In this section, we will discuss data-driven models of technology change. Developing a data-driven model requires identifying an equation that describes the variation observed in a given dataset. The most common equations used to describe trends in technological change are linear, polynomial, or exponential relationships, among others. To select among them, we’ll fit different equations (or "curves") to the dataset. Then we’ll measure how well the model captures the variability in the data to select the one that fits best. The way we fit a particular model to the data is to choose the variables and the parameters. Suppose, for example, we are using a linear model. Then our equation will have the form _y = mx + b_. In this equation, _x_ and _y_ are variables, which might represent cost, time, or production. The parameters are _m_, which represents the slope of the slope of the line, and _b_, which is where the line hits the y-axis. We want to choose _m_ and _b_ so that the resulting line best fits the data. These parameters may be meaningful—for example, _m_ may tell us about the rate of change and _b_ might be the initial cost (depending on exactly what we're modeling). # Moore's Law In 1965, Gordon Moore, director of research and development at Fairchild Semiconductor, made a bold prediction in a special issue of the journal _Electronics_. He started by observing that the number of transistors on microchips had approximately doubled each year for some time, and he predicted that this would continue for at least the next decade. By that prediction, microcircuits would contain an astonishing 65,000 transistors per microchip. He was both overly optimistic and overly cautious: By 1975, the rate had slowed so that it was doubling in a bit over two years rather than one—but that doubling continued not just for a decade, but into the 21st century (Lotha, 2023). And transistors are far from the only technology that has improved roughly exponentially over long periods of time. Moore's Law has been generalized to apply to other technologies and other performance metrics. This more general form of Moore's Law states that a technology's performance shows exponential improvement with time. In other words, with each step forward in time, a technology improves by a fixed percentage. As an equation, Moore’s Law can be expressed as $y = ae^{-bt}$, where the variables are $t$, the time that units have been being produced, and $y$, whatever is being produced (in the original example, this would be the number of transistors on a microchip), and the parameters are $a$, the cost of the initial unit, and $b$, the improvement rate. Let's look at an example of fitting Moore's Law to data on how the cost of coal-fired power plant capacity changed over time. ![[Pasted image 20250317125410.png]] > [!Figure] > Capital cost of coal-fired power plants. Source: Adapted from McNerney, J., Doyne Farmer, J., & Trancik, J. (2011). Historical costs of coal-fired electricity and implications for the future. _Energy Policy_, _39_(6), 3042-3054. [URL](https://www.sciencedirect.com/science/article/abs/pii/S0301421511000474) Note that the y-axis is logarithmic while the x-axis is linear, so a straight line approximation represents an exponential function. How well does Moore's Law fit the data in the figure above? Many of the data points fall some distance from the fitted curve. But it also seems to be a reasonable fit for this trend, given that the data points generally follow the curve. However, to really determine how well this model does in describing the data we would need to perform further quantitative analysis. > [!note] > Intuitively, considering time as a determinant of technology change makes a good deal of sense. Although innovation may be driven by human effort, technological improvements are constrained by the physical and informational knowledge of a given period in time. For that reason, we might posit that time is a good candidate for a proxy for the processes driving technology change. > [!info] > Read Gordon Moore's original 1965 paper ["Cramming more components onto integrated circuits."](http://american.cs.ucdavis.edu/academic/ecs154b/154bpdf/gordon_moore_1965_article.pdf) # Wright's Law The more you do something, the better you get at it. That's the essence of Wright's Law. Unlike Moore's Law, which tracks progress over time, Wright's Law tracks progress as a function of cumulative production. So the more of something has been made, the more efficiently it can be made. For this reason, it's often called the "experience curve" or "learning curve". Wright's Law was developed by aeronautical engineer and MIT graduate Theodore Paul Wright, who started tracking the cost of producing airplanes compared to the total number of airplanes produced in 1922. In 1936, he published these graphs in his paper _Factors Affecting the Cost of Airplanes_: ![[Pasted image 20250317125704.png]] > [!Figure] > Source: Adapted from Wright, T.P. (1936, February). Factors affecting the cost of airplanes. _Journal of the Aeronautical Sciences, 3_(4), 122–128. [URL](https://pdodds.w3.uvm.edu/research/papers/others/1936/wright1936a.pdf) ![[Pasted image 20250317125757.png]] > [!Figure] > Source: Adapted from Wright, T.P. (1936, February). Factors affecting the cost of airplanes. _Journal of the Aeronautical Sciences, 3_(4), 122–128. [URL](https://pdodds.w3.uvm.edu/research/papers/others/1936/wright1936a.pdf) Mathematically, Wright's Law states that for each 1% increase in cumulative production, we observe a fixed percent improvement in performance. In other words, it is a power law as a function of cumulative production. As an equation, it can be expressed as $y = ax^{-b}$, where the variables are $x$, the total number of units produced cumulatively, and $y$, the cost per unit, and the parameters are $a$, the cost of the initial unit, and $b$, the rate of increase. A key difference between Wright's Law and Moore's Law is that Wright's Law implies that we have the power to change the rate of technological development. Cumulative production does not necessarily follow a fixed rate with time—we can make choices about how fast we produce a technology. If we invest more and produce more, Wright's Law predicts that we'll also get better and more efficient at it more quickly. Both Wright's and Moore's Laws imply that technology will only improve. Knowledge will never be forgotten. This is because both cumulative production and time can never move backward. If annual production levels drop from one year to the next, Wright's Law predicts that improvements will slow, but they can't be reversed. So both these laws adopt the perspective that "knowledge is forever." Of course, in reality, we do sometimes forget, and technologies can backslide to some extent. This model (and the others discussed here) are simplified representations of the real world. But this isn't all bad: As we will discuss in the next section, their simplicity can be an advantage when developing a forecast of future technological performance change. ![[Pasted image 20250317130027.png]] > [!Figure] > Cumulative production by wholesale price of photovoltaic modules. Source: Adapted from Johnson (2002); Dunay (2003) as cited in Kuik et al. (2006). _Innovation dynamics induced by environmental policy_. Institute for European Environmental Policy. [URL](https://ec.europa.eu/environment/enveco/policy/pdf/2007_final_report_conclusions.pdf) Figure above displays an example in which Wright's Law has been fitted to data on the price of photovoltaic modules versus the cumulative production of photovoltaic energy capacity. How well does Wright's Law fit these data? Once again, we can see that many of the data points fall at some distance from the trend line, but as a whole the data appear to follow the curve quite closely. Note that the pretty good fit observed in the previous figures above the last one does not offer strong evidence that Moore's or Wright's Laws apply more generally. To draw such conclusions we would need to quantitatively test the goodness of fit on a large dataset, covering many different technologies. > [!info] > Cumulative production would seem to be a reasonable proxy for the processes driving technology change. As previously mentioned, Wright's Law is sometimes called the experience curve. As an industry accumulates experience in the provision of a product or service, improvements tend to accumulate as well, moving the whole production process toward a better-performing state. Alongside increases in production levels, one can reasonably expect that learning by doing, economies of scale, and research and development investments will all lead to accumulating improvement. > [!note] > Read T.P. Wright's original 1936 paper ["Factors Affecting the Cost of Airplanes."](https://pdodds.w3.uvm.edu/research/papers/others/1936/wright1936a.pdf) # Goddard's Law In 1982, C.T. Goddard, an engineer at Bell Labs, argued that learning wasn't really the right thing to focus on to explain technological improvements in his paper _Debunking the Learning Curve_. Instead, he said, the more fundamental driver of technological improvements is economies of scale. “Learning is still in the picture,” he wrote, "but it stems not so much from what was done in the remote past as from the growing cadre of development personnel and manufacturing engineers which the increase in revenue can support." In other words, economies of scale are determined by the level of production now, not the cumulative production for all time. So, he argued, technological improvement should be viewed as a function of the annual rate of production, not the cumulative production. In essence, Goddard argued that the cost-benefit from learning diminishes over time and is constrained by physical, technical, and economic boundaries. Mathematically, that implies that for each 1% increase in annual production, costs should decline by a fixed percentage. Goddard's Law, like Wright's, is effort-based. However, unlike both Wright's and Moore's Laws, it allows for the possibility of losses of progress. If a technology's annual production goes down, Goddard's Law would predict that the cost per unit service would go up (in accordance with the economies of scale decreasing). Figure below shows an example where Goddard's Law has been fit to data on photovoltaics module prices. The data are shown on a log-log plot of the average price versus annual production. If Goddard's Law holds, we would expect to see the data following a straight line. ![[Pasted image 20250317130624.png]] How well does Goddard's Law describe the data displayed in the figure above? As with the previous examples, most of the data points in this set fall at some distance above or below the trendline. By eyeballing the data, the curve looks like it may fit the data somewhat better than the curves in the previous sections. However, to say something more definitive, we will need to quantitatively test the goodness of fit of these curves. > [!info] > We know that economies of scale can drive down prices. As companies increase their production capacities, larger production facilities make more efficient use of machinery and other investments and bulk purchasing saves money. Thus, annual production can be a good proxy for the processes driving the cost improvement in technologies, perhaps especially for technologies whose costs are particularly affected by economies of scale. > [!note] > Read C. Goddard's 1982 paper ["Debunking the Learning Curve."](https://ieeexplore.ieee.org/document/1136009) # Limits