requirement of being additive as regards process. In order to satisfy their criteria for a measure of complexity, “the complexity of assembling a car starting from scratch” must be equal to “the complexity of assembling the car from the parts, plus the complexity of assembling the parts starting from scratch” (Lloyd and Pagels 1988: 190). As that quotation vividly illustrates, it is this feature of thermodynamic depth (i.e., that it is additive as regards process) rather than the fact that it measures complexity or constitutes a quantity of information that makes it so acutely (and indeed uniquely) relevant to the topic of objective surplus arising in the context of commodities being produced by means of commodities. However, that relevance is not readily grasped because the language with which thermodynamic depth is described seemingly leads the nature of thermodynamic depth to be misunderstood by readers.
Broadly, the misunderstandings fall into two categories. One category is to misunderstand thermodynamic depth as nothing more than one possible measure among many of the complexity of commodities. This misunderstanding leads to the inference that thermodynamic depth is simply just another concrete property of commodities, like their mass, or their albedo, or their proximity to Nairobi, or whatever such concrete property one may seek to measure. Such concrete properties of commodities are irrelevant to the question of objective surplus arising from the production of commodities by means of commodities since production of heterogeneous commodities is perfectly capable of making things lighter rather than heavier, or less rather than more reflective, or farther away from rather than closer to Nairobi, and likewise it can make things simpler as well as more complex. However, thermodynamic depth is different. Being additive as regards process, it is always increased by production, and this is one of the key features that it has in common with embodied labor. It is not possible for production to reduce the amount of labor that a commodity embodies, and likewise it is not possible for production to reduce a commodity’s thermodynamic depth.
The other category of error is to misunderstand thermodynamic depth as nothing more than one physical quantity among several that are always increased by production, such as energy transformed or entropy arising. In general, these quantities are not embodied by units of output cycling back round to reenter production such that the amount used up can be netted off against the gross amount produced. Alternatively put, there is always an aggregate increase in them but no surplus. The energy transformed by and entropy arising from production generally bear an arbitrary relation to the energy and entropy possessed by the commodity qua production output because generally (i.e., aside from production processes where energy in some form is itself a production output) an arbitrarily large proportion of the energy transformed by and entropy arising from production ends up a property not of the production output but, rather, of the immediate production environment in the form of waste heat. Thus, while these properties are additive as regards process overall, production can and often does reduce the amount of them possessed by commodity itself. Thermodynamic depth, however, is different. It is the object itself that possesses the increased thermodynamic depth that a process results in. This is another key feature that thermodynamic depth has in common with embodied labor. A surplus of embodied labor may be computed because the embodied labor from production cycles back into production as a property of commodities themselves, so as to be netted off to yield a surplus, and (in contrast to entropy arising, energy transformed, and so on) the same is true of thermodynamic depth.
It may be noted that the concept of thermodynamic depth has been criticized on the basis that it is not objective: It depends on an observer’s choice as to the degree of detail in which the macroscopic state of the system in question is defined (Crutchfield and Shalizi 1999). However, that concern does not arise in the case of commodities because that choice is taken out of the observer’s hands. If a particular socially recognized standard of brick is required only to be a rigid oblong of clay with flat sides in order to undergo exchange in the market, it is not for the observer seeking to distort the brick’s thermodynamic depth to demand idiosyncratically that it also be striated with a specific intricate clay banding and reject the ones that do not exhibit that feature.
Nonetheless, a related concern about how the macroscopic state is defined does arise when dealing with multiple units of the same commodity, and it is of the essence of commodities that they are encountered in multiple instances. Having multiple instances of a thing does not mean that one can multiply the thermodynamic depth of a single instance by the number of instances to yield the thermodynamic depth of the totality. Indeed, it was precisely to avoid the easy proliferation of complexity that would arise from a measure of complexity that is additive as regards the parts from which a totality is assembled that Lloyd and Pagels (1988) sought a measure of complexity that is additive as regards process instead. The reason that thermodynamic depth cannot be summed in this way, expressed in the descriptive terms adopted above, is that bringing seven instances of a thing into being does not necessarily require seven times the information constraint (Lloyd and Pagels deploy the example of an organism for which, when it reproduces to produce multiple instances of itself, much of the necessary information constraint is already in place). This outcome has the consequence that, although an individual worker can add a determinate quantity of thermodynamic depth to a single commodity, and all of the workers operating all of the means of production in the world are adding thermodynamic depth to the totality of commodities, the quantities involved in the former phenomenon do not simply scale up to yield the quantity involved in the latter phenomenon. This feature is something that the modeling of surplus thermodynamic depth needs to take into account.
