It is often difficult to imagine the real consequences of exponential growth. To illustrate this, an ancient story related to the invention of chess is usually used. According to the legend, when the creator of the game presented his invention to the king, he was so impressed that he offered him the reward he wanted. The inventor asked for something seemingly modest: that a single grain of rice be placed in the first square of the board, two in the second, four in the third, and so on, doubling the amount in each square until the 64 squares were completed. The king accepted without hesitation, believing that the reward was negligible. However, he soon discovered that the humble request hid a huge growth: when the counting reached the last squares, the amount of rice became so enormous that it exceeded all the production of the kingdom for centuries. What seemed like a trivial exercise ended up revealing the surprising power of exponential growth, a phenomenon that, although it begins almost imperceptibly, can lead to astronomical figures in a few steps.
If we accumulate the rice corresponding to the squares of the board, the sum Sn=2n−1 exceeds the current annual world production when we complete box 55. The reference we take is the FAO’s most recent estimate for 2025/2026, at around 563.4 million tonnes, assuming that one kilogram of rice includes approximately 50000 grains. This result shows that the accumulated exponential growth exceeds even global productive magnitudes in a few stages.
We needed 55 doublings to reach a meaningful physical limit. In the field of electronics and telecommunications, this type of two-factor growth is usually described by using the term “octave”, taken by analogy from music, where each octave also represents a doubling of the frequency. We will keep this data: in the case of rice, world production covers an exponential growth of 55 octaves.
How long does exponential growth take to double a magnitude? The following rule of thumb is usually provided: growth of x% per year doubles the magnitude in 70/x years. A growth of 2% per year would require 35 years to double, and a growth of 7% per year would require only 10 years. Proving this rule is not difficult [1. footnote].
To understand the speed of exponential growth, it is often explained to us by imagining a petri dish inoculated with a single microorganism. Under optimal laboratory conditions, many fast-growing bacteria—such as Escherichia coli—have a doubling time of about 20 minutes. If we start from a single cell and it divides successively, the population follows the 2n progression, so that, after about 30 duplications, the order of one billion cells is reached, which is approximately the capacity of a petri dish. This implies that, supposedly, the plate would be fully colonized in about 10 hours (30×20′).
Although this example is common in popular science, it is also misleading if interpreted literally. It could be said that, if a bacterium doubles its population every 20 minutes, the plaque will be half full just 20 minutes before it is completely saturated, and 12.5% full only one hour before total collapse. This conclusion is correct only under the strict assumption of unlimited exponential growth, but that assumption is false in any real biological system. In a petri dish—as in any finite environment—the population does not continue to double until the final instant: long before the surface is completely covered, bacteria begin to experience nutrient reduction, waste accumulation, competition for space, and metabolic constraints that drastically slow the rate of growth. In these circumstances, the dynamics cease to be exponential and are better described by a logistic model, in which growth slows down progressively as the population approaches a limit or carrying capacity. Therefore, the image of an unstoppable growth that remains intact until the last moment of “collapse” is a useful simplification to illustrate the power of duplications, but it does not reflect the real behavior of a microbial culture, or of many other physical systems (in fact, of the vast majority).
A little bit of math analysis (not much)
Let’s express mathematically what has been said so far.
Exponential growth is one in which the temporal variation of a magnitude is proportional to that magnitude. For example, a growth of 1% per year is exponential growth. Analytically, let’s assume a function f(t):
Where a is the growth rate (e.g., 1% per year, 0.01). In this case, the function f(t), with t expressed in years, is expressed as follows (to do this we have to solve the previous differential equation):
Where f0 is the value that f takes at the instant t=0 [2. footnote].
Exponential behavior has another very special particularity. If a magnitude grows exponentially, the rate of change of growth will also be exponential (growth of growth: the derivative of f), and total cumulative growth (the integral of f) will also be exponential [3. footnote]. This means that if a magnitude has an exponential behavior, this behavior will also manifest itself in other directly related quantities.
Now let’s look at logistic behavior—the kind of growth of a population of bacteria in a petri dish. The simplest expression of a logistic function (making different coefficients equal to 1, for simplicity; it is the standard logistic function) is as follows:
In solving this differential equation, f(t) starts from very small values (f(0) almost 0), and never exceeds the value 1. The following figures show the behavior of the logistic function as a function of time (Source: Logistic function, wikipedia).
