The Kaya Identity Crisis

July 24th, 2014 by Roy W. Spencer, Ph. D.

There have been several posts over at WUWT regarding whether the Kaya Identity equation is useful, or mathematically trivial, or just a tautology.

The Kaya Identity is a specific application of the more general IPAT (I=PAT) equation which estimates the global environmental impact I based upon what are believed to be the main drivers of I, usually put in terms that economists find useful and can estimatepopulation, per capita GDP, etc. You can read more about it here.

To get total global CO2 emissions with the Kaya Identity, you multiply together (1) population , (2) GDP per person (affluence term), (3) energy used per GDP (energy intensity) and (4) the amount of CO2 released per energy used. Again, the terms used are ones economists work with, and so it is more useful in economics and policymaking circles than in, say, climate science.

As Willis Eschenbach pointed out, simply as an algebraic equation, you can cancel out terms in the Kaya equation and get the trivial result that CO2 = CO2. This is what seems to have generated much of the hoopla over at WUWT.

But the same as true of just about any equation where the physical units must balance on both sides: say, the equation to estimate the miles driven if you know the average speed and the total time driving:

Miles = [hours]x[miles/hour]

You can cancel out the hour terms in the above equation, and get the seemingly trivial result that miles=miles but the equation is still useful.

The same is true of the Kaya Identity. It is a useful tool, to the extent that the individual terms on the right hand side really are the main economic-related drivers of the quantity on the left hand side…and the units match.

Also, as Willis points out, you can put all kinds of silly terms in an equation with the units on both sides simplifying to the same thing. But the unit matching is only a necessary but not a sufficient condition for an equation to be physically meaningful.

The bottom line is that I don’t see anything wrong with the Kaya Equation.

74 Responses to “The Kaya Identity Crisis”

Toggle Trackbacks

    • David Springer says:

      Dr. Spencer your example is wrong.

      It fails to properly label the factors.

      The left side of the equation is miles traveled not simply miles.

      The first hours term on the right side should be hours traveled not simply hours. And the second instance is hour not hours.

      Properly then MilesTraveled = HoursTraveled * Miles/Hour.

      There are no terms which can be cancelled.

      Try again.

      • TonyM says:

        Fits nicely, I thought.

        Average speed = miles/hour = miles travelled/hours travelled

        Perfect; we would not want to spoil the perfect simple solution to CO2 emissions.

  1. Johan says:

    Mathematically speaking, there is nothing wrong with the KAYA equation. How Willis could botch that up, is beyond comprehension.
    There are some caveats. There is no reason to believe that the four terms on the right hand side are the only driving forces, or for that matter, that they are “fundamental”; but more importantly, the terms on the right hand side are not independent of each other. This is recognized by IPCC.
    The far bigger problem is how KAYA is being used for policy purposes, or rather “abused” or “misused”.

    • MRW says:

      Willis doesn’t understand double-entry accounting or the fact that the Kaya expression is an accounting artifact, not a measurement. Many people over at WUWT who think like Willis have the same problem discussing the deficit and the national debt. It never dawns on them to look at the other side of the ledger.

      • MRW says:

        For example: the National Debt, that huge $17+ trillion figure, is nothing more than the total amount of US money in every pension fund, university trust, business and household bank account, savings bond, and mattress and piggy bank TO THE PENNY. (Why is it to the penny? Because the US govt, sole monopoly creator of the currency, has an accounting record of every penny it has ever produced since 1791, minus those destroyed, which are called taxes.)

        If you eliminate the national debt, you completely and utterly impoverish the people, render them penniless. Not one red soux in their pocket.

        That’s what the other side of the ledger means.

  2. slp says:

    Except that in your example the [hours] numerator is not identical to the [hour] denominator. With the Kaya identity they are dividing the exact values on the denominator of one “factor” to the numerator on another, so units are immaterial.

    • wfoster says:

      You are mistaken, and if you think about it you will see Mr. Spencer is correct.

