Complementarities, Weak Links, AI, and Economic Growth
Using growth theory to discipline our thinking on AI and economic growth
Recently, Brian wrote a long post examining some rather provocative claims about AI and economic growth. The post is long, but worth reading in detail. At the same time, it is important to recognize the underlying purpose of the post. When we start thinking about a speculative world, we need something to discipline our thinking. We need a model. Thus, what I want to do this week is to provide some very basic lessons for framing the debate around AI and economic growth.
Complementary Tasks and Production
Production often requires a number of complementary tasks. Construction might require framers, carpenters, plumbers, electricians, and bricklayers. Restaurants produces meals that require cooks, servers, bartenders, and dishwashers. Manufacturing is often organized using an assembly line, in which each worker on the line is assigned a particular task. In these types of production, each task is important to the production of the final output good.
Michael Kremer’s influential paper, “The O-Ring Theory of Economic Development,” is useful for organizing our thinking about these types of production. The motivation for that paper is to think about the explosion of the space shuttle Challenger. Although the space shuttle consisted of thousands of components, the failure of just one component, the O-rings, led to its explosion. As Kremer pointed out, this concept seems fairly generalizable to many different types of production. Furthermore, once we start thinking about production in this way, we can learn a lot of lessons.
Consider a simple model. Suppose that every worker has a type. The worker can either be a high-skilled type or a low-skilled type. The difference between types can be understood in terms of the percentage of time that the worker successfully completes a task without error. Let this percentage be denoted as q. To keep the math simple, suppose that for the high-skilled worker, q = 0.75, and for the low-skilled worker, q = 0.25. Now, imagine that production, y, can be characterized by the following production function.
Where k is capital and f(k) is a function such that f ’, -f ’’ > 0, f(0) = 0, and f(1) = 1. Note that production is increasing in the number of workers and the amount of capital. However, production also depends on the skills of the workers. To see why this is important, assume that the amount of capital used in production is normalized to 1 and assume that two workers are used in production.
When two high-skilled workers are employed together, total production is 1.125. When one high-skilled worker is matched with a low-skilled worker, production is 0.375. Finally, when two low-skilled workers are matched together production is 0.125. Note the importance of the skill levels of the workers. Skilled workers are 3 times more productive than low-skilled workers, as measured by their ability to successfully complete a task. Yet, when two high-skilled workers are matched together, they produce 9 times more than when two low-skilled workers are matched. Furthermore, although high-skilled workers are 3 times better than low-skilled workers, using 3 low-skilled workers and 1 high-skilled worker is not equivalent to 2 high-skilled workers. To see why, plug those values into the production function. Such a firm would produce ~0.05 units of production. This is lower than the 2 low-skilled workers alone. Quantity isn’t a substitute for quality.
So why does this matter?
Well, one important implication is that if one considers a firm that has this sort of production function, one would find that wage that firms are willing to pay is an increasing function of the skills of the workers they already employ. Given that, one would expect high-skilled talent to be collected at particular firms since those firms will have the highest willingness to pay for high-skilled workers. In other words, we should not observe the intermediate example of firms in the numerical example above. We should expected high-skilled workers to be matched with other high-skilled workers. We should also expect a secretary at Goldman Sachs to get paid a higher salary than a secretary that works for a local bank. This should be true even if their job responsibilities are the same. Similarly, we should expect different levels of skills at jobs based on the quality of the product offered. Fast food restaurants don’t hire world famous chefs. Similarly, and perhaps most importantly for the discussion to come, this conclusion can provide an explanation for why we see such large productivity and wage differences between rich and poor countries.
The lesson of the model is that complementarity in production is important. Quantity is not a substitute for quality. The importance of complementarity has broader implications.
Weak Links
Building on this work, Chad Jones has a paper on “weak links” and intermediate goods. The basic premise of his paper is as follows. The typical neoclassical model of economic growth has capital, labor, and technology being used to produce an output good. There is a multiplier effect associated with capital. More investment in capital leads to more output, which itself leads to more capital.
There is reason to believe that intermediate goods, which are absent from this standard model, have the same sort of multiplier effect. This is potentially empirically important in explaining just why rich countries are so much richer than poor countries. For example, the multiplier effect associated with capital is inversely proportional to the labor share of income. Since the labor share is 2/3, the multiplier is 3/2 or 1.5. Thus, if total factor productivity doubles in the economy, real GDP should increase by about 2.8 (2^1.5). This isn’t big enough to explain the differences between countries. The inclusion of intermediate goods implies that the multiplier should equal the product of the inverse of the labor share and the inverse of the intermediate good share of gross output. Since the intermediate goods share of gross output is about 1/2, this implies a multiplier of 1.5 * 2 = 3. Thus, when total factor productivity doubles, real GDP should increase by a factor of 8 (2^3). Now, we are getting somewhere.
These are back-of-the-envelope calculations based on a modified standard growth model. The precise calculations aren’t that important. Instead, what is important is the general multiplier effect. Intermediate goods matter a great deal for production in ways that are likely to multiply. More intermediate goods means more output, which means more intermediate goods. The same can be said in reverse. (This multiplier effect is really what is behind the Diamond-Mirrlees result in economics that one shouldn’t tax intermediate goods, as Brian has discussed with regard to tariffs.) For example, electricity is an intermediate good in the production of both the construction industry and the banking industry. Thus, if the electrical infrastructure is good, you get more production in construction and banking. At the same time, more production in construction and banking means it is easier to build and finance electrical infrastructure.