The upper left graph (“position”) shows the time evolution of the function f. It is S-shaped; therefore, it is also called sigmoid function. The start of the growth of the curve is exponential. It can be seen that if f(t)<<1 (as happens at the beginning), the resulting differential equation is identical to that of exponential growth. The lower left graph (“velocity”) shows the growth rate of the function. The start-up is also exponential, but it reaches a maximum and from there the growth slows down, until it becomes almost zero. It is bell-shaped. This makes the function f tend to 1, and never exceed that value. The graph above right (“acceleration”) shows the rate of change in growth. It starts exponentially, while we go up the growth bell its value is positive, at the top of the bell it becomes zero, and in the descent of the bell its value becomes negative (the growth decreases, until it becomes almost zero).
Logistic behavior is very simple to model and analyze. A multitude of phenomena in nature subject to growth processes that are in turn limited by external factors can be roughly modeled by means of variants of the logistic function. Systems grow exponentially at first, until the external limits (availability of space, material resources, energy, waste sinks, structural limits) end up imposing a limit to growth. [4. footnote].
Examples in Nature of exponential growth with a tendency to saturation
The number of cells in the human body evolves from a single cell (zygote) after fertilization to about 30 billion in adulthood. In the first few days, growth is exponential, going from 1 to about 100–200 cells in the blastocyst. During embryonic and fetal development the number increases several orders of magnitude, reaching about 1012 cells at birth. After birth, growth slows down until it stabilizes in adulthood, where there is a dynamic balance between cell division and cell death. In total, the growth in the number of cells in the human body encompasses approximately 45 octaves over 18 years, although the first 40 octaves are covered during the first 5 months of gestation. From that moment on, the growth in the number of cells is much more moderate, and some cells grow, above all, in size. The figure below shows this evolution.
Note that the vertical axis of the graph is logarithmic scaling, rather than linear. In a representation with a logarithmic vertical axis, exponential growth appears as a line, which reflects that the magnitude is multiplied by the same factor at equal intervals of time.
Another example of exponential growth with a tendency to saturation is in the field of electronics, specifically in the behavior of a diode. The diode is a pn junction of semiconductor material, and it is the most basic device in microelectronics. By applying a positive voltage difference (V) between its contacts, an electric current (I) circulates through the diode. The equation that relates current and voltage in a pn junction (a diode) combines principles of quantum physics, statistical mechanics and classical thermodynamics; it was proposed by Shockley (one of the inventors of the transistor) in 1949 and presents an exponential behavior. In an ideal diode:
Where IS is the diode saturation current, VT is a temperature-dependent parameter (around 26 mV at room temperature), and n is an ideality factor between 1 and 2. This exponential behavior causes a diode to conduct very high currents that would lead to its destruction, if the applied voltage reaches just over one volt. But in a real diode there are other phenomena that make the current that flows through it never too great. The various sections of the diode have small ohmic resistances (ΣR) to the passage of current that limit the voltage applied to the junction:
In the case of a typical silicon diode such as 1N4007, the most popular in our electronics teaching laboratories, the current increases exponentially with the voltage applied from currents on the order of nanoamperes to around one ampere. From that point on, the series resistance of the device introduces an ohmic limitation that makes the growth no longer exponential. The exponential growth of the current in this diode therefore extends over approximately 30 octaves, and is self-limited by other mechanisms inherent to the operation of the device (for example, the existence of ohmic resistors).
Exponential growth is not exclusive to biological systems or abstract mathematical models. It also appears very clearly in the development of microelectronics. The best-known example is the so-called Moore’s law. Formulated in 1965 by Gordon Moore—who along with seven other scientists would leave Shockley’s diode-manufacturing company for its authoritarian style and found the companies Fairchaild and Intel at the end of the 50s of the last century—this law establishes that the number of transistors that can be integrated into an integrated circuit doubles approximately every two years. Since the number of transistors is directly related to processing power, this doubling implies an exponential growth in computing power. In other words, every two years a technological “octave” is produced: the processing capacity of microprocessors is multiplied by two. This behavior has been maintained, with remarkable regularity, for more than half a century, constituting one of the most spectacular and sustained examples of exponential growth in a technological system.
How many octaves does Moore’s law have behind it? To quantify this growth, we can take a simple benchmark. The first commercial microprocessor, the Intel 4004 (1971), contained about 2,300 transistors, while NVIDIA’s R100 AI GPU has 336 billion transistors, which is equivalent to a growth of 27 octaves.