3. Physical Surplus
As is widely recognized, a problem that is encountered when seeking to model the objective surplus that arises when commodities are produced by means of commodities is the transformation problem. The problem is most commonly encountered in circumstances in which that surplus is sought to be computed by reference to the quantities of labor that may be treated as embodied in commodities by reference to average labor under given conditions of production. It arises from the fact that some sectors will be more capital intensive than others, thereby having the consequence that profitability as measured by reference to embodied labor will differ between sectors—a result at odds with the tendency of the market to equalize profitability across sectors. Exponents of the embodied labor approach to computing surplus will therefore generally seek to apportion between sectors a systemwide average profitability that is computed by reference to embodied labor but is apportioned between sectors by means of prices that depart from their embodied labor equivalents to yield equal profit rates between sectors (e.g., see Marx 1981: 254–72).
The problem that arises when seeking to effect this apportionment is that computing the appropriate adjustment requires that capital in each sector be valued by reference to the price that was paid for it rather than the labor it embodies. Thus, in order to transform embodied labor into prices, one needs to already have transformed embodied labor into prices—a chicken-and-egg problem that is, strictly speaking, incapable of a solution, since there are more unknowns than independent equations (Sinha 2016).
However, the problem does not need to be solved if the quantities of embodied labor are simply dropped from the model, and this was the revolutionary advance made by twentieth-century political economist Piero Sraffa. Sraffa showed that, given a systemwide wage rate, relative prices can be derived between sectors to yield a uniform profit rate systemwide directly from the quantities of inputs required per unit of output in each sector. There is no need for quantities of embodied labor to play a role in the model at all (Sinha 2016). It is Sraffa’s approach that is adopted here. In other words, physical surplus is modeled by reference to physical quantities of each commodity.
A clear and simple articulation of that approach can be found in Hahnel (2017), whose model is adopted and elaborated on here. Hahnel begins his exposition by showing the quantity of each good in a two-sector economy and the quantity of labor required to make each good, as follows:
In the above chart, a(ij) is the amount of good i required to produce a unit of good j, and L(j) is the number of worker-hours required to produce a unit of good j. The figures posited yield a surplus of each good. Hahnel shows that it is sufficient to posit a wage rate, and the model will yield a profit rate across both sectors and relative prices between sectors. For example, if the wage rate is 0.5 units of good 2 per hour, the profit rate is 12.6 percent, and the price of good 1 relative to good 2 is 1.19.
The Sraffian approach treats wages as a distribution out of profits, which means that there is no need to specify a wage basket composition (the wage rate of 0.5 units of good 2 per hour merely deploys good 2 as a nonmonetary numeraire for wage rate rather than specifying a wage basket). There is consequently no need to posit, determine, or even constrain sector sizes. Nonetheless, it is possible to posit a wage basket composition, and if one does so, it is possible to posit (within certain consequent constraints) relative sector sizes in terms of number of workers such that the output will satisfy the input needs of both sectors as regards means of production and also fill the workers’ wage baskets. That being the case, if one proceeds to posit a total number of workers, it is possible to determine total output across both sectors, and also total input across both sectors, including both means of production and wage goods expressed in terms of physical quantities of both goods.
For example, suppose a wage basket contains equal quantities of both goods and suppose there are two workers in sector 1 for every worker in sector 2. Accordingly, 2 units of each good are produced per hour for every three workers. Across both sectors, 1 unit of good 1 and 1.2 units of good 2 are used up as means of production to produce these 2 units of each good. In order for a wage basket containing equal quantities of both goods to amount to a price equal to 0.5 units of good 2 per hour, where the price of good 1 relative to good 2 is 1.19, the wage good input per worker-hour is 0.2283 units of each good. Accordingly, the wage good input for three workers is 0.6848 units of each good, bringing the total of good 1 used up for every 2 units produced to 1.6848 and the total of good 2 used up for every 2 units produced to 1.8848. The surplus across those three workers and 2 units per sector of output (corresponding to that 12.6 percent profit rate) is therefore 0.3152 units and 0.1152 units of goods 1 and 2, respectively.
Next, it is a simple step to allocate a thermodynamic depth to each good, but a complicating factor is the feature of thermodynamic depth whereby summing thermodynamic depths will yield figures far in excess of the thermodynamic depth of the totality (as to which, see above). One way to reflect this feature of thermodynamic depth is to raise the number of units to a power between zero and 1—say 0.1—and use the resulting figure as a multiplier instead of the unadjusted number of units. Adopting that approach requires also adopting total numbers of workers for the model rather than doing computations on a per-worker basis—say 180 workers in sector 1 and 90 workers in sector 2. If the thermodynamic depth of good 1 is 20,000 joules per kelvin per unit and the thermodynamic depth of good 2 is 18,000 joules per kelvin per unit, the total output of this economy in terms of thermodynamic depth (i.e., the depth of each unit multiplied by number of units raised to the power of 0.1) is 63,872 joules per kelvin per hour. The thermodynamic depth of the means of production used up and the wage baskets consumed is 63,122 joules per kelvin, giving rise to 750 joules per kelvin of net surplus thermodynamic depth.
