Tyler Cowen has just posted a conversation with Nicholas Bloom, a Stanford economist interested in economic growth. That gives me a chance to revise a working paper I’d written last year, Stagnation and Beyond: Economic growth and the cost of knowledge in a complex world [2], which is based on a paper Bloom had published along with three colleagues, Are Ideas Getting Harder to Find? [3]. I took two of their case studies, Moore’s law and drug discovery, examined them in an informal manner and concluded that (much of) the increasing difficulty of discovering new ideas can be attributed to the cost of finding out more about the world.
At the time I wrote that paper I was unaware of Paul Romer’s 1992 article, Two Strategies for Economic Development [4]. He develops his argument with a toy model that is similar in spirit to the idea I’d advanced in my working paper. The objective of this post is to update that idea in view of Romer’s model by deploying some toys of my own.
First I set the stage with some passages from Cowen’s conversation with Bloom. Then I introduce Romer, after which I offer my elaborations. First I introduce some simple diagrams through which we can visualize the relationship between our conceptual systems (maps) and the world itself (territory). I conclude by anchoring that story in time and space making it one that is, in principle, about the evolution of human society over time.
Bloom’s conversation with Cowen
Bloom sets the stage:
The big picture — just to make sure everyone’s on the same page — is, if you look in the US, productivity growth . . . In fact, I could go back a lot further. It’s interesting — you go much further, and you think of European and North American history. In the UK that has better data, there was very, very little productivity growth until the Industrial Revolution. Literally, from the time the Romans left in whatever, roughly 100 AD, until 1750, technological progress was very slow.
Sure, the British were more advanced at that point, but not dramatically. The estimates were like 0.1 percent a year, so very low. Then the Industrial Revolution starts, and it starts to speed up and speed up and speed up. And technological progress, in terms of productivity growth, peaks in the 1950s at something like 3 to 4 percent a year, and then it’s been falling ever since.
Then you ask that rate of fall — it’s 5 percent, roughly. It would have fallen if we held inputs constant. The one thing that’s been offsetting that fall in the rate of progress is we’ve put more and more resources into it. Again, if you think of the US, the number of research universities has exploded, the number of firms having research labs.
Thomas Edison, for example, was the first lab about 100 years ago, but post–World War II, most large American companies have been pushing huge amounts of cash into R&D. But despite all of that increase in inputs, actually, productivity growth has been slowing over the last 50 years. That’s the sense in which it’s harder and harder to find new ideas. We’re putting more inputs into labs, but actually productivity growth is falling.
Cowen responds by saying, “Let’s say paperwork for researchers is increasing, bureaucratization is increasing. How do we get that to be negative 5 percent a year as an effect?” A bit later he’ll offer:
Doesn’t the explanation have to be that scientific efforts used to be devoted to public goods much more, and now they’re being devoted to private goods? That’s the only explanation that’s consistent with rising wages for science but a declining social output from her research, her scientific productivity.
In his response to Tyler, Bloom makes two suggestions that are consistent with my hypothesis:
Why is it happening at the aggregate level? I think there are three reasons going on. One is actually come back to Ben Jones, who had an important paper, which is called, I believe, “[Death of the] Renaissance Man.” This came out 15 years ago or something. The idea was, it takes longer and longer for us to train.
Just in economics — when I first started in economics, it was standard to do a four-year PhD. It’s now a six-year PhD, plus many of the PhD students have done a pre-doc, so they’ve done an extra two years. We’re taking three or four years longer just to get to the research frontier. There’s so much more knowledge before us, it just takes longer to train up. That’s one story.
A second story I’ve heard is, research is getting more complicated. I remember I sat down with a former CEO of SRI, Stanford Research Institute, which is a big research lab out here that’s done many things. For example, Siri came out of SRI. He said, “Increasingly it’s interdisciplinary teams now.”