Jones then combines this insight with Kremer’s work on complementarity. However, he generalizes the method. Rather than assuming a production function like Kremer’s above, he assumes that intermediate inputs enter the production function via a constant elasticity of substitution. This provides additional flexibility to the model. If the elasticity of substitution is greater than 1, then the intermediate goods are substitutes. On the other hand, if the elasticity of substitution is less than 1, then the intermediate goods are complements. Jones assumes that the elasticity of substitution is less than 1. This allows him to consider what happens when there are “weak links” in the production process. If one thinks of the intermediate goods as things like electricity and transportation, this makes a lot of sense. You cannot substitute transportation for electricity. Instead, it is much more natural to think of them as complements in production.
Growth and AI
These types of growth models give us something to work with when we think about AI. For example, what these models suggest is that the critical parameter is the elasticity of substitution. Let’s think about why this is.
The world has a number of general purpose technologies, such as electricity or the internet. We would expect these general purpose technologies to have a multiplicative effect on output. Electricity makes construction more productive. As construction becomes more productive, one can build new and better electrical infrastructure, which further increases output.
At the same time, these general purpose technologies tend to be complements in the production process. Think about transportation and the internet. A business with a presence on the internet will expand its market. In order to get its products to market, it will need transportation infrastructure and services. The internet and transportation are complements rather than substitutes.
The complementarity has potentially important implications because it creates “weak links” in the production process. The production of an internet-based business is limited by the potential weak links of poor transportation infrastructure and/or services. It is also limited by poor infrastructure for electricity production. If electrical power is down frequently, then people will not have access to the website storefront.
A natural question to ask is whether the development of AI should be seen as a complement or a substitute to other intermediate goods. The more provocative predictions about AI seem to imply that AI will lead to greater automation and that automation via machines, robots, and other forms of capital will substitute for boring, old human labor. As this automation becomes more and more important, economic growth will explode and the capital share of income will go to 100%.
A natural question to ask, however, is what happens when some things cannot be automated — or even if certain things are just much harder to automate than others. What if automation is subject to comparative advantage?
Let’s consider this scenario. Suppose that there are two intermediate goods used in the production of the final output good. Also, assume that certain types of intermediate goods production cannot be automated. Finally, assume that these intermediate goods are complements in production.
The constant elasticity of substitution production function can be written as
where y is output, h is the input that is hard to automate, e is the input that is easy to automate and sigma is the elasticity of substitution.
Since we are assuming that these inputs are complements, let’s make things really simple and set the elasticity of substitution equal to 1/2. This implies that our production function can be written as
Now, suppose that automation means the production of the intermediate good for which production is easy to automate becomes infinitely large over time because of the rapid growth that comes from automation. It follows that total production will just be equal to the production of the hard-to-automate input (y = h).
Complementarity creates weak links that limit the ability of production to explode as a result of automation. Furthermore, in this case, the share of income that goes to the easy-to-automate intermediate good would go to zero as the production of this intermediate good becomes infinite. Thus, complementarity implies that the weak link input will reap all of the rewards from automation.
This is in some sense a dramatically oversimplified example. Nonetheless, it does seem to highlight the importance of complementarity and weak links in the production process — characteristics without which it would be hard to explain dramatic differences in productivity and wages across countries as well as patterns of production.
A more serious exercise would be to take the model of Chad Jones and Christopher Tonetti. They start with a constant elasticity of substitution production function in which output is a function of capital and labor. They then allow a fraction of tasks to be automated and allow automation to change over time. In their empirical exercises, the elasticity of substitution less than 1 proves to the important factor. It creates a weak link. As a result, they show that when tasks are automated through rapid improvements in capital productivity, overall economic growth is still constrained by labor, which is improving at a much slower rate. The reason is that these labor tasks are the weak links in the production process. As a result, in their baseline case, although economic growth increases in their numerical simulations, it is only 19% higher in 2060 than the counterfactual without automation. In addition, the capital share in their model remains relatively constant. The reason for this latter result comes from the fact that as automation makes capital more productive, less capital is needed for a given level of “effective capital.” This helps to keep the capital share of income relatively constant.
Thus, the crucial factors are (a) how much can be automated, and (b) whether we should think of AI as a complement to other inputs in the production process.
Some Concluding Thoughts
When we think about economic growth, it is hard to explain the vast differences in productivity and wages between rich and poor countries without appealing to things like complementary inputs and weak links in production. These represent important building blocks when thinking about growth.
When thinking about AI, it is important to discipline our thinking by starting out with a model. Ideally, that model should capture some important characteristics of reality. In other words, we shouldn’t just pick any old model off the shelf. We need to think about models that have something to say about issues presented by AI.
Historically, general purpose technologies have dramatically changed the lives of human beings. However, when we look at the long-run trend in economic growth in places like the U.S. or the U.K., the trend seems pretty linear over a long period of time. In other words, it is hard to identify the effects of general purpose technologies like electricity and the internet in observed growth rates.
Why is that?
It is possible that these general purpose technologies have a large effect that declines over time. Perhaps the reason we don’t see this in the data is that the creation of general purpose technologies cause an increase in the growth rather that tapers off over time. However, another general purpose technology comes along and thus it is hard to detect the speed-up and slowing-down of growth as the technology is diffused throughout society.
Another possible explanation is that general purpose technologies tend to be things that are complementary to other inputs in the production process. When that is the case, these other inputs potentially create weak links that limit the ability of growth rates to change very much.
When we think about AI, there is very much a “this time is different” mantra surrounding the technology. AI is simply going to eat the world. But what if AI cannot automate everything? What if AI is once again a general purpose technology that is complementary to other inputs in production that cannot be automated? If that is the case, AI is likely to have a significant impact on growth. However, the wildly speculative and provocative claims will prove incorrect.
This is why it is important to discipline our thinking with well-worn theory. Perhaps the theory will be proven wrong. Nonetheless, it at least can help define the terms of the debate and identify exactly why predictions differ.