These figures yield an ergomochleusis (i.e., a quantity of thermodynamic depth produced per period of average labor) (Quentin and Butler-Cole 2023). That 63,872 joules per kelvin of gross thermodynamic depth was produced by 270 workers in an hour, which works out to 0.0657 joules per kelvin per worker-second.
4. The Effect of Changes in Conditions of Production
Capitalist competition compels capitalists to pursue technical innovation (Marx 1976). One consequence of such advances will be the same output produced with less labor. For that reason alone, ergomochleusis will generally tend to increase, although it is far from the only reason that labor savings will cause ergomochleusis to increase. This is because labor savings almost invariably rely on newly developed equipment, and since thermodynamic depth is additive as regards process, newly developed equipment is going to be of greater thermodynamic depth than the equipment used to create it. The consequence is that the thermodynamic depth of aggregate output will increase upon the adoption of new production techniques irrespective of the reduction in labor requirement.
However, as an ultra-conservative assumption, the analysis here proceeds without such additional increase beyond the increase in ergomochleusis necessarily arising from a reduced labor requirement. This approach is noted here in order to draw attention to that conservative assumption at an appropriate juncture and also to reinforce the point that thermodynamic depth is additive as regards process. In the case of, say, energy use or resource depletion as a measure of output, wherein some new process might be vastly more energy- or resource-efficient than the previous one, technical advances might cause output to drop when computed on a per-worker-second basis by reference to those measures. This will not happen in the case of ergomochleusis (i.e., thermodynamic depth produced per worker-second), whatever efficiencies are introduced. Ergomochleusis will always rise upon the successful introduction by capitalists of labor-saving production technology.
The model set out above may be deployed to illustrate the effect on ergomochleusis of an introduction of labor-saving technology. Suppose that the quantity of labor required to produce good 2 in Hahnel’s example drops from 0.5 to 0.3333 because of an improved production technique in sector 2. If the wage rate is still 0.5 units of good 2 per hour, the profit rate is now 20.4 percent, and the price of good 1 relative to good 2 is 1.3194. One-third of the workers in sector 2 have been laid off so that the same number of units of goods 1 and 2 are being produced. To return to ergomochleusis, with the same 180 workers in sector 1 but now only 60 in sector 2 and the same thermodynamic depth attached to each, the hourly output of thermodynamic depth is still 63,872 joules per kelvin, but that now represents 0.0739 joules per kelvin per worker-second.
The reduction of labor costs through technical improvements under the compulsion of capitalist competition means, then, that the per-worker-second production of thermodynamic depth will tend to rise. With two maximally simplifying assumptions, it is possible to go even further than that. The first assumption is that labor-saving technologies are introduced at regular intervals. The second assumption is that the number of worker-seconds required to produce the output of the system is reducing by a constant proportion for each labor-saving technology introduced (clearly it cannot be a linear reduction since that would cause the number of worker-seconds required to produce the output of the system to drop below zero). On the basis of these assumptions, systemwide ergomochleusis will necessarily increase exponentially over time.
5. Value
Quantities of labor treated as embodied in commodities by reference to average labor under given conditions of production may be referred to as value (Marx 1976). As already mentioned, value so defined is not a required term for the purposes of modeling objective surplus, and if it is used for that purpose, it gives rise to an insoluble technical problem. Marxists are nonetheless generally inclined to retain the concept of value since it is central to the Marxian critique of political economy (Marx 1976).
It must be emphasized that value—defined as quantities of labor to be treated as embodied in commodities by reference to average labor under given conditions of production—is a social property of commodities under capitalist social relations and not a physical one. To speak of a commodity as embodying a quantity of human labor is not to speak of any concrete property of the commodity as a physical object but to refer to the fact that, as at exchange, the commodity may be treated as representing a determinate quantity of average labor as deployed in conjunction with the means of production required to produce that commodity under prevailing conditions of production.
