Learning in Times of Change: A Dynamic Explanation for

Apr 30, 2009 - Synopsis. Technology growth and cost can often be approximated by exponential functions over time: together they are essentially equiva...
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Environ. Sci. Technol. 2009, 43, 4002–4008

Learning in Times of Change: A Dynamic Explanation for Technological Progress F. FERIOLI† AND B . C . C . V A N D E R Z W A A N * ,†,‡ Policy Studies Department, Energy Research Centre of The Netherlands, Amsterdam, The Netherlands, and Lenfest Center for Sustainable Energy, Earth Institute, Columbia University, New York, New York

Received January 28, 2009. Revised manuscript received April 14, 2009. Accepted April 15, 2009.

This paper analyzes the dynamics for growth and cost reduction of innovative products and techniques in the energy sector. We provide examples showing that simple exponential relations can be used to describe growth and cost reduction as a function of time for many types of technologies. These two simplemodels,fortechnologicalgrowthandprogress,respectively, taken together are shown to be de facto equivalent to the wellknown learning curve. The main novelty of the stylistic computational component-based model we present is that it introduces time in the learning curve methodology. While there may be additional explanatory variables, we argue that accounting for time improves the understanding and use of learning curves.

1. Introduction Cost reduction appears to be a necessary step to achieve commercial success for almost every new product or technology that enters the market. Mass penetration of cars and consumer electronics, for example, was enabled by dramatic cost reductions that brought innovations such as the T-Ford and the PC within reach of the average consumer. Interestingly, growth and cost reduction appear to be two related phenomena, as discussed, among others, by Arrow in his seminal 1962 paper (1). Today, the issues of market diffusion and cost reduction are probably nowhere more pressing than for energy technologies. As calls for a radical change toward a more sustainable energy system multiply, nonpolluting energy technologies remain in most cases considerably more expensive than conventional fossil fuelbased ones. Given the now widely recognized challenges associated with global climate change, researchers and policy makers alike increasingly focus on the double objective of promoting market penetration and cost reduction for lowcarbon energy technologies. In this paper we examine a simplified model that links growth and cost reduction for such technologies, and assess the implications of this model for the diffusion of innovations in the energy sector. Our aim is to develop a conceptual framework through which public policies for promoting alternative energy technologies can be balanced and benchmarked. It was observed in the 1930s that the number of labor hours necessary to produce an airframe (i.e., the main support * Corresponding author e-mail: [email protected]. † Energy Research Centre of The Netherlands. ‡ Columbia University. 4002

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structure or body of an airplane) decreases as the cumulative production of the plant under consideration increases (Wright, 1936 (2)). More exactly, Wright found that both the labor required and correspondingly the production cost incurred decrease with a fixed percentage every time cumulative production doubled. In other words, under the assumption that the number of work hours employed is a proxy for the total production costs, a power law relation appeared to exist between the cost of an airframe and cumulative production: C(x) ) C(x0)

() x x0

-L

(1)

in which x is the cumulative output, L is a positive parameter, C(x) is the airframe cost at cumulative output x, and (x0, C(x0)) is a suitable normalization point. For every doubling of cumulative production the cost decreases by a constant fraction, LR ) 1 - 2-L, called the learning rate (LR). This behavior was subsequently observed for many other innovative products and technologies, from the fabrication of electronics to the production of paper and pulp (3). For a wide array of industrial products, cost appears to obey the law expressed by eq 1 over several orders of magnitude of cumulative output. Taking the logarithm of both sides of this equation, one easily shows that a linear relation exists between the logarithm of cost and that of cumulative production. For example, Figure 1 shows the price of photovoltaic (PV) modules plotted versus their total globally shipped capacity on a double-logarithmic scale (data from Harmon (4) and linear fit by the authors). So far no satisfying theoretical explanation has been found for this apparent empirical rule, but it is plausible to ascribe the phenomenon to productivity gains resulting from technological learning (see the Supporting Information (SI) for more details). Learning curves have been used for many purposes, e.g., to determine the cost gap associated with the effort of riding the learning curve (5). Learning curves have also been used extensively for planning at the firm level (3) and are increasingly employed as tool for informing energy policy making (6, 7). Learning curves have been developed for many energy technologies (8), among them a range of low-carbon energy technologies such as wind power (9, 10) and PV modules (Figure 1) and alternative fuels such as sugar cane ethanol (11). In particular for the design of energy policy, learning curves are used to quantify implementation and diffusion support measures, like subsidies and feed-in tariffs (6). A plethora of computational integrated assessment models simulate anticipated energy use or the future costs of the energy system on the basis of learning curves, in highly

