Factorized equations are a way to express in a simple, usually linearized fashion, the empirical dependence of a phenomenon on a number of variables. Maybe the most famous one is Fermi’s “four-factors” formula, to express the neutron criticality of a mass of fissile material as the product of four microscopic quantities, each defining a different process by which neutrons are produced and consumed. I wrote quite some time ago in this letter, a discussion about the Buckingham theorem, the magical trick by which non-dimensional equations help us discover fundamental relations among physical quantities. A factorized equation is something similar, however without the requirement of keeping dimensions homogeneous. In short, you can put in the equation any multiplicative quantity you like, a bit like tossing ingredients in American pizza.

Yoichi Kaya was a Japanese economist who studied energy policies. About thirty years ago, the man wrote down an equation that was to became famous, but certainly not because of its complexity. In fact, Kaya’s equation starts from an identity, such as X=X. The fact that X is equal to X is something on which, I presume, we are all in fair agreement. Now, let us posit the quantity G as the total World’s emission of greenhouse gases in one year. If we now make the bold statement that X=G, the former identity becomes G=G. Fantastic, isn’t it?

Now, let us posit E as the total World’s energy consumption in one year, and write E/E=1. This is where high mathematics comes on the rescue: by multiplying the identity by 1, we are sure that it will remain an identity, so let us write G=(G/E)*E. That is encouraging.

As a third move, let us put P as the World’s total gross product in one year (it could be its value in whatever currency), and by pushing forward our luck, write again P/P=1, and make the next step in the still identical identity as G=(G/E)(E/P)*P.

You got the idea. Now we introduce M as the World’s total population, and get the full form of Kaya’s equation, as:


And now you could ask, why such an equation (which, by construction, should rather be called an identity) is so important?

In this simple form, the equation says that the total greenhouse gas emissions all over the World can be understood as the product of four factors: G/E, the gas emission per fraction of energy consumed; E/P, the amount of energy needed to produce one unit of GDP; P/M, the amount of goods and services per capita, or else the average individual income; and M, the World’s population.

Let us follow the notion that in order to calm down the heating trend of the Earth’s atmosphere, G should be reduced at least by a factor of 3 by the year 2050, which means a yearly reduction of the order of 3-4%. (It is not important to have a precise value, which may be subject to adjustment according to the continued research effort in global warming etc, it is enough to use just a reference figure.) Of course, an obvious solution could be to reduce the World’s population M. Sometimes, especially after reading the morning news, I myself tend to think that this should be a beneficial action for a number of other reasons, besides gas emissions. However, it does not seem to be the most straightforward one to implement. Instead, even the most cautious estimates of social scientists and demographers have M to increase by about 25%, from here to 2050. Which means that the other three factors in Kaya’s equation must instead decrease by a total of 3,75, instead of just 3.

Another not-so-obvious solution would be to reduce our standards of living, that is, work on the P/M ratio. While such an attitude is indeed considered as a possibility, especially by some minority groups in some developed countries, the largest part of the World’s population is in fact still striving to do the opposite, that is, to attain the highest possible standards of living, of which our “western” society appears to be the most recognizable representation. But even in our “developed” countries, the inversion of the P/M trend does not seem advisable, for example if we want to avoid the collapse of our social pension systems (strictly true for countries adopting the European mutualistic system, somewhat less but still partly true for countries such as USA or Canada). On the other hand, the perspective of unlimited growth of P/M must face the exhaustion of non-renewable resources, starting with fossil oil but looking forward to raw materials, minerals, phosphates, and counting. It is likely that if such consumption of resources were to be factored in the P/M calculations, we might surprisingly find that the World is already following a downward trend. However, if we stick to the global predictions that give an average 2% yearly increase of P/M worldwide, we get another factor of 1,8 over 30 years. Therefore, the two remaining terms in Kaya’s equation must now account for a 6,75 reduction, which more realistically could be taken as a factor of 8.

