Marx was well aware of, and even expressed admiration for, the fact that compared to all previous economic systems capitalism had greatly increased the pace of technological change. He assumed that individual capitalists are hard driven to adopt any new technology that lowers their cost of production because this would give them a temporary advantage over their competitors, who, in turn, would be quick to adopt cost-reducing changes for fear of being driven out of business. He also viewed capital-using, labor-saving technical change, i.e. capital deepening, or automation, as a means for capitalists to replenish the ranks of what he called the “reserve army of the unemployed” to keep wages down. However, Marx thought that what was in the interest of individual capitalists temporarily might not necessarily serve their collective interests in the long run.
As we have seen, Marx traced the source of profits to the properties of one input capitalists purchase, labor power, which Marx believed has the unique ability to produce more value when used than the number of hours it takes to produce it. However, when we apply this reasoning to capital-using, labor-saving technical change a problem appears to arise.
Go back to our example from chapter 1 of a capitalist making shirts. When we calculated his rate of profit we found that when his employees work 20 hours making shirts for wages sufficient to purchase a subsistence bundle of goods it takes 15 hours to produce, using sewing machines it took 50 hours to produce and cloth it took 30 hours to produce, and when everything exchanges for the number of hours it took to produce it, his profit rate will be: (50+30+20) minus (50+30+15) all divided by (50+30+15) which gives 5.263%. But what if a shirt capitalist discovers that by equipping his employees with better sewing machines, machines that can sew cloth faster, each employee can turn more cloth into more shirts every hour? If these higher quality sewing machines (which for convenience we continue to assume last only a year) take 60 hours to produce, and if employees, working 20 hours with faster machines can turn cloth it took 40 hours to produce into twice as many shirts, what will happen? The cost of making the shirts has risen from (50+30+20) = 100 to (60+40+20) = 120, or 20%. But the number of shirts produced has doubled, i.e. increased by 100%. This means that the cost per shirt using the new technology is less than when using the old technology, and would be adopted by profit maximizing capitalists. However, if everything eventually goes back to selling for the number of hours it took to make it, shirt capitalists’ rate of profit will be: [(60+40+20) − (60+40+15)]/(60+40+15) = 4.348%. What are we to make of this? There are two possibilities:
1. One possibility is that this is not what happens to the rate of profit when capitalists engage in capital deepening – which was the reaction of most economists after Capital was published. Most economists reasoned that if capitalists engage in capital deepening it must be because it increases labor productivity, in which case, if the real wage remains constant it should increase, not decrease, the rate of profit in the economy – and therefore something must be wrong with Marx’s theory if it suggests otherwise.1
2. The other possibility is that the rate of profit will eventually decline as capitalists engage in capital deepening, which is what Marx believed to be the case. In short, Marx stuck with his theory and predicted that such a tendency would eventually manifest itself. This became known as the “tendency for the rate of profit to fall” (TRPF), which is often referred to by Marxists as an “internal contradiction” in capitalism, and is cited by some Marxists as one reason capitalism is prone to crisis, or contains the seeds of its own destruction.
To his credit Marx recognized that if capital deepening increases labor productivity this would change how much profit could be gained by hiring a given amount of labor power because it would decrease the amount of labor time it takes to produce the same real wage bundle of goods, and therefore decrease the “exchange value” of labor power which capitalists must pay. In our example, the amount of time it takes to make the shirts in our subsistence wage bundle is now less than it used to be, and therefore capitalists would only have to pay something less than 15 for 20 hours of labor power. Marx called this effect an increase in the “rate of exploitation,” and treated it as “counteracting” the “tendency for the rate of profit to fall” produced by a rise in what he called the “organic composition of capital.”
Marx’s rate of profit, r(M), based on the assumption that all commodities including labor power exchange according to their labor values, reduces to r(M) = s’(1−q). The “rate of exploitation,” s’, is the ratio of the fraction of an hour an employee works generating profits for her employer to the fraction of an hour she works for herself so to speak. The “organic composition of capital,” q, is the ratio of “constant capital” – expenditures on non-labor inputs purchased from other capitalists according to the total number of hours it took to make them – to “total capital” – expenditures on all inputs, including labor power, all paid for in accord with the total number of hours required to produce them.
From the formula r(M) = s’(1−q) it should be apparent that no matter how much capital deepening might increase q, if it simultaneously increases s’ sufficiently it need not reduce Marx’s rate of profit. Nonetheless, Marx opined that eventually capitalism would kill the goose laying its golden eggs by reducing the amount capitalists spend hiring labor power relative to buying produced inputs for them to work with from other capitalists.2
Some Marxist economists3 continue to search for evidence that capital deepening is finally lowering the rate of profit in the economy, i.e. for empirical evidence that the “tendency” has finally won out over any “counteracting tendencies.” We return to this question after presenting the Sraffian theory of technical change.