FIGURE 1. Linear fit of price data for PV modules plotted against their cumulative production (data from (4)). 10.1021/es900254m CCC: $40.75

 2009 American Chemical Society

Published on Web 04/30/2009

aggregated or very detailed form, as a means to estimate the total energy technology costs or the overall emission abatement expenses required to address environmental challenges like global climate change (e. g., (12)). The learning-by-doing concept and eq 1 have drawn criticism for various reasons and their practical usefulness for forecasting and decision-making is regularly questioned (see notably (13), as well as the SI). For example, learning has been shown to sometimes inadvertently stop, and technologies may unexpectedly disappear from the market altogether (as pointed out in (14)). Learning curves have been proposed for the manufacturing costs of wind turbines, as well as the installation costs of wind power and the total generation costs of wind electricity (9, 10). It can be readily observed, however, that it is not possible to add up power law functions of the form of eq 1 and obtain a learning curve for the final product or overall service. This is not to say that it is impossible to obtain a good or reasonable power-law fit (on a double-log scale) of the data points regarding the total system. This may still be feasible, but can be explained by statistical fluctuations that cover up the real nature of the underlying function (for a specific case see (15)). Related to this is the observation that the learning curve literature is often unable to account for other factors contributing to cost reductions that cannot be attributed to learning-bydoing, such as R&D spending. For an extensive account, see (16), complemented by (17). The former notably point out that distinguishing the effects of R&D and market deployment is often not easy, and that the learning literature regularly misses explaining R&D expenses while the R&D literature not rarely ignores learning effects. It has also been noticed that learning rates may vary over time (8, 18) and are sometimes negatively affected by subsidies (as they distort incentives to innovate). Abstaining from fabricating or using a certain technology during extended periods of time may lead to efficiency losses or cost increases as a result of “forgetting-by-not-doing”. All these observations suggest that the dimension time may need to be reintroduced in analyses of technology costs and their evolution. Indeed, other functional relations for cost and diffusion seem to hold for many industrial products and economic activities (see the SI, and (19) and (20)). Exponential growth and exponentially decreasing costs are two of several models that can be used to approximate economic activity. Goddard observes that for a number of products the growth of cumulative production over time can also be fitted with a power law (21, 22). Also, most analysts would agree that ultimately annual growth tends to slow down for mature products and economies, which in the longer run results in growth patterns that are less pronounced than exponential but, e.g., follow a power law model (as proposed by Goddard (21)) or the well established logistic growth model (see the SI or, for example, (23)). We come back to this issue later in this article (Section 4). For new technologies during the initial stages of market penetration (to which learning curves are most often applied) exponential growth and cost reduction seem the right approximations. In a recently published paper we argue that more generally it is often hard to apply learning curves to mature technologies (24). This raises the question, for scientists and policy makers, when precisely one can speak of the “initial” versus “mature” stages of market penetration. We restrict ourselves here by remarking that the answer depends both on the type of technology considered and on the share it has obtained relative to the maximum deployment achievable. In this paper we proffer a critical analysis of the learning curve methodology. The starting point of our study is the realization that cost reduction for a firm’s technology is usually achieved through several small steps that improve a part or component of the final product. In Section 2 we

discuss functional relations that describe the growth in output and gain in productivity associated with technological innovation. The realization that cumulative production and cost reduction can be expressed as simple functions of time allows us to develop a simple model for progress at first sight equivalent to, but the deeper meaning of which is fundamentally different from, the learning curve methodology. A more refined numerical model is presented in Section 3 to demonstrate how many small improvements can be aggregated to reproduce a cost-production relation equivalent to eq 1. The implications of this model are discussed in Section 4, while Section 5 summarizes our main conclusions and recommendations.