However, there is a tension in this concept since commodities, labor, and prevailing conditions of production are all material phenomena, and accordingly, value is a social quantity with a material substrate. This has led to extensive debate among Marxists around certain pressure points having to do with the relation between value and its material substrate. Perhaps the most salient such pressure point for analytical purposes in the context of actually existing capitalism today is the fact that many outputs associated with high levels of capitalist profit (and therefore presumptively embodying significant quantities of value) behave like commodities but are not quantitatively constrained by the resources allocated to production in the way that physical goods are. In the case of such “immaterial” outputs, the link between the commodity-like object over which exchange is taking place and a determinate quantity of average labor under prevailing conditions of production is broken. A diversity of positions have been adopted by Marxists to deal with this phenomenon, including (a) the claim that value itself is undergoing a crisis of measurability (Part II of Pitts [2018] surveys this tendency); (b) emphasis on the analytical purposes that may be served by the concept of value as a social relation even absent a determinate quantitative relation between the commodity and average labor under prevailing conditions of production (Part I of Pitts [2018] surveys this tendency); and (c) reliance on the traditional bifurcation of total labor into a category of labor that is productive of surplus and an “unproductive” category of labor that is remunerated out of surplus, this latter category being analyzed to include the labor implicated in immaterial commodity-like outputs (Rotta and Teixeira 2019; Quentin 2024).
The degree to which the conclusions drawn herein are likely to be of relevance to work in the Marxist tradition will vary according to the position the work in question adopts with regard to the relation between value and its material substrate. Some positions Marxists may take on that relation will have the consequence that the conception of value articulated here (i.e., quantities of labor treated as embodied in commodities by reference to average labor under prevailing conditions of production) is simply not the same conception of value as the one that they are using in their work. However, for those who share the conception of value articulated here, the argument in this article has the following important implications.
While value (so defined) is a social quantity, it bears a relation to a physical quantity (i.e., the thermodynamic depth of commodities), that relation taking effect via ergomochleusis. This is trivially true where conditions of production are treated as static since under static conditions of production there will simply be a per-worker quantity of thermodynamic depth produced every second of average labor, and the relevant ergomochleusis will function as a conversion factor. Accordingly (again, assuming no change in conditions of production), the impulse on the part of value to valorize itself (i.e., to create a surplus of value) (Marx 1976) may be understood as mapping directly onto an impulse on the part of the thermodynamic depth of commodities under capitalist social relations to create a surplus of thermodynamic depth.
This is, as we say above, trivially true and would not be interesting if it were the case that ergomochleusis varies arbitrarily with changes in conditions of production. If ergomochleusis varied arbitrarily with changes in conditions of production, the only thing preventing the relation between value and thermodynamic depth being as arbitrary as the relation between value and any other physical property of commodities would be the assumption that conditions of production remain static. However, ergomochleusis does not vary arbitrarily with changes in conditions of production. It increases as production technology advances, and (as shown above) has a prima facie tendency to do so exponentially.
On the basis of the analysis in this article, this dynamic appears to be an inevitable structural tendency of physical matter falling within circuits of capitalist production. Such a tendency would require countervailing tendencies if it were not to result in an accelerating feedback loop of exponentially accumulating thermodynamic depth. For example (as in the model above), the number of workers operating means of production might have a tendency to fall as a proportion of the working population and/or the overall population. Alternatively, (a) consumption habits might change to soak up additional output; (b) there might be a tendential shift of investment to sectors producing low-depth outputs; or (c) there might be a composite and changing bundle of countervailing tendencies co-evolving with the system and displaying all sorts of spatial and sociological specificities as history unfolds, opportunistically making use of phenomena orthogonal to simple class antagonism, such as racial- or gender-based structural hierarchies.
However, for the purposes of stating the conclusions of this article, it does not matter what these countervailing tendencies are; the point is that the production of commodities in the capitalist mode appears, on the physical level of the thermodynamic depth of those commodities, to be one of those physical/material systems in which there is an exponential tendency and countervailing tendencies, yielding an equilibrium or oscillating result. These sorts of systems are well known in various branches of the natural sciences—a classic example is when an otherwise potentially exponentially increasing population of prey is kept in check by a population of predators.
To do political economy in respect of capitalism as a social system using value as an objective measure of social surplus, as Marxists do, is seemingly therefore to simultaneously investigate something that is both (a) a historically specific mode of social production, and (b) a material system in which there is an exponential tendency and countervailing tendencies, yielding an equilibrium or oscillating result on the level of its physics. That physical/material behavior of the system is reliant on the existence of social relations under capitalism, but the fact that those relations are social does not make them any less material in their material effects. And one of their material effects, so this article seeks to demonstrate, is to bring into being a system wherein the thermodynamic depth of matter falling within its circuits has taken to making more of itself, with a tendency to do so exponentially.
Footnotes
Declaration of Conflicting Interests
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The authors received no financial support for the research, authorship, and/or publication of this article.