to technology improvement: S-curves, performance limits, and cost floors Each of these models suggests that improvement could continue forever. But is this realistic? Surely, we might wonder, there must be some ceiling to how much technology can improve. Or costs of materials might impose a floor to how low technology prices can go. Mathematically, that would mean adding a constant to each of the functional forms discussed above. Such a model is known as an "S-curve." The figure below displays one version of a conceptual model of technology development: As improvement in a given technology begins to tail off, that creates the opportunity for a new technology to break in. The new technology starts small and as it improves, it eventually takes over and replaces the old one. Intuitively, the idea of a ceiling for improvement makes sense, and much literature has discussed the applicability of S-curves to technological innovation. But the data often doesn't reflect this. In some of the figures above, for example, the data shows a consistent exponential rate of improvement and even an increasing rate over more than a century. Even as new technologies enter the marketplace, older technologies continue to improve. It is possible that zooming into a particular technology would show a leveling off of performance, but the jury is still out. Despite its popularity as a conceptual model of technological change, the S-curve so far has shown up only rarely in data on technology performance change. ![[Pasted image 20250317131350.png]] > [!Figure] > S-curve model. Source: Adapted from Mazouz, A., Alnaji, L., Jeljeli, R., & Al-Shdaifat, F. (2019). Innovation and entrepreneurship framework within the Middle East and North Africa region. _African Journal of Science, Technology, Innovation & Development_. [URL](https://www.researchgate.net/publication/331976444_Innovation_and_entrepreneurship_framework_within_the_Middle_East_and_North_Africa_region) # Effort-based vs. time-based models The most common data-driven models of technological change fall into two categories, time-based and effort-based models. Moore's Law is a time-based model: It considers time as the independent variable, and cost intensity (or another measure of performance intensity) as the dependent variable. Wright's Law and Goddard's Law are effort-based models since both cumulative production and annual production are determined by how much effort we put into it. In this type of model, some measure of effort is the independent variable, where effort can be measured in terms of the number of units produced, investment in research and development, or another type of investment in technology production and development. Time-based models imply that there is little that decision-makers can do to increase rates of technological change, whereas effort-based models suggest that the decision to invest more or less in a particular technology will affect its future performance. In both time-based and effort-based models, the independent variables are meant as proxies for much more involved processes. In other words, time is a marker for a variety of changes that are taking place, such as growing knowledge in related fields and accumulating investments in technology production and research. In an effort-based model considering, say, cumulative production as the independent variable, production is meant as a marker for changes to production plant size, and investment in research and other technology development activity. Studying a proxy means analyzing one phenomenon that can be used to infer changes in another. For example, in neuroscience, researchers use an fMRI machine to collect data on the increase in oxygenated blood to various parts of the brain when presented with stimuli (Glover, 2012). The increase in blood is not the target of the research but rather stands as a proxy for brain activity. That is, researchers observe increased blood flow but infer from this observation an increase in activity. In a similar fashion, we will track time or measures of effort as proxies for a richer set of investments in researching and producing a technology. # Conclusion We have discussed three data-driven models of technology change: Moore's, Wright's, and Goddard's Laws. Moore's Law anticipates progress at a predictable rate with the passage of time. Wright's and Goddard's Laws anticipate progress according to the production of the relevant technology. Wright's Law states that improvements accompany cumulative production, which can only increase. Goddard's Law states that progress is contingent on annual production, which may also decrease. # Additional Information - Goddard, C. (1982). Debunking the learning curve. _IEEE Transactions on Components, Hybrids, and Manufacturing Technology_ _5,_ 328–335. [URL](https://ieeexplore.ieee.org/document/1136009) - Mazouz, A., Alnaji, L., Jeljeli, R., & Al-Shdaifat, F. (2019). Innovation and entrepreneurship framework within the Middle East and North Africa region. _African Journal of Science, Technology, Innovation & Development, 11_(6). [URL](https://www.researchgate.net/publication/331976444_Innovation_and_entrepreneurship_framework_within_the_Middle_East_and_North_Africa_region) - McNerney, J., Doyne Farmer, J., & Trancik, J. (2011). Historical costs of coal-fired electricity and implications for the future. _Energy Policy_, _39_(6), 3042-3054. [URL](https://www.sciencedirect.com/science/article/abs/pii/S0301421511000474) - Moore, G. E. (1965). Cramming more components onto integrated circuits. _Electronics_, _38_(8). [URL](http://american.cs.ucdavis.edu/academic/ecs154b/154bpdf/gordon_moore_1965_article.pdf) - Nagy, B., Farmer, J., Bui, Q., & Trancik, J. (2013). 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