2. Exponential Growth and Cost Reduction Much attention has been paid over the past two decades to how technology development feedback and inducement may be included in theories of economic growth (for an overview see (25)). This “endogenous growth theory” may benefit greatly from the abundance of studies in the more detailed technological innovation literature (e.g., (26), in which the different stages in technology development are characterized and the notions “initial” and “mature” are specified). As these technology assessments confirm, arguably the simplest model to mathematically represent (the early phase of) technological diffusion assumes a fixed percentage growth, β, each year. Hence, total output increases by a factor R ) 1 + β each year, and the output in year n + 1, yn+1, is simply yn+1 ) Ryn (with yn the output in year n). Time is uniformly normalized by dividing by the unit of one year in order to ensure dimensional consistency of our equations. Without loss of generality, we assume an output of 1 in year 1 s it is readily demonstrated that the main results in this paper also hold for nonnormalized output growth data. The annual output as function of time can thus be written as: y(t) ) Rt

(2)

The cumulative production, x(t), is found by integration of eq 2, which (unsurprisingly) is again an exponential in t (with an extra multiplicative factor and a constant introduced by integration; see eq 3 in Appendix I of the SI). Therefore the growth model for cumulative production can generally be approximated by a simplified exponential relation of the form x(t) ) abt

(4)

in which a and b are fitting parameters. The parameter b of eq 4 is thus not exactly identical to the annual growth parameter R in eq 2. For several products, companies, and technologies, growth in cumulated production can be accurately approximated by this reduced exponential model: some examples s with data from Goddard (21) s are shown in Figure S1 (SI). Likewise, examples of energy technologies include PV modules, wind turbines, sugar cane-based ethanol production in Brazil, and large stationary fuel cells. In Figure S2 the historic cumulative production data for these technologies and products are fitted with eq 4 (data, respectively, from 4, 10, 11, 27). The fits in Figures S1 and S2 are good to excellent. Having developed a simple model for technological expansion, we now focus on the dynamics of progress. We assume that cost reduction occurs through increases in productivity such that the unit cost is reduced by a fixed number of percentage points every year. Hence, the cost in year n + 1 equals Cn+1 ) γ Cn with γ ) 1 - δ and δ the (positive) productivity gain. Cost as function of time can thus be expressed by an exponentially decaying relation of the form: VOL. 43, NO. 11, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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C(t) ) df t

(5)

in which d and f are again fitting parameters (f < 1). Figure S3 shows that, for several technologies, cost (or price) as function of time can be properly approximated by eq 5 (data from, respectively, 21, 4, 11). The elimination of time from eqs 4 and 5 leads to a power law relation between cost and cumulative output (see Appendix I in the SI for the derivation): C(x) ) d

( ax )

ln f / ln b

(6)

Equation 6 is de facto equivalent to the learning curve of eq 1. Parameters a and d are equivalent to the normalization point (x0, C(x0)) in eq 1, and since f < 1 (that is, costs are decreasing over time) and b > 1 (output is increasing over time), the exponent in eq 6 is negative, so that indeed eq 6 represents a learning curve (with learning rate LR ) 1 2ln f/ln b).

3. A Simple Numerical Model for Progress To test the arguments put forward in the previous Section, we develop a simple stochastic model for growth and progress. For this purpose we use the numerical computation language Octave. With a normalized annual output of 1 in year 1, the annual growth factor R is simulated by generating random numbers. We assume these numbers are normally distributed around a mean growth rate µ > 1 to mimic market fluctuations for a typical growing industry. To calculate the output in year t we generate a series of t random numbers Ri. The mean µ and standard deviation σ of our normally distributed set of Ri are chosen so as to generate growth patterns and fluctuations comparable to those observed in practice (such as for the energy technologies of Figure S2). Output fluctuations that are too drastic to be realistic (i.e., Ri , 1 or Ri . 1) are rejected (to avoid negative values for Ri we could have used a log-normal distribution instead of a Gaussian distribution s we find no good reason and quantification, however, for the asymmetry (skewness) of such a function). Thus, the annual output in year t becomes: t

yt )

∏R

i

(7)

i)1

This model proves to generate a good approximation for growth of a firm or industry in real life (see also Section S3 in the SI). Only slight differences between simulation results and observable economic growth occur, for example because growth rates might in reality not be exactly normally distributed, or since they could depend to some extent on performances in recent years. We calculate cumulative production, typically over a range of a few decades, by summing the annual outputs of eq 7 to obtain: t

xt )