His third suggestion repeats one of Cowen’s:
Then finally, as you say, I suspect regulation costs, various other factors are making it harder to undertake research. A lot of that’s probably good. I’d have to look at individual regulations. Health and safety, for example, is probably a good idea, but in the same way, that is almost certainly making it more expensive to run labs…
I am most interested in pursing the notion that ideas are getting harder to find because that’s just how the world is. Even assuming that I am correct, one must still show that that factor is stronger than the one’s Cowen favors and, for that matter, other factors one might suggest. That’s a different kind of argument and one I won’t pursue here.
Romer’s model
Let’s start Paul Romer’s toy model from 1992. Romer sets it up with one that is standard in economics, that of a factory (p. 67):
One of the great successes of neoclassical economics has been the elaboration and extension of the metaphor of the factory that is invoked by a production function. To be explicit about this image, recall the child's toy called the Play-Doh Fun Factory. To operate the Fun Factory, a child puts Play-Doh (a form of modeling compound) into the back of the toy and pushes on a plunger that applies pressure. The Play-Doh is extruded through an opening in the front of the toy. Depending on the particular die used to form the opening, out come solid Play-Doh rods, Play-Doh1-beams,or lengths of hollow Play-Doh pipe.
We use the Fun Factory model or something just like it to describe how capital (the Fun Factory)and labor (the child's strength) change the characteristics of goods, converting them from less valuable forms (lumps of modeling compound) into more valuable forms (lengths of pipe).
But Romer isn’t interest in the production of physical goods. He’s interested in the production of ideas. The child’s chemistry set proves useful (p. 68):
Another child's toy is a chemistry set. For this discussion, the set can be represented as a collection of N jars, each containing a different chemical element. From the child's point of view, the excitement of this toy comes from trying to find some combination of the underlying chemicals that, when mixed together and heated, does something more impressive than change colors (explode, for example). In a set with N jars, there are 2^N–1 different mixtures of K elements, where K varies between 1 and N. (There are many more mixtures if we take account of the proportions in which ingredients can be mixed and the different pressures and temperatures that can be used during mixing.)
As N grows, what computer scientists refer to as the curse of dimensionality
sets in. The number of possible mixtures grows exponentially with N, the dimension of this system. For a modestly large chemistry set, the number of possible mixtures is far too large for the toy manufacturer to have directly verified that no mixture is explosive. If N is equal to 100, there are about 10^30 different mixtures that an adventurous child could conceivably put in a test tube and hold over a flame. If every living person on earth (about 5 billion) had tried a different mixture each second since the universe began (no more than 20 billion years ago), we would still have tested less than 1 percent of all the possible combinations. […]
Continuing on, Romer observes (pp. 68-69):
The potential for continued economic growth comes from the vast search space that we can explore. The curse of dimensionality is, for economic purposes, a remarkable blessing. To appreciate the potential for discovery, one need only consider the possibility that an extremely small fraction of the large number of possible mixtures may be valuable.
What interests me is that this vast search space of possibilities is mostly empty of useful ideas, combinations of chemical elements in this case. Moreover I want to distinguish between the space itself and the strategy we have for searching that space. That strategy is based on our theory or theories about that space, or domain as I sometimes like to call it. I want to examine the relationship between the world, on the one hand, and our ideas of it on the other. To do so I must enter the den of the metaphysician, to borrow a phrase from Warren McCulloch.
On the relationship between the world and our ideas of it
The background space in the following simple diagram represents the space of potential chemical combinations (in this case) while the black dots indicate the useful ones.
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| Figure 1: Useful ideas in the field of all possible ideas |
Notice that some areas are less sparsely populated than others. Now let us superimpose a mental map on it – the orange grid is that map:
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| Figure 2: Our theory of the domain (in orange) |
We can now see that while, yes, some regions of the territory are less populated than others, on the whole the mental map – our theory of the domain – is highly congruent with the territory. Most “bins” in our map contain a useful idea, though it isn’t in the same place in each bin. If we search each bin, then, we will very likely find a useful idea. Some searches will take longer than others, but most searches will be rewarded.