The Sraffian framework is well suited to analyzing if any new technological discoveries will be adopted or rejected by capitalists, as well as the effects of technological changes which are adopted on prices, income distribution, and economic productivity.
To know whether capitalists will replace an old technique with a new one, we simply compare the cost of producing a unit of output using the old and new technologies at current prices and the current wage rate. If the new technology lowers production costs it will be adopted, and if not it will be rejected. In this respect Sraffa and Marx made the same assumption about how individual capitalists go about deciding to adopt or reject a new technology, which is also what other economists have always assumed.
For example: Suppose when r = 0%, w = 0.691, p(1) = 1.273, and p(2) = 1 a capitalist in industry 1 discovers the following new capital-using, labor-saving technique:
a’(11) = .3
a’(21) = .3
L’(1) = .8
Will capitalists in industry 1 replace their old technique with this new one? The new technique is capital-using since a’(21) = 0.3 > 0.2 = a(21). But it is labor-saving since L’(1) = 0.08 < 1.0 = L(1). The extra capital raises the private cost of making a unit of good 1 by: (0.3−0.2)p(2) or (0.3−0.2)(1) = 0.1000. The labor savings lowers the private cost of making a unit of good 1 by: (1.0−0.8)w or (0.2)(0.691) = 0.1382. Which means that under these conditions this new capital-using, labor-saving technology lowers the overall private cost of producing good 1 and would be adopted by a profit maximizing capitalist in industry 1. Moreover, all other capitalists in industry 1 would rush to adopt the new technique as well in order not to be underpriced and outcompeted.
However, this does not mean that the new technique would be adopted under different circumstances. For example, suppose r = 20.8%, w = 0.400, p(1) = 1.137, and p(2) = 1, and suppose capitalists in industry 1 discover the same new technique. As before, the extra capital raises the private cost of making a unit of good 1 by: (0.3−2)p(2), or (0.3−0.2)(1) = 0.1000. But now the labor savings lowers the private cost of making a unit of good 1 by: (1−0.08)w, or (0.2)(0.400) = 0.0800, which means that under these conditions the new technique raises rather than lowers the overall private cost of making a unit of good 1, and would not be adopted by profit maximizing capitalists.4
Sraffa (1960) himself clarified how new technologies affect relative prices by distinguishing between basic goods which either directly or indirectly enter into the production of all goods, and non-basic goods which do not: Sraffa demonstrated that technical changes in a basic industry will necessarily affect the entire relative price system. While technical changes in a non-basic industry will simply lower its own relative price, and the prices of any other non-basics if it should happen to enter into their production. To keep things as simple as possible, we assume throughout that all industries are basic, as both industries are in the example we have been using. And therefore any technical change introduced in any industry will affect the entire relative price structure of the economy.
To continue with the same example, what will happen to relative prices if the original conditions are r = 0, w = 0.691, p(1) = 1.273, and p(2) = 1 and the new technology is adopted by capitalists in industry 1? Writing the new price equations for the economy where the new cost reducing technology is being used by all capitalists in industry 1 we have:
(1 + r’)[.3p(1)’ + .3(1) + .8w’] = p(1)’
(1 + r’)[.2p(1)’ + .4(1) + .5w’] = 1
As always, before we can calculate relative prices and the rate of profit in the Sraffian model we must stipulate a value for the hourly wage rate. Assume the real wage rate remains the same, i.e. that w’ = 0.691. Under this assumption we have:
(1 + r’)[.3p(1)’ + .3(1) + .8(.691)] = p(1)’
(1 + r’)[.2p(1)’ + .4(1) + .5(.691)] = 1
Which can be solved for: p(1)’ = 1.232 and r’ = 0.8%. Not surprisingly, once all capitalists in industry 1 are using the new technology the price of good 1 relative to the price of good 2 changes: Now a unit of good 1 exchanges for a little less of good 2: p(1)’ = 1.232 < 1.273 = p(1).