      Usually, we want to back out miles/hour from the observations of miles traveled and hours duration. Miles (plural)divided by hours (plural), give us the result: [miles/hour] = [miles]/[hours]. But you could also want to know the miles traveled from an observation of duration and observations of velocity, or [miles] = [hours]x[miles/hour].

      Economists, in an accounting mode, use identities like the Kaya identity all the time to decompose changes in some variable of interest into separate observables. For example, why did some guy travel more miles than another fellow? To what degree is the difference due to different durations of the trips or to different velocities?

      The real potential problem with the Kaya identity is the possible illusion of control over all of the separate components of the decomposition, rather than the decomposition itself.

      • Okiami says:

        The denominator in “miles per hour” is 1. It does not cancel out plural hours. The example therefore is inapposite to the Kaya identity.

        • Jere Krischel says:


          X miles = Y hours * Z miles/1 hours

          The hours *units* cancel, but Y does not cancel with 1. It can’t unless magically, it’s 1.

          The “Kaya Identity” omits the variables.

  3. MRW says:

    In double-entry accounting, the two columns must net to 0 (zero). That doesn’t mean the assets and liabilities listed in each column are trivial.

  4. JohnKl says:

    Hi Roy,

    You stated:

    “The bottom line is that I dont see anything wrong with the Kaya Equation.”

    Unfortunately, I don’t see much right with it.

    You went on to claim:

    “To get total global CO2 emissions with the Kaya Identity, you multiply together (1) population , (2) GDP per person (affluence term), (3) energy used per GDP (energy intensity) and (4) the amount of CO2 released per energy used.”

    In this finite world we live in we can truly only hope to understand anything at least to the extent we can measure it. For example, while population, CO2 emissions and energy consumed or used can be to some extent measured and/or estimated and quantified affluence really cannot. How do you measure it? By the quantity of fiat currency held by some given individual? From what I’ve heard Switzerland and perhaps Lebanon both gave up gold backed currencies (I’m unsure about Lebanon but I’m not sure). If they did then no country on the planet has a reality based currency system. In which case the quantity of wealth held by any given individual becomes murky at best. GDP per person tells us little about how efficiently the population uses their resources or their impact on nature. A very poor country can have little wealth but the population may have very poor habits of resource utilization like the disposal of trash in the phillipines. A wealthy country like Austria and/or Norway may be very wealthy but efficiently utilize resources. Other wealthy countries may have poor resource utilization like some parts of the U.S. Being trained in economics I may sound picky to some but my objection stands. Moreover, returning to the wealth question many people may own considerable wealth in land and resources but generate commensurately little in the way of GDP or active production. Many more problems could be brought up with the affluence part of this equation and indeed the other measurable components, but alas I’m out of time and will write more later.

    Have a great day!

    • Johan says:

      Solved at WUWT by the SCOTT BENNETT equation
      where L = available natural resources.

      To lower CO2/GDP, one lowers – ceteris paribus – (L/GDP), which means that for the same amount of GDP, one uses less natural resources (or conversely, with the same amount of natural resources, one produces more GDP or “wealth”).

      Isn’t that nice 🙂

      • Dr. Doug says:

        Johan, the first three terms on the right-hand-side of your equation all cancel out together, which means that you have a circular argument embedded in your equation. That does not lead to useful analysis.

        • Johan says:

          GDP = GDP1 + GDP2, where GDP1 is “primary sector” and GDP2 the rest (secondary and tertiary sectors).
          The equation (not identity) would become

          CO2 emissisons per unit of GDP = output of primary sector per unit of GDP * population per unit of output of the primary sector * GDP per capita * energy intensity * CO2 emissions per unit of energy

          or y = x1*x2*x3*x4*x5, where x1 to x5 are different variables (albeit not independent of each other), and those variables *might” to some extent be controlled by policies. For example, let’s extract less minerals (or extract them more efficiently) to produce the same unit of total GDP. Of course, GDP1/GDP = GDP1/(GDP1+GDP2), so lowering GDP1 in the numerator would also lower GDP1 in the denominator, but who says we cannot at the same time increase GDP2 by producing more efficiently in terms of natural resources (other than being more energy efficient, already captured in E/GDP)?