∑y

i

(8)

i)1

The output of our simple model is presented in Figure S4 for four different cases with varying assumptions regarding the simulation parameters µ and σ for the annual growth rate. The randomly generated data are fitted with an exponential curve of the form of eq 4. Note that the integration operation involved in calculating cumulative production (eq 8, equivalent to eq 3 in the SI) has the effect of smoothing year-to-year variations in the simulated annual growth rate Ri. For this reason, growth charts similar to those presented in Figures S1 and S2 (and hence learning curves based on them) appear less scattered when cumulative production, rather than annual output, is chosen as the independent 4004

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variable. The plots in Figure S4 show that the data randomly generated through eq 8 can be accurately represented by a functional relation of the form of eq 4, for a wide range of values of parameters µ and σ. For the cost part of our numerical model, we consider the overall cost of a product as the sum of the costs of its components (as in (24)). For example, the cost of individual parts plus the cost of assembling and marketing may make up the total cost of a given consumer product. As with output growth, cost reduction takes place through a stochastic process. In particular, cost reduction occurs in a number of finite steps that reduce the cost of each component by a fraction. The magnitude and number of the cost reduction steps are not fixed, but vary randomly for each step respectively by component. The purchase of a new automated machine, for example, may in one step reduce the cost of manufacturing part of a final product by x%. Cost reduction for each component at each step materializes randomly in time, that is, in our model not as a function of cumulative production. Theoretically one could suppose improvements to continue indefinitely. With an infinite number of steps, however, the cost of each component would asymptotically tend to zero. In a separate paper the authors argue that progress will ultimately be halted as technologies reach maturity (24), so that costs do not tend asymptotically to zero as time tends to infinity. In particular the cost of raw materials always provides a floor to technology prices, as recently demonstrated for instance by a spike in the cost of PV modules caused by insufficient supply of monocrystalline silicon (see Figure S9; the availability and price of an input resource like oil constitutes another evident generic example). Hence we introduce a finite number of cost reduction steps: given the finiteness of this number, or alternatively the limited time interval we consider, costs always remain nonzero. Thus, for the initial period of market penetration we assume finite progress, which is a realistic approximation for an innovative product. This model of progress is more detailed than the learning-by-doing hypothesis normally used to justify learning curves. In particular, it defines explicitly how improvements for components contribute to cost reduction for the final product, and in particular how small improvements for components can be added to obtain the learning curve in eq 1 for the final product. Still, since cost reduction steps occur with random frequency, the model remains abstract enough to realistically describe the generic dynamics of progress. To render this model explicit numerically we define the cost of a final product as the sum of n components with respective costs ci. The number of components is considered large (with n taking typically a value of around 100 in our simulations) and the cost of each component is assumed to constitute only a small fraction of the overall cost of the final product (so that any single component has limited influence on the cost of the final product). Consistent with our discussion on technology growth, we suppose without loss of generality the cost to be 1 in year 1, so that: n

∑c ) 1 i

(9)

i)1

The costs of all components in later periods are obtained by the generation of random numbers. For each component a number of cost reduction steps mi is defined. The number of steps varies randomly for each component: typically the value of mi is between 10 and 100. For each cost component ci and number of cost reduction steps mi, random positive cost reduction coefficients γij < 1 are determined. At each step, the cost reduction for each component is obtained by multiplying its cost with the corresponding γij coefficient. The final total cost can thus be expressed by:

n

C)

t

∑ ∏cγ

i ij

i)1

(10)

j)1

The coefficients γij s initially generated as random numbers with values between 0 and 1 s are transformed and renormalized in such a way as to produce realistic results, that is, so as not to involve values that make costs fall too quickly to insignificant levels. It can readily be shown, through repeated simulation, that the type of initial distribution of the generated random numbers or the algorithm used to transform and renormalize the coefficients do not affect our final results as long as n and mi are large enough numbers. Again, as with most learning curves in the literature, processes of cost reduction are assumed to take place over a period of several decades. Figure 2 illustrates schematically our model for cost reduction and depicts the evolution of normalized total costs, based on step-by-step cost improvements of individual components, for a technology or product over a period of, e.g., 25 years. The main parameter of our cost model is the total cost reduction reached at the end of the simulation: it is controlled by normalizing the coefficients γij in such a way that the desired overall rate of cost reduction is attained, that is, every coefficient γij is multiplied by a factor in order to obtain a cost reduction (at the end of the simulation) that corresponds to a cost improvement such as could have been observed in practice. The model’s output is shown in Figure S5, in which the overall cost versus time is plotted for a period of 25 years. The parameter values, adopted to make the simulations leading to the total cost evolutions in the three cases demonstrated in Figure S5, were chosen so as to generate overall cost reductions in the range of those observed in practice (such as those shown in Figure S3 for integrated circuits, PV modules, and Brazilian ethanol production). If the number of cost components n and the number of cost reduction steps mi are large enough, the total cost as function of time can be approximated by: C(t) = nci(γij)kt