How the introduction of cost reducing technical changes might affect the rate of profit in the economy puzzled political economists for over a hundred years. It long appeared that the answer to this question even in a simple framework where homogeneous labor is the only primary input was very complicated, and quite possibly not definitive. A capitalist in a particular industry would not adopt a new technology unless it was less costly and therefore more profitable than the existing technology in the short-run, i.e. unless it was cost reducing at current prices and the current wage rate. However, once all capitalists in the industry adopt the new, lower-cost technology, absent barriers to entry and exit the entire price system would presumably adjust to eliminate “super profits” in that industry. In which case who could say whether at these new prices, p’, the new uniform rate of profit in the economy, r’, would turn out to be higher or lower than the old uniform rate of profit, r. In our example it turned out that the new uniform rate of profit is higher: r’ = 0.8% > 0.0% = r. But who is to say if this will always be the case? Just because the “counteracting tendencies” outweighed the “tendency for the rate of profit to fall” in this particular example of a capital-using, labor-saving technical change, does not mean they will in every case, or most cases – or at least that is what many Marxist economists thought.
However, in 1961, Nobu Okishio proved a theorem that should have put matters to rest once and for all. Okishio proved that any technological change that reduces costs at current prices and the current wage rate, and therefore would be adopted by profit maximizing capitalists, would either raise the uniform rate of profit in the economy, or leave it unchanged as long as the real wage remained constant.5 In other words, the result in our example is no accident: If the new technology is cost reducing at current prices and the current wage rate, and if the real wage remains the same, the new uniform rate of profit in the economy will either be higher or remain the same, but never be lower. In Marxian terms, Okishio proved that what had always been referred to as “counteracting tendencies” necessarily, and always, must overcome any “tendency” for the rate of profit to fall caused by capital deepening. In short, it turns out that the theory that greedy capitalists pursuing greater profits through automation would thereby witlessly kill the goose laying their golden eggs simply does not fly. To mix metaphors, it was a red herring.6
Returning to our example, the remaining question is how technological changes affect productivity. This question did not seem to concern Marx himself, nor most of his followers. Perhaps this is because like many others they assumed that any capital deepening that lowered production costs, and therefore would be adopted, would necessarily increase labor productivity. They also were not terribly interested in criticizing capitalism for its lack of efficiency, concentrating instead on “internal contradictions” leading to “crises” and the fact that labor is “exploited.” In any case, it is ironic that this is the one place where labor values can be of help.
In the simple framework we are using if the labor value of every good is lower after a new technology has been adopted by all firms in some industry, this means it takes us fewer hours in grand sum total, i.e. both directly and indirectly, to produce each and every good. In short, if all labor values fall, labor productivity is higher. Conversely if all labor values rise, the new technology has decreased labor productivity. Moreover, we don’t have to worry that a technical change in one industry might lower the labor value of some goods but increase the labor value of other goods. In the industry where the change took place the labor value of the good it produces either fell, rose, or remained the same. If it fell and it enters into the production of another good, either directly or indirectly, the value of that good must also fall since there was no change in the input coefficients for the production of that good. If it does not enter into the production of another good, either directly or indirectly, the value of that good will remain the same. Similarly, if the labor value of the good where the technical change was adopted increases, the value of other goods must either increase or remain the same. And of course if the new technology did not change the labor value of the good it produces then no values will change.
So did the technical change we have been discussing in our example increase or decrease labor productivity? In chapter 1 we calculated the labor values for the original technologies in our example to be: V(1) = 1.843 and V(2) = 1.447. We can now calculate the new labor values after the new technology is adopted by all capitalists in the first industry by solving the new value equations:
V(1)’ = .3V(1)’ + .3V(2)’ + .8
V(2)’ = .2V(1)’ + .4V(2)’ + .5
Which yield: V(1)’ = 1.750 and V(2)’ = 1.417.
Since both labor values are now smaller than before, the technical change clearly increases labor productivity. However, comparing labor values before and after any technical change only tells us if labor productivity has increased or decreased. It does not tell us quantitatively how much productivity has changed.
Part iii of theorem 18 proved in Hahnel (2017) now provides an easy way to calculate the size of changes in labor productivity in the economy as a whole stemming from any technological change: The size of the change in overall labor productivity is ρ(l) = (1−ß’) where ß’ is the dominant eigenvalue of the matrix (A’+b̃L’), where {A’, L’} defines the technology for the economy after the technical change is adopted, and b̃ is an hourly real wage bundle large enough to reduce the rate of profit in the original economy, prior to adoption of the new technology, to zero.