          BTW, I do not care for either the KAYA or SCOTT BENNETT identy / equation, for the simple fact that neither in KAYA nor in SCOTT BENNETT the variables on the right hand side are independent of each other.

          • Dr. Doug says:

            It could certainly be useful to treat L within the category of GDP. You could model differing economic structures (resource intensities) and their implications for the E/GDP ratio. But you can’t break out L apart from GDP. See below.

        • Johan says:

          Dr. Doug, in the original KAYA identity / equation (whatever), decreasing energy intensity E/GDP (or equivalently increasing energy efficiency GDP/E) will affect GDP and hence GDP per capita. The causal relationships between energy intensity / efficiency and GDP have been the subject of so many studies, that whole forests have been chopped down to fill libraries with books devoted to that phenomenon.
          So although you’re correct that SCOTT-BENNETT contains dependencies among the RHS variables, so does the original KAYA identity. But for reasons beyond my comprehension you keep on defending KAYA, on every possible thread where it is mentioned.

          • Dr. Doug says:

            Johan, the issue is not ‘dependencies among the RHS variables’, which are handled properly in the Kaya Identity, but rather ratios that are interdependent, which exist in your identity but not in the Kaya identity. The ratios in the Kaya identity can vary independently. Yours cannot.

            ‘every possible thread’?? Only here and at WUWT.

          • Johan says:

            Dr. Doug, if GDP depends on E/GDP, than GDP per capita depends on E/GDP. So I have no idea what you’re talking about.

          • Dr. Doug says:

            Johan, again what I’m talking about is that your identity includes, in series,
            These ratios together have to equal 1 — the elements all cancel out.
            Thus you cannot hypothesize a change in L/GDP without simultaneously hypothesizing an exactly offsetting change in the other two ratios. According to your equation, a variation in L cannot have any effect on CO2. This runs counter to the point that you wish to make with your equation. Your equation simply does not work!
            By contrast, in the Kaya Identity, the ratios can vary independently, so the equation does work for its intended purpose.

          • Johan says:

            Yes, Dr. Doug, thank you for pointing out the obvious. With GDP=GDP1+GDP2, the correct decomposition would of course be CO2=P*GDP1/P*E/GDP*CO2/E + P*GDP2/P*E/GDP*CO2/E. And why stop there? One might call E=E1+E2 with say E1 renewables and E2 all the non-renewables. Simple algebra, although it might get a little tedious in the end. I am very much aware of what decomposition methods are, and even used the more sophisticated logarithmic mean Divisia index myself at one time or another.
            My problem is you’re using this as a Straw Man fallacy. The simple fact that the original KAYA doesn’t contain such a circular argument, does not in itself prove that E/GDP is independent of GDP/P, or that CO2/E is independent of GDP/P, of for that matter, that E/GDP is independent of CO2/E. Why do you keep insisting that it does? Even IPCC admits that the 4 terms on the right hand side are not independent of each other.
            (qoute) Most important, the four terms on the right-hand side of equation (3.2) should be considered neither as fundamental driving forces in themselves, nor as generally independent from each other.(/quote)
            Hello, Dr. Doug, if even IPCC is smart enough to recognize that, why aren’t you?


          • Dr. Doug says:

            First, I take it that you are withdrawing your claim that your ‘Scott Bennett’ equation ‘solves’ anything or that it improves upon the Kaya Identity. You acknowledge that it has problems that the Kaya Identity does not have.
            Second, I left out nuances for the sake of clarity. The ratios in the Kaya Identity are mathematically independent of each other (my point recently), but they indeed have practical interdependencies. These matter.
            For example, a rise or fall in the energy intensity of the economy (E/GDP) is likely to have a simultaneous impact on the carbon intensity of energy (CO2/E) — in part due to (perhaps) a relative shift to low-CO2 energy sources, and in part due to equipment-vintage effects in increasing the proportional use of the most modern, energy-efficient and low-emission equipment. I alluded to these issues once or twice recently on WUWT.*
            These interdependencies do not render the Kaya Identity useless. Indeed, the Kaya Identity offers a framework for thinking about how they work–for bringing analytical order to the data. Moreover, the interdependencies usually result in relatively small adjustments to the values of the ratios.
            *(Note for economics nerds: I noted that the marginal values of the KI ratios, for changes in GDP etc., would typically differ from the preceding average values.)