(11)

in which ci is the average initial cost of a component, γij is the average cost reduction and k is the average number of cost reduction steps per year. In other words, the overall cost decays exponentially over time. The fits through the data points of Figure S5 show that the results of our simple numerical model can be interpolated with good accuracy by the exponential relation of eq 5. We thus illustrate how small random improvements, per component and over time, can be aggregated to produce an overall cost reduction trend that can be approximated by an exponential function, in a similar way as presented in Figure S3.

We thus have two basic realistic generators of data for technological growth and progress (Figures S4 and S5, respectively). Once the various parameters for the two halves of this simple numerical model are appropriately chosen, the calculated trends for output growth and cost reduction simulate observed evolutions of these variables for many products and technologies (e.g., those in Figures S1, S2, and S3). We now demonstrate that, taken together, these two submodels are equivalent to learning curves reported in the literature (e.g., the one in Figure 1). This is shown in Figure S6, in which data points calculated by our model for cost reduction and cumulative production are combined and plotted on a double-logarithmic scale. One can see that over several orders of magnitude of cumulative production the results of our simulations can be interpolated by a linear fit with accuracy comparable to that of published learning curves. The main parameters for our simulations are the average growth rate in annual output, the standard deviation of this growth rate, and the total cost reduction achieved at the end of the simulation (other assumptions in the model such as the number of components, n, and cost reduction steps, mi, do not influence the simulation as long as they are large enough numbers s note that normally we do not have empirical estimates for these quantities). For the simulations depicted in Figure S6, the values of these main parameters were chosen so as to reproduce observed learning curve relations between cumulative output growth and cost reduction as known for energy technologies. As additional veracity check to validate our model we compare our learning curve simulations directly to costversus-output data as available in the literature. Figure 3 shows two arbitrary examples of reported learning curve data, for cost and cumulative output of PV modules (4) and Brazilian ethanol (11), respectively. The numbers for PV, collected over more than two decades, represent an average annual growth rate during this period of as high as approximately 40%/yr (4). During this interval the cost of PV modules decreased by more than an order of magnitude. For Brazilian ethanol the average growth rate was significantly lower, approximately 10%/yr, during a little less than two decades. Ethanol production cost reduced by about a factor of 3 during these years (11). We contrast these historic data with results from our simulations. As before, the main assumptions in our numerical model concern the average annual growth rate over the whole period under consideration, the standard deviation of this yearly growth, and the final total cost reduction, which are all taken so as to stay close to real-life values of these quantities (the three parameters are in fact extrapolated from the material reported in references (4) and (11)). We see that our model can reproduce well the learning curves published for these two cases. It can be readily shown that any other learning curve can be simulated in a similar way through our model.

4. Predictive Value of Cumulative Production

FIGURE 2. Illustration of our cost reduction model. The overall technology consists of n components, the cost of each decreasing stepwise over time on the basis of a multiplication of positive coefficients γij < 1.