As far as the ratio E/P, there is little doubt that it has a decreasing trend. Since the beginnings of the industrial revolution, in the early XIX century, the energy intensity of the economic system has been constantly decreasing: as far as the technology level increases, the amount of energy to produce economic value decreases. When coupled to the observed increase of energy consumption, this trend explains the humongous increase of capital value in the past century, and continuing in the first quarter of this century. The energy consumption increased by about a factor of 3 between 1800 and 1900, and by more than a factor of 9 between 1900 and 2000 (see e.g. https://www.encyclopedie-energie.org/en). This is an increase of about 2,15% per year in the last century, to be compared to an estimated increase of the population of about 1.4% per year. This larger increase of energy usage per capita, coupled to the higher efficiency brought about by technology, means only that the total economic output (in dollars, euro, or whatever) has expanded much more than linearly, in fact it is rather close to 3% per year (https://ourworldindata.org). However, according to the best economic models, the main responsible for the economy growth is not just the positive coupling between energy and technology, but rather the increase and improvement of global trade (to which new technologies and innovation certainly provide a substantial contribution).

Further reducing the energy intensity is desirable, however it is going to be increasing difficult. Think for example of the good progress made in lighting, by moving from incandescent bulbs (Joule heating) to LEDs (semiconductor excitation): after such a change, which could mean energy savings by a factor of 5 or 6 if all bulbs in the world are replaced (lighting amounts to 20% of energy consumption, and 6% of CO2 emissions), it would be very difficult to get another similar gain by a next technology switch. If we want to give an optimistic estimate, let us assume that the E/P ratio will decrease by another factor of 2 over the next 30 years.

Then, we move to the last factor, G/E, that should therefore decrease by at least a factor of 4, to keep with the required global decrease of G by a factor of 3. Here the choice is very limited: the only way to reduce G/E is to reduce, if not eliminate, energy sources that are accompanied by greenhouse gas emissions. The real question is, how realistic is this perspective? In the course of history, and most notably the history of technology, there are no examples of energy sources that have disappeared: we only have examples of new energy sources that added to existing ones, but never observed the dismissal of one of the old sources being fully replaced by a new one. Oil has not replaced coal, instead the world consumption of coal has been constantly increasing for the entire XX century, and has slowed down (but not decreased) only in the last ten years. Countries such as USA, Russia, China, India, Australia, are the major producers of coal as well as some of the major end-users of energy. Coal is rarely exported but is typically used close to the production sites: this implies that it is difficult to ask a country to stop using part of its wealth, if not with a corresponding compensation.

The Kaya identity plays a core role in the development of future emissions scenarios in the IPCC Special Report on Emissions Scenarios. The scenarios set out a range of assumed conditions for future development of each of the four inputs. The importance of Kaya’s equation lies in its ability to point out the factors that regulate the amount of gas emissions, together with the regulators that can act on each factor. While M is a variable that is in the hands of demographers, they can do very little (and probably don’t want) to modify it. On the other hand, the ratio P/M is eminently political: lawmakers, economists and political representatives have various tools at their disposal to modify the distribution of wealth and consumption of goods and services in their countries, for example by taxation and industrial regulations. The first two ratios, G/E and E/P, are the realm of scientists and engineers.

One important objection to the equation is that there is no distinction between production and consumption: it is taken for granted that each unit produced is immediately consumed. Unfortunately, this is not at all the case. A lot of the total production is not used, but goes wasted for a number of reasons (as a physicist, I can start seeing some entropy and irreversibility to creep in). In all areas of the global economy, it is a sad rule that we produce more, and much more, than we actually need, notwithstanding the accompanying fact that such excess is distributed very much non-homogeneously, so that there are people trashing food while other people starve. In fact, we could better write the factor P/M as the product (P/C)(C/M), that is production/consumption times consumption per capita. The ratio P/C is necessary greater than 1, since we cannot consume what we do not produce; however, it is generally much larger than 1, and this is certainly an area in which much could be done, trying to bring it as close as possible to 1. For example, there is a waste of at least 30% in agricultural (over)production, and a waste of at least 50% in transportation (individual vs collective transport, moving goods across long distances instead of favouring local consumption,…), a large waste in equipment (we routinely dispose of almost new or repairable devices, a trend helped by the so-called “programmed obsolescence”), as well as a large waste in building lots of new housings that often remain unused. It is easy, but certainly unfair, to point out that in the “good old world” all such issues had a much smaller impact, we produced only the necessary food, used only the necessary tools, lived in the same home generation after generation. However, it appears very clearly that even before massively investing in trying to change, with little predictability, the structure of energy sources in ways that have no parallels in human and technology history, a similar gain can be obtained by politically inducing industries and citizens to respectively adjust their production and consumption attitudes, something that could be done without forcing a drastic devaluation of our lifestyles (the C/M factor) or, more drastically, by waiting for a stray meteorite to kill 1/3 of the world’s population.

The four (or five) factors of our future

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