Suppose, for example, we find that ß’ equals 0.95. In this case adoption of a new technology will increase labor productivity by ρ(l) = (1.00−0.95) = 0.05, or 5%. If people next year work the same number of hours as they did this year they will produce 5% more goods. Or, if people next year consume exactly what they consumed this year they could work 5% fewer hours than last year. On the other hand, suppose we find that ß’ equals 1.05. In this case labor productivity has decreased by ρ(l) = (1.0−1.05) = −0.05, or 5%, and people will either produce 5% less goods, or have to work 5% more hours. Returning to our particular example, we discovered in chapter 2 that if workers are paid a real wage bundle of 0.333 units of good 1 and 0.266 units of good 2 per hour they work, the rate of profit in the economy will initially be zero. This allows us to formulate (A’+b̃L’) and calculate its dominant eigenvalue, which turns out to be 0.984166. Which means that if capitalists in industry 1 adopt the new technology in our example, overall labor productivity will increase by ρ(l) = (1.00−0.984166) = 0.015834, or 1.5834%.7
We now know that replacing the old recipe for making good 1 with the new one increases labor productivity. We know this because it reduces the labor values of both goods, and because ρ(l) = + 0.015834, meaning that labor productivity has increased by 1.5834%. We also know that profit maximizing capitalists in industry 1 would adopt this new technology if p(1) = 1.273, w = 0.691, and p(2) = 1, and in so doing would have served the social interest by promoting what we might call dynamic efficiency. But we also know that profit maximizing capitalists in industry 1 would reject this new technology if p(1) = 1.137, w = 0.400, and p(2) = 1. In which case capitalists in industry 1 would obstruct the social interest by rejecting a new technology that increases productivity. What are we to make of this?
Adam Smith actually envisioned two, not one, invisible hands at work in capitalist economies: One invisible hand promotes static efficiency, and the other promotes dynamic efficiency. Smith not only hypothesized that the micro law of supply and demand would lead competitive capitalist economies to allocate scarce productive resources to the production of different goods and services efficiently at any point in time, i.e. achieve static efficiency; he also believed that competition would drive capitalists to search for and adopt new, more productive technologies thereby raising economic efficiency over time, i.e. achieve dynamic efficiency. Smith assumed that all new technologies that reduce capitalists’ costs of production – and only technologies that reduce capitalists’ production costs – improve the economy’s efficiency. We have just discovered that apparently Smith’s second “invisible hand” is imperfect just like his first!8 In some circumstances capitalists will serve the social interest by adopting new technologies that increase productivity, but in some circumstances they will not – they will commit what we might call “sins of omission.” It turns out that capitalists will also sometimes commit what we might call “sins of commission.” In some circumstances capitalists will serve the social interest by rejecting new technologies that decrease productivity, but in some circumstances they will adopt technologies that lower productivity. How are we to make sense of this?
To solve the puzzle we start with what we know: We now know the new technology in our example made the economy more efficient. We know the new technology was capital-using and labor-saving. And we know capitalists in industry 1 embraced it when the wage rate was 0.691 (and the rate of profit was zero), but rejected it when the wage rate was 0.400 (and the rate of profit was 20.8%). The reason for capitalists’ seemingly contradictory behavior is now clear: When the wage rate is higher, the savings in labor costs because the new technology is labor-saving are greater – and great enough to outweigh the increase in non-labor costs because the new technology is capital-using. But when the wage rate is lower, the savings in labor costs are less, and no longer outweigh the increase in non-labor costs. Apparently the price signals {p(1), p(2), and w} in the economy in the first case lead capitalists to make the socially productive choice to adopt the new, more productive technology, whereas different price signals in the second case lead capitalists to make the socially counterproductive choice to reject the new, more productive technology.
No matter how efficient, or socially productive a new capital-using, labor-saving technology may be, it is clear that if the wage rate is low enough (because the rate of profit is high enough) such an efficient technology will become cost-increasing, rather than cost-reducing, and capitalists will reject it. Similarly, no matter how inefficient, or socially counterproductive a new capital-saving, labor-using technology may be, if the wage rate is low enough (because the rate of profit is high enough) the inefficient technology will become cost-reducing, rather than cost-increasing, and capitalists will embrace it.9 In other words, Adam Smith’s second invisible hand works perfectly when the rate of profit is zero but cannot be relied on when the rate of profit is greater than zero. Moreover, as the rate of profit rises (and consequently the wage rate falls), the likelihood that socially efficient capital-using, labor-saving technologies will be rejected, and the likelihood that socially counterproductive capital-saving, labor-using technologies will be adopted by profit maximizing capitalists increases.
Marxists and Sraffians are in agreement that if and only if a new technology reduces costs of production will profit maximizing capitalists adopt it. Marxists and Sraffians are also in agreement that after a technical change is adopted by all in an industry there will be a general adjustment in relative prices to eliminate super profits in the innovating industry, and once again yield a uniform rate of profit throughout the economy. After which, there is a parting of the ways in the Sraffian and Marxian analyses of technical change.