The arguments presented in Sections 2 and 3 illustrate that learning curves can be interpreted as an envelope for two underlying and simultaneous processes: exponential growth and exponential progress. We provide evidence that learning curves for any product or technology can be dissected between two such exponential relations. If our “doubleexponential model” indeed reflects the true mechanisms behind learning-by-doing, then learning rates depend on trends for growth and cost reduction and are thus not necessarily constant. In this case, learning curves can at best be employed for the extrapolation of current growth and cost trends. This is an important observation that so far has received little attention in the scientific literature and poor consideration in spheres of policy making. A related point VOL. 43, NO. 11, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 3. Comparison of learning curve data for PV (4) and Brazilian ethanol (11) (() with data generated by our model simulations (0). is that, once the points discussed in Section 3 are taken into account, in an environment of exponential growth for a product or technology (or company, industry, or sector for that matter), every improvement that reduces costs, whatever its nature, can in principle result in a power law relation between costs and cumulated production. In other words, every type of cost-reducing progress, whether learning-bydoing proper, or process automation, economies-of-scale, or learning-by-searching (R&D), can contribute to the overall “learning process” for the product or technology under consideration, and can thus be assembled under the common denominator of the learning curve. We think this point is not always properly addressed in the learning curve literature (and cannot be immediately derived from eq 1 alone). Alternatively, our model does not proffer insight into what the nature of technology behavior may be in periods when (economic) growth is absent, or even negative, during which public R&D efforts are sometimes stimulated and technical progress may follow as a result (but with typically a delay in time due to the nature of the research process). Since learning-by-doing results from a sequence and accumulation of smaller improvements viz. cost reductions, both in time and of parts (see also (24)), learning curves can also be developed for individual components like for the final product. Because market forces usually level prices through competition between manufacturers, learning curves can similarly be derived for single firms, whole industries, and entire economic sectors. Condition in all these cases, however, as argued in the previous sections, is that they are characterized by growth in an exponential fashion. One could reason that the hypotheses of exponential growth and exponential productivity increase remain just empirical observations, not necessarily better substantiated than the learning curve of eq 1. Still, we observe that in economics far more examples are available of exponential growth and progress relations than of well-behaved learning curves. As remarked before, growth and productivity of a company or national economy are customarily measured in relative terms, i.e., in percentages with respect to the results achieved in the year prior. The learning curve of eq 1 does in principle not contain any additional information in comparison to the expressions of eqs 4 and 5. The former clearly does suggest, however, that cumulative production is the explanatory variable, while strictly speaking, we claim, it cannot be concluded that an increase in cumulative production is in general the underlying origin of observed cost reductions, even when this in specific individual instances may be the case. Indeed, we suspect that cumulative production may not be the best explanatory variable for cost reduction trends. Rather, time as variable has an important role to play, as well do patterns of phenomena like economic growth and technical progress as a function of time. Equations 4 and 5 are merely expressions to produce curved fits of data for production and costs as function of time. Naturally, the fitting parameters a, b, d, and f are not uniform across technologies, nor are they generally an 4006

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intrinsic constant property of a given product, especially when long periods of time are considered and stages of technical diffusion are reached beyond the phase of maturity. Rather, the parameters depend on the particular data set considered, in a similar way as the learning rate is normally not constant: in the majority of cases the best linear fit of costs versus cumulative production on a double-logarithmic scale depends on the length of the time series considered (see, e.g., (8) and (18)). Another way of demonstrating this is to consider the simplified example of a firm whose annual output grows at a constant rate β with simultaneously a constant annual increase in productivity. Let us also assume that the firm is a start-up, so that the cumulated production is small during the first few years s certainly a reasonable assumption for innovative products. The growth rate in cumulative output changes rapidly during the first years and initially differs considerably from the (constant) annual output growth rate. Under these assumptions, cost versus cumulative output data can generally be fit with good accuracy by a power law of the form of eq 1, hence implying a proper learning curve. For such a learning curve, however, the learning rate is not a constant and changes significantly over time. Figure 4 shows the learning curves for this simplified model after 10 and 35 years in the case of a 10%/yr growth rate and a 4%/yr productivity gain (both constant). In each of these two cases a well-behaved learning curve can be determined, but the difference in slope of the linear fit in the corresponding double-logarithmic plots is substantial: the learning rate is found to be 11% after 10 years and 17% after 35 years. Figure S7 shows the value of the learning rate as function of time for this straightforward example and demonstrates that in this case the learning rate tends to a constant value close to 20%. Even after 40 years of deployment, however, the learning rate still changes non-negligibly. This example illustrates well the kind of fundamental intricacies one experiences when determining and using learning curves. Others have described some of the difficulties in determining learning rates from historical cost and capacity data and using them to estimate future costs, among whom are Neij (9) for wind power, Schoots et al. (28) for hydrogen production technology, and McDonald and Schrattenholzer (8) and Ferioli et al. (24) for energy technologies at large. Many studies, however, brush over these fundamental issues. The few that take note of it provide support for our view that a good fit of cost-capacity data does not necessarily imply a constant learning rate. Another insightful exercise in this respect is to use cost and cumulative capacity data, e.g., for PV from Harmon (4), and calculate the learning rate after periods of say 5, 10, 15, and >20 years (see Figure S8). The variations in the calculated learning ratio prove to be significant. The misleading implications of relating costs to cumulative production, and thereby disregarding time as relevant variable in the development and use of learning curves, are also effectively exemplified by recent price data