Now consider the next diagram. The shaded area represents the territory that we have in fact explored:
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| Figure 3: Territory we have explored (gray) |
There are still a good many useful ideas that we have not yet discovered. Many of them are in bins immediately adjacent to ones where we have come up with a useful idea. In fact, I count more empty bins (9) in territory we’ve searched than in territory we haven’t (5).
However, our economy appears to be stagnating. If the relationship between our ideas and the world is as I have depicted it in Figures 2 and 3, then we should be coming up with new ideas fairly regularly. The effort of discovery should be roughly the same for each idea [5]. It isn’t. Rather, the effort is increasing. Perhaps the fit between our conceptual grid and the world itself isn’t so regular as those diagrams suggest.
Consider this rather different diagram:
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| Figure 4: Well-explored territory (gray) |
Most of the useful ideas have been found – that is, they are in territory we have explored – and most of the territory we haven’t explored is empty of useful ideas. There are still good ideas out there; but they aren’t close to us, they’re distant. We’re going to have to expend more effort to find them. That expenditure shows up as a decline in productivity. Our economy has entered a period of stagnation.
That, I believe, is the situation we are in.
History and evolution: What does this mean, ideas are “close to us”?
Those diagrams are all well and good. But they imply the existence of a transcendental point of view that simply isn’t available to us. That is, it is easy enough for us to examine the diagram, and see:
- the distribution of ideas in territory, as in Figure 1,
- the conceptual grid we lay over the territory, as in Figure 2, 3, and 4 and
- to compare the two in each case.
In the real world, however, we don’t have access to that point of view – a problem that bothered Immanuel Kant, among others.
However, we, the human race, have been living in this real world for a long time. And over that long time our ideas about the world have changed rather considerably, often in ways that improve our well-being. I suggest that it is the very fact of change that gives credence to those simple diagrams. We change our ideas and, in consequence, can construct better garments, more secure dwellings, have access to a greater variety of foodstuffs, can travel further and with greater safety, and so forth. This process isn’t so simple as that sentence implies, often enough we blunder into the unexpected, and that forces us to change our ideas, and so forth. But it will do for my present purposes.
So, let us go back in time some three million years or so. Our ancestors are very clever apes living somewhere in, I believe, East Africa, but my argument doesn’t depend on just where or even on the idea that our ancestors come from one single region. Wherever they are, they have certain capacities of perception and movement, capacities that have evolved over millions of years and so are reasonably well suited to their physical environment.
Let us stipulate that those things, of whatever kind, to which we have easy access (as a function of our perceptual, cognitive, and motor capacities) are close to us – perhaps physically close, but mostly close in a more abstract, more metaphysical sense, if you will. By extending ourselves, though, we can reach things that are somewhat distant from us, perhaps even quite distant. At some point, for example, our pre-human ancestors discovered how to fabricate stone tools of various kinds, arrow and spear heads, knives, and hand axes. This involves knowledge of fabrication techniques and knowledge of how to use the resulting tools, in hunting, food preparation, and so forth. It also involves knowledge of where the best raw stone is located, which may not be where people lived. These various kinds of knowledge increased our ancestors control over their world and their lives, but did have a cost associated with them. It took time to figure these things our, to reduce them to routine, and to pass those routines on to others.
Consider another diagram:
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| Figure 5: Point of origin |
This is the same as Figure 4, but with the addition of that green spot. That green spot is where those ancestors of ours were located in the world. It is, if you will, an anchor tying us to a particular historical trajectory. That’s how evolutionary processes are, tied to specific sequences of events in time and space.
Note, however, that I don’t mean only physical location (East Africa) but…What should we call it, this relationship between our capacities (at any given moment) and the world-in-itself? For the moment I’m content to think of it as our metaphysical position in the world – as I've already indicated, where our physical location in the world is but one aspect of this metaphysical position.
I can’t say that I have a great deal of affection for that term; the usage certainly is not standard. Nor do I dislike it either. Perhaps a new term is needed, but I haven’t the time for that now.