For several decades, Moore’s law has been an almost ideal example of sustained exponential growth. However, in recent years physical and economic limitations have appeared that have slowed the rate of doubling (in fact, we have been saying this for almost 30 years now). These limitations have not stopped growth, but have forced the emergence of new solutions: first it was the reduction of the size of transistors and the use of new materials, but later came new transistor architectures, parallelism (multicore architectures), three-dimensional integration (stacking microprocessors) or heterogeneous systems (combining CPUs, GPU and now AI accelerators in a single system). These innovations do not eliminate physical limits, but they displace or circumvent them, allowing technological evolution to continue through a succession of phases of approximately exponential growth. The following graph (with a logarithmic vertical scale) clearly shows the exponential growth in the number of transistors in commercial microprocessors over five decades.
Source: OurWorldinData.org
The exponential growth in microelectronics is not limited to Moore’s law. Ray Kurzweil, an American inventor and futurist known for his work in pattern recognition and for his role in the development of digital technologies (Ray Kurzweil has led Google’s engineering team so that its language models not only process data, but also understand the semantic and conceptual context of the human mind). He proposed a generalization of this idea under the name of the Law of Accelerated Returns. According to Kurzweil, Moore’s law is just a particular case of a broader phenomenon: technological progress is not only exponential, but the rate of growth itself also tends to increase over time. Each new generation of technology allows for the development of more powerful tools, which in turn accelerate the development of the next, giving rise to a process of positive feedback. To quantify this phenomenon, Kurzweil uses indicators related to the efficiency of information processing, typically expressed as calculation capacity per unit of cost (e.g., operations per second and per dollar). This type of magnitude makes it possible to compare very different technologies over time—from the first electromechanical systems to today’s integrated circuits—and to observe an exponential trend sustained for more than a century.
Source: Kurzweil Library. In this indicator, growth accelerates: despite having a logarithmic vertical scale, the slope of the curve is not constant (a line), but tends to increase.
If we translate this evolution into our language of doubling, the result is remarkable: the cumulative growth in computing capacity per unit of cost from the beginning of the twentieth century to the present is in the order of 53 octaves. In other words, a phenomenon comparable, and even superior, to that observed in Moore’s law or in the exponential behavior of electronic devices such as the diode, and approaching the 55 octaves of the growth margin of world rice production.
Kurzweil’s conclusion from this trend is even more surprising. If exponential growth—and the acceleration of that growth—continues, a point will be reached where technological progress will be so rapid and profound that it will be impossible to extrapolate the behavior of the system. He calls this point a technological singularity, and places it around the middle of the twenty-first century. The singularity would not be an abrupt or unexpected phenomenon, but the natural consequence of a prolonged process of sustained exponential growth for decades that would lead to artificial intelligence surpassing human capacity, radically transforming civilization by irreversibly merging our minds with digital technology.
Personally, I’m more than skeptical of Kurzweil’s singularity. Starting with the indicator he proposes to demonstrate his law of increasing returns (operations per second and per dollar). The denominator of this indicator is the economic cost, and as ecological economics rightly points out, we must be tremendously skeptical about this type of indicator. Economic cost calculations are blind to an immensity of negative externalities that are hidden from conventional economic calculation. Many externalities are difficult to assess, and when some economists evaluate them, they assume a principle of substitutability that is not applicable to many concepts. It would be much more appropriate to evaluate this performance of computing capacity not based on its conventional economic cost, but considering its ecological footprint. Within this ecological footprint, which must inevitably be multidimensional, an important weight must be played by its material footprint and its energy footprint, understood as the essential material and energy support to carry out this calculation. I am convinced that the temporal evolution of this indicator would lead to long-term exponential growth, but I am also convinced that the evolution would be close to that of a logistic function after a while—let us remember once again that the logistic function has an exponential start. Why? Because any process with a material support—AI has it, as our thinking has it—requires, at the very least, the use of materials and energy consumption. Our planet is what in thermodynamics is called a closed system: the availability of materials on our planet is finite—the materials used can be recovered after use, but that comes at an often very significant energy cost—and the flow of incoming energy (solar energy) is also limited. Sooner or later, these energetic and material limits will come into play. In fact, there are numerous indications that global industrial society is already approaching—and in some cases surpassing—those biophysical limits. One of the best-known frameworks is that of planetary boundaries, which identifies critical thresholds in fundamental processes of the Earth system (climate, biodiversity, nitrogen and phosphorus cycles, freshwater, etc.). The most recent assessments show that several of these limits have already been exceeded, increasing the risk of abrupt and irreversible changes in the systemic functioning of the planet.