All Sraffians recognize the validity of the Okishio theorem, which lays to rest any concerns that capital deepening will reduce the rate of profit in the economy if the real wage is held constant. Some Marxist economists persist in futile attempts to resuscitate a “tendency for the rate of profit to fall,” which we will address in the next chapter. Marxist economists have been largely uninterested in the effects of technical change on productivity – apparently willing to accept that Adam Smith’s second invisible hand does work. Consequently it has fallen to Sraffians to emphasize that capitalists can not be trusted to always serve the social interest with regard to adopting and rejecting new technologies. The higher the rate of profit and lower the wage rate, the more likely it becomes that capitalists will commit both sins of omission – fail to adopt capital-using, labor-saving technologies that are more productive – and sins of commission – adopt labor-using, capital-saving technologies even though they are less productive.
1 As we discover below, the implicit assumption shared by Marx and most economists that technological changes that increase labor productivity always reduce production costs, and technological changes that reduce production costs always increase labor productivity does not hold. But this is a different problem from the one discussed here, which is the effect of capital deepening on the rate of profit in the economy.
2 It is interesting how a simple failure in mathematical reasoning can continue to fuel a misconception. When properly interpreted the formula r(M) = s’(1−q) clearly indicates that even if production were completely automated and therefore q approaches one and (1−q) approaches zero, r(M), Marx’s rate of profit, would not have to decline as long as s’ increases infinitely – which it could if the number of hours it takes to produce a daily subsistence bundle approaches zero. Yet many TRPF disciples remain convinced that since a rising q displaces ever more labor power from production, and labor power is the sole source of profit, this must eventually depress the rate of profit. Many in the steady-state and de-growth movements make a similar mistake when thinking about the relationship between growth of output produced and growth of environmental throughput. They reason that continued growth of output produced must increase environmental throughput no matter how much we “decouple.” But the simple mathematical truth is that as long as the rate of growth of throughput efficiency is as high as the rate of growth of output, throughput will not increase but remain constant. See part 2 in Hahnel 2017.
3 A recent example is Fred Moseley 1991.
4 Astute readers may already be curious about two issues this result raises: (1) Are there circumstances under which the new technology would neither lower nor raise the cost of production? Yes. In this example if w = .500, in which case p(1) = 1.190 and r = 12.6%, it will cost the same amount to produce a unit of good 1 using the new and the old technology. More importantly: (2) Does this new technology increase or decrease economic productivity? If it increases productivity, what are we to make of the fact that in the second circumstance capitalists would reject it? If it decreases productivity, what are we to make of the fact that in the first circumstance capitalists would adopt it? We will shortly address this issue.
5 See Okishio 1961 for the original proof. For a proof accompanied by an explanation of the intuition behind the result, as well as clarification of a little noticed conundrum the Okishio theorem raises, see part 1 in Hahnel 2017. Note that this does not mean the rate of profit might not fall. It simply means if it does fall it must be due to some other cause, such as a rise in the real wage, not because capitalists introduced new capital-using, labor-saving technologies.
6 A number of Marxist economists have attempted to save TRPF theory from the death blow delivered by the Okishio theorem to no avail, as discussed in the next chapter.
7 The components for A’ are a(11)’ = .3, a(12)’ = .2, a(21)’ = .3, and a(22)’ = .4. The components for b̃L’ are b(1)L(1)’ = .33(.8) = .26, b(1)L(2)’ = .33(.5) = .16, b(2)L(1)’ = .26(.8) = .213, and b(2)L(2)’ = .26(.5) = .13. Adding corresponding components yields a(11)’ + b(1)L(1)’ = .566, a(12)’ + b(1)L(2)’ = .366, a(21)’ + b(2)L(1)’ = .513, and a(22)’ + b(2)L(2)’ = .533. Programs to calculate eigenvalues for matrices are now available online. I used the Comnuan calculator at http://comnuan.com/cmnn01002/. Entering these four coefficients of our 2×2 matrix into the Comnuan calculator yields a dominant eigenvalue of .984166.
8 Smith’s first invisible hand fails whenever there are externalities, i.e. effects on “external parties” to a market exchange. This problem was originally pointed out by Alfred Pigou, who was also the first to propose what are still called “Pigovian” taxes in his honor to correct for inefficiencies caused by external effects. See Hahnel 2007 and 2014b for reasons to believe this problem is far more serious than most mainstream economists are willing to admit.
9 For proof that in a simple Sraffian model if and only if the rate of profit is zero will there be a one-to-one correspondence between technological changes which increase labor productivity and cost reducing technological changes see theorem 4.9 in Roemer 1981. For proof that as the rate of profit rises the likelihood that capitalists will commit ever more sins of omission and sins of commission see theorem 4.10 in Roemer 1981.