Putting that aside, the point I am making is that, in a world constituted as ours is, the effort expended to acquire an idea is an effective measure of that idea’s (metaphysical) distance from us. Correlatively, the effort required to acquire a second idea, Fred, after having acquired a first idea, Amanda, is a measure of the distance between Fred and Amanda. I am, of course, only interested in the effort directly related to uncovering the idea. The effort required, e.g. to fill out forms and jump through bureaucratic hoops is extraneous and should not be taken into account in thinking about this issue, even if it is annoyingly intrusive in point of actual fact.
One final remark. Cultural growth does not seem uniform over the long term [6]. In my working paper, Stagnation and Beyond, I lay out some ideas that David Hays and I have worked on [7], a theory of cultural evolution over the longue durée – see pp. 3-4, 15-28 in [2] for their relevance to the problem of stagnation. The following diagram illustrates that long-term trajectory:
The story we tell is one of cultural paradigms existing at four levels of sophistication, which we call ranks. In the terminology of current evolutionary biology, these ranks represent major transitions in cultural life. Rank 1 cognitive architectures emerged when the first humans appeared on the savannas of Africa speaking language as we currently know it. Those architectures structured the lives of primitive societies that emerged perhaps 50,000 to 100,000 years ago. Around 5,000 to 10,000 years ago Rank 2 architectures emerged in relatively large stable human societies with people subsisting on systematic agriculture, living in walled cities and reading written texts. Rank 3 architectures first emerged in Europe during the Renaissance and gave European cultures the capacity to dominate, in a sense, to create, world history over the last 500 years. The late 19th century saw the emergence of Rank 4 architecture.
My suspicion, then, about the current stagnation is that we’re at a plateau, marking time before ascending another growth curve. To make that ascent we need new conceptual frameworks, frameworks which will, in effect, bring us closer to a realm of new ideas currently invisible and inaccessible to us.
Our story is not over. We are preparing a new beginning.
References
[1] Tyler Cowen, Nicholas Bloom on Management, Productivity, and Scientific Progress, https://medium.com/conversations-with-tyler/nicholas-bloom-tyler-cowen-productivity-economics-b5714b05fc2b.
[2] William Benzon, Stagnation and Beyond: Economic growth and the cost of knowledge in a complex world, Working Paper, August 2, 2019, https://www.academia.edu/39927897/Stagnation_and_Beyond_Economic_growth_and_the_cost_of_knowledge_in_a_complex_world.
[3] Nicholas Bloom, Charles I. Jones, John Van Reenen, and Michael Webb, Are Ideas Getting Harder to Find? American Economic Review 2020, 110(4), https://doi.org/10.1257/aer.20180338.
[4] Paul M. Romer, Two Strategies for Economic Development: Using Ideas and Producing Ideas, The World Bank Economic Review, Volume 6, Issue suppl_1, 1 December 1992, pp. 63–91, https://doi.org/10.1093/wber/6.suppl 1.63, or here: sci-hub.tw/10.1093/wber/6.suppl_1.63.
[5] Since we don’t search for ideas one at a time, but rather in research programs aimed at discovering as many ideas as possible I suppose we should be talking about the marginal effort of discovery for the next idea, or some such thing. We can set that nicety aside for the informal purposes of this argument.
[6] See Stagnation and Beyond, pp. 11–13.
[7] Here is our basic paper, William Benzon and David Hays, “The Evolution of Cognition”, Journal of Social and Behavioral Structures 13(4), 297-320, 1990, https://www.academia.edu/243486/The_Evolution_of_Cognition. Hays has written a history of technology from this point of view, David Hays, The Evolution of Technology Through Four Cognitive Ranks (1993), http://asweknowit.ca/evcult/Tech/FRONT.shtml. See also posts at New Savanna under the labels, tech evol, https://new-savanna.blogspot.com/search/label/tech%20evol, and cultural ranks, https://new-savanna.blogspot.com/search/label/cultural%20ranks.
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