FIGURE 4. Learning curves after 10 years (left) and 35 year (right) in the case of a constant 10%/yr growth rate and 4%/yr productivity increase. for PV modules as shown in Figure S9 (see SI, based on data from (29)). We do not claim that our model is more accurate, or possesses a greater predictive value for the future costs of a product, than the traditional learning curve model s we leave this discussion for further research. Rather, we would like to caution against too much confidence in extrapolations based on the hypothesis of a constant learning rate. Our model has, in our opinion, the advantage of highlighting the role of other relevant variables, like time, growth rate, and productivity, which elucidates the determination of the cost reduction potential for a product or technology. A relevant stylistic model to illustrate the link between cost, growth, and productivity is the well-known CobbDouglas production function that relates output to capital and labor, usually written as: y ) AK RL1-R

(12)

in which y is annual output, K is the capital employed in production, L is the level of labor involved, A is the productivity, and R is an appropriate exponent. Equation 12 expresses that additional production obtained through an increase in capital and/or labor will by itself not result in increased productivity and thus, according to this relation, not in cost reduction. It has been observed, however, that as new and more efficient capital goods replace obsolete ones, productivity increases (1) and that an increased level of investments in the production process significantly promotes productivity gains (30). The latter study analyzed the case of a steel mill owned by a large Swedish company that was left to operate during the 1930s and 1940s without significant capital investment or technological change (30). Still, the author observed, output per man-hour employed in this plant rose by almost 2%/yr. In other similar plants where significant capital investments were made in upgrading existing equipment output rose by almost 4%/yr. These analyses portray a model of progress similar to that described in this paper, with sequential investments renewing the capital stock and reducing costs through small incremental improvements. In particular, the study of the Swedish steel mill suggests that even a minimum level of maintenance required to keep a factory in operation may be enough to increase productivity, albeit perhaps modestly in comparison to the total industry’s potential. During economic downturns, however, overall productivity normally decreases (31). Also, as discussed by Goddard (22), the costs for a firm or an industry rarely decrease if annual output contracts. Indeed, even if the costs of raw materials are likely to decrease during recessionary periods, investments and innovation are halted so that lasting improvements cannot usually be realized. Commodity price reductions during economic recessions in principle may have beneficial effects on technology costs, but we think this effect operates within certain bounds. When this bound is exceeded,

e.g.,during a severe economic downturn, learning comes to a halt because the financial capital and overturn of physical capital is not available to implement innovations in the production process s technical progress thus halts. It can hence be argued that neither the exponential growth model nor the learning curve are reasonable approximations to describe a period of economic recession or other changes in the fundamentals of the global economy or regional sections of it, such as related to commodity price surges, natural resource constraints, or environmental crises like climate change. By definition, exponential growth regards temporary contractions only as short-term departures from a long-term growing trend s hardly a suitable model to analyze real recessions. On the other hand, since cumulative output growth slows down during recessions, the traditional learning curve model implies a slower productivity increase as function of time in order for the learning rate to remain constant. This result is broadly in line with observations since slower growth will result in a lower level of investment and, ultimately, in reduced productivity gains. While the conventional learning curve model thus captures an important feature of the relation between capital investment and productivity, the hypothesis of a constant learning rate amounts to a particular and unwarranted nonlinear relation between growth and progress. Since it provides no further explanation of this relationship, the learning curve appears too simplistic to accurately describe reality alone.

5. Discussion In this article we demonstrated that technology growth and cost can usually be approximated by exponential functions over time, and that these two functional relations taken together are essentially equivalent to the learning curve. Since it accounts explicitly for the variable time, our new model supplies additional information about the future cost reduction potential for innovative energy technologies. Our analysis shows that the learning rate may well not be an intrinsic characteristic of a technology, as it is often suggested in the literature, and, most importantly, that it may vary considerably over time, with the concomitant consequences for the use of learning curves for private strategic planning and public policy making in the energy sector. In the SI we discuss our results further and put them into the perspective of some recent publications in the field, among which Wene (32) and Neuhoff et al. (33).

Acknowledgments This paper is dedicated in memory of Leo Schrattenholzer.

Supporting Information Available Many figures and explanatory text that support the main arguments in this article. This information is available free of charge via the Internet at http://pubs.acs.org. VOL. 43, NO. 11, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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