          • Johan says:

            Dr. Doug,
            Yes, of course, the so-called SCOTT-BENNETT identity is a joke, as obviously (CO2/GDP)=(GDP1/GDP)*(P/GDP1)*(GDP/P)*(E/GDP)*(CO2/E) is exactly the same as (CO2/GDP)= 1 * (E/GDP) * (CO2/E). It certainly was never intended to “solve” a mathematical problem with KAYA, as there is no mathematical problem with KAYA. And “refining” KAYA in the sense of including structural effects would basically mean introducing summation signs, as demonstrated in my previous post.

            Now you seem to be making a very subtle difference between “mathematical” and “practical” dependencies. To me, if y=f(x1,x2,x3,x4), where for argument’s sake x2=g(x3,x4); I would say that x2 is very much “mathematically” dependent on x3 and x4 (again, it does not matter what x2,x3 and x4 stand for).

            But to cut a long story short, my main objections to KAYA are perfectly stated in the following reference:
            Rosa E.A. and Dietz Th., Human drivers of national greenhouse-gas emissions, Nature Climate Change 2, 581586 (2012)
            In a nutshell, the authors identify two main problems with both IPAT and KAYA:

            1) IPAT and KAYA assume “unit elasticity”. In other words, a 1% change in one of the RHS variables (say GDP per capita) would mean a 1% change in the LHS variable (say CO2 emissions). That is not what empirical research tells us.

            2) They do not take explicit account of culture and institutions.

            And the authors conclude: “IPAT and KAYA have been useful starting points, but progress requires a new generation of models that estimate, rather than assume, the effect of each driver, net of the others”.

            Now, you’re not going to argue with Nature, are you? 🙂

          • Dr. Doug says:


            If researchers use the identity in ways that rule out important practical interdependencies among the variables (or ratios), then they will err. Pielke’s application, as referenced in WUWT, does not appear to suffer from this problem — due perhaps partly to the specific question that he uses the Identity to address. I can well imagine that some researchers do it wrong, but I don’t recall that anyone here or at WUWT has yet offered an example of that. My posts have been directed against arguments that the KI is wrong in principle; I have allowed that the identity can be misused in practice.

            Reflecting on your arguments, I also imagine that the KI framework is likely to be cumbersome for handling something like the effects of CO2 regulations on GDP. In that case a more purpose-built model would be more economical, while a KI framework could perhaps handle these effects only through Ptolemaic epicycles, to use Thomas Kuhn’s metaphor.

            Regarding the criticism that you cite here (from Nature Climate Change), do the authors give examples of researchers who ignore simultaneous changes in multiple RHS variables? That would be rather like using a partial derivative when a total derivative is called for.

            I have two objections, by the way, to what the NatureCC authors say (as you summarize it). First, the Kaya Identity exhibits unit elasticity of the LHS variable to the sum of RHS percentage changes, not to the change of one RHS variable only. And this is not an assumption, but a mathematical fact, due to KI being an identity. Perhaps the real problem the authors address is that some KI users neglect that more than one variable is changing.

            Second, KI as such can’t neglect culture and institutions, although KI users could certainly neglect how culture and institutions feed into the values of the KI ratios. This issue applies to any model, using KI or not.

            Finally, I regret that I misinterpreted your intention in offering your own identity. I still do not know what your intention was. If it was to ridicule the Kaya Identity, then it misfired, because it suffers from its own problems that the Kaya Identity does not share.