I am going to focus now on the energetic aspect, which I know more closely. And I’m going to allow myself a self-citation. In 2008 I wrote the book Energia urriko mundu baterako gida, published by Manu Robles-Arangiz fundazioa. The central idea of the book was that a sustainable future scenario of our civilization requires an energy transition from fossil fuels to the exploitation of solar energy flows; but this transition, if it wants to be sustainable and fair, has to manage energy scarcity. By analysing the potential of the different technologies that exploit solar energy flows, I carried out in the book a simple prospective exercise of what could be a future evolution of photovoltaic solar energy in the world from the year 2000 to 2030. The results are synthesized in the two graphs that I show below, and which I will now explain.
The graph on the left shows the evolution of global photovoltaic power between 2000 and 2030. The evolution is shown both in installed power and in economic cost. Both the annual installed power and the accumulated installed power are also shown. Note that the vertical axes (both in power, GW, and money, €) are on a logarithmic scale. The almost straight behavior (two segments, until 2020 and from 2020) make it clear that my model assumed an exponential growth, which would take from 2 GW of photovoltaic systems installed in the year 2000 to 2 TW of installed power in 2030 (2000 GW). Specifically, my model assumed an annual growth in global PV system production of 30% between 2000 and 2010, 35% between 2010 and 2020, and 20% from there to 2030. The model assumed a “learning effect” of 18% in the economic sphere, based on historical experience. This means that a doubling of installed power brings with it an 18% reduction in costs. This would lead to an 84% cost reduction in 30 years. The model also considered an energy learning effect, which implies a 15% reduction in the energy return time (an indicator related to the energy profitability of the installation, i.e. the energy cost invested in the construction of the system for each unit of energy generated subsequently). Photovoltaic systems demand a substantial amount of energy in their manufacture, which puts them in a situation of energy deficit when they start generating electricity. During the first years of exponential growth, in addition, the accumulated electricity generation is very low. This means that the net balance of available energy is negative in the first few years, and it takes two decades to exceed 70% of total generation (graph on the right). But after a time of exponential growth, photovoltaic systems reach a huge generation park thanks to the fact that they are very cheap and very profitable in terms of energy.
Like the vast majority of prospective studies, mine has also turned out to be (slightly) wrong. While I was forecasting 2 TW of PV systems in 2030, the latest data tells us that we have just surpassed 3 TW globally. My forecasts of 20% growth in installed capacity from 2020 onwards have been clearly exceeded, and in the last three years annual growth has been around 35%. My forecast of cost reduction (85% in 30 years) has also been exceeded, reaching that level 5 years earlier, and with room to continue reducing costs. If we take as a reference the first installation connected to the grid in Europe (Lugano) in 1982, of 10 kW, the 3 TW of installed capacity currently implies a growth of 28 octaves [5. footnote].
Does this mean that we can expect the growth of PV generation to continue to grow indefinitely? Not at all. To begin with, just as in a human adult cell growth serves fundamentally to replace cell tissue that is dying, in the future the productive capacity of photovoltaic systems will be aimed at replacing equipment that reaches the end of life—hopefully with a very high degree of recycling. We currently have an annual manufacturing capacity of 1.8 TW of PV systems in the world. If we assume a life span of around 25 years, this would maintain (very approximately) a stationary PV park of 45 TW. That’s just 4 more octaves to add to the 28 calculated above. At the same time, the current exponential growth we are experiencing will also be affected by other limits that will slow it down: even if the recycling and reuse of materials increases, they will never be 100%, and in any case, growth in absolute terms needs new primary materials; land requirements also impose constraints; and other technological limits, such as the need for electrical storage (at night there is no sun), will also impose limits. Sooner or later, the current exponential growth will be transformed into a logistic behavior.
The contribution of photovoltaic solar energy will be essential in any sustainable future scenario. Assuming an annual hourly productivity of 1000 hours (a reasonable world average, assuming a huge deployment), 45 TW could supply 45000 TWh per year. This is more than the world’s annual electricity consumption, which is barely over 30000 TWh, but it is also much lower than the total primary consumption, which is close to 180000 TWh. We are therefore returning to a scenario of energy scarcity, which will have to be managed with social and environmental justice.
To recap
Exponential growth appears naturally in the development of living systems, understood in a broad sense, within the dynamics of complex systems: we are not referring only to biological organisms, but also to social, economic or technological systems that have the capacity for adaptation, reproduction or accumulation of complexity. A clear example is human civilization.
In all of these systems, initial behavior is usually dominated by growth. This growth is very often exponential, since the very structure of the system makes future development dependent on its present state: the larger or more complex the system, the greater its capacity to continue growing. The embryonic development of a human being is a paradigmatic example: the number of cells grows practically exponentially during the early stages of gestation. However, this initial growth is not the “normal” state of the system, but a transitory phase. In the case of a human organism, after birth the growth in the number of cells slows down progressively and gives way to other types of processes: growth in cell size, functional differentiation, learning, acquisition of experience, cultural development, spiritual development. This pattern is general. Living systems do not grow indefinitely, but tend toward states of dynamic equilibrium. Growth is limited by internal and external factors: availability of resources, processing capacity, structural stability, accumulation of waste. These limits are not accidents external to the system, but an inevitable consequence of its own dynamics. Precisely for this reason, exponential growth naturally gives way to logistic-type behavior. In this context, it is appropriate to introduce the concept of homeostasis. In a living system, the fundamental objective is not to grow without limit, but to remain in stable operating conditions in the face of external shocks—to be resilient. Homeostasis describes this capacity for self-regulation: the maintenance of key variables (temperature, available energy, internal composition, etc.) within certain margins compatible with the survival of the system. Growth should be understood as an initial phase that allows the system to reach a scale at which homeostasis can operate effectively. It must therefore be a time-limited process. It is not an end in itself, but a stage within a broader dynamic. Trying to prolong indefinitely a phase of exponential growth in a finite system is not only unrealistic; it introduces instabilities that end up manifesting themselves in one way or another. Carcinogenic processes in living organisms are an example of uncontrolled growth that breaks the homeostasis of the system and ends up compromising its viability.
Lessons for today’s civilization
If we accept that exponential growth is a natural but transitory phase in the evolution of complex systems, the immediate question is how this idea fits into the development of contemporary industrial civilization. The global economy of recent decades—in reality, of the last two centuries — has been strongly based on dynamics of sustained growth, in many cases of an exponential nature, both in material production and in the consumption of resources and energy. Industrial civilization interprets this growth not as a phase, but as a goal in itself. Continued economic growth—measured through indicators such as GDP—has become a fundamental organizing principle, coming into direct tension with the biophysical reality of a finite planet.
From this perspective, a key lesson is that growth cannot be the ultimate foundation of a viable economic system in the long term. It should be understood, in any case, as a necessary phase to reach certain levels of development, but not as a permanent state. The challenge is to steer that growth toward a transition that leads to a stationary economy, capable of staying within the biophysical limits of the planet. This idea raises profound questions about the dominant economic model. Contemporary capitalism, in its different variants, has been structured around mechanisms that require continuous growth: capital accumulation, return on investment, expansion of markets. In the absence of growth, these mechanisms tend to generate internal tensions (financial crises, unemployment, debt problems) that hinder the stability of the system. Therefore, the transition to a stationary logic is not simply a technological challenge, but an institutional and cultural one. It involves rethinking fundamental aspects such as the distribution of wealth, tax systems, the role of work or the very definition of well-being. It means, ultimately, shifting the focus from the quantity of production to the quality of life of all people, within a viable ecosystem.
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[1. footnote] The starting condition can be formulated as follows:
Where x is the annual growth, and n is the number of years. Taking logarithms:
On the other hand, we know that if x<<1:
Taking logarithms:
Replacing:
[2. footnote] Exponential growth can be modelled by discrete growths (a growth of 1% per year) or by the continuous function derivative, df(t)/dt. In calculus we work with continuous functions, but in the numerical models of a computer algorithm we will work with discrete growths. Both approaches are equivalent. The time change of a discrete function with an annual growth of 1% will be:
Where t takes discrete values 1, 2, 3… This can be found to be equivalent to the following continuous function:
An annual growth rate of 1% is equivalent to an infinitesimal growth rate of 0.995%.
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[3. footnote] In our first year of engineering, classmates told a joke (very bad, I apologize in advance): There are two functions in space, and one says to the other: “I’m going to derive you!” The other responds: “I don’t care, I’m exponential of x!”
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[4. footnote] Logistic behavior arises naturally when using the modeling of complex dynamic systems based on flows and stocks. Let’s imagine that we are modeling the growth of a population. Population is the stock, and growth is the flow. It is reasonable to assume that growth (flow) is proportional to the population (the stock), on the one hand, and to the distance between that population and the carrying capacity (maximum population that the environment can support). Bringing this behavior to the mathematical field leads us to a logistic function. An example of these modellings, carried out with VENSIM, can be seen in As if we were ants. Collapse is not inevitable in complex societies.
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[5. footnote] Even if it is infinitesimally, I feel part of the enormous success of the development of solar energy. From 1994 to 2008 I collaborated as a pre- and postdoctoral researcher in the improvement of industrial manufacturing technologies of crystalline solar cells, the dominant ones in the market.
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