          • Johan says:

            Dr. Doug,
            I should have provided this weblink before, so you can judge for yourself

            It seems to me we’re on very different wavelengths.
            I do not see an identity, I see an equation, namely y=x1*x2*x3*x4. The RHS variables are “independent” variables. In principle, you can assign any value to each of them (within certain limits). When assigning a value to one independent variable, nothing in the equation tells you what values to assign to the other independent variables. The equation does not prevent you to “assume” that x1 will increase by a factor of 2; and at the same time x2 by a factor of 3, while x3 and x4 remain equal. So, what does that tell you, that an increase of the RHS by a factor of 6 implies an increase of y by the same factor of 6? The RHS is a product of variables; multiplying only one of the variables on the RHS by a certain factor is exactly the same as multiplying the whole RHS by that same factor.

            Since this is my last post on this subject, I’ll summarize my position on IPAT and KI:

            1) Ehrlich, Holdren, Kaya et al. should simply have written down a list of what they consider “main driving forces”. They could then have said to the world: “discuss”. The article I’m referring to is interesting in that it clearly shows that there is a lot more to be said about their and other “drivers” than one can find in IPAT and KI.

            2)They should never have written down an equation, derived from an identity. Whether you like it or not, “unit elasticity” really is the (implicit) assumption in both IPAT and KI. They (implicitly) assume: twice the population, twice the CO2 emissions; or twice the affluence, twice the CO2 emissions. IPAT and KI say nothing about how x1, x2, x3 (and x4) are related to each other, other than: well, if you keep the two or three other independent variables constant, then a 10% increase in the independent variable under consideration means a 10% increase in y. You may say time and again, well, doubling population may actually mean more than doubling CO2 emissions, because you also have to change the other independent variables. But which ones, by what amount, and in what direction (up or down)? The equation doesn’t tell you! You can all make it up as you go along.

            As for what I intended with what you call “my” equation? Nothing. My equation is mathematically correct – a number multiplied by 1 is the same number. You can consider as many variables as you like, remain mathematically correct, and still it wouldn’t mean a thing. So again, and for the last time, you can all make it up as you go along.

          • Dr. Doug says:

            Well, then, I’d say that Ehrlich & Holdren are misusing the Identity. I don’t endorse it.

            I’m also not surprised. It was Ehrlich’s Population Bomb that gave me my first “lesson” in economics, at about age 13. I later learned better.

            As an economist, my educated intuition always looks for slippages, offsetting effects, and (in some cases but not this one) positive feedbacks. I also look for unintended consequences of perverse incentives.

            We come out in the same place even if I call it misuse of the identity and you call it the use of a faulty identity.

            Thanks, Johan, for the link and the thoughtful explanation.


      • Dr. Doug says:

        To be more specific, if P and GDP stay the same, the change in P/L will simply offset the change in L/GDP. If GDP grows, then exactly the same offset happens in two steps.

      • Dr. Doug says:

        By the way, inserting L into the analysis is double-counting anyway. More or less resource use changes the structure of GDP and therefore the energy intensity of GDP. In principle, more L could either raise or lower E/GDP.

  5. graphicconception says:

    Using initials, the Kaya Identity states:

    C = P * G/P * E/G * C/E

    An economist will now clearly see that as affluence increases so carbon emissions will RISE.

    However the correct identity is:
    C = G * P/G * E/P * C/E

    As any economist will now clearly see, as affluence increases so carbon emissions will FALL.

    • Dr. Doug says:

      Your “correct” identity incorporates fallacies: that population depends on GDP in the short run and that energy use depends on population irrespective of prosperity. As an economist, I assure you that I will clearly see that your identity is rubbish.

      • graphicconception says:

        So there we have it. My “correct” identity produces exactly the same answer as the real one but mine is rubbish and the other is not!

        I will even let you keep G over P if you think it helps.

        C = G * 1/(G/P) * E/P * C/E

        The problem as I see it is that the formula can include anything you want to draw attention to. Just because it appears as a term does not imply any scientific relevance.

        It has all the appearance of someone wanting to draw attention to some points and putting them into a formula to give it some faux credibility.

  6. Josualdo says:

    Let’s go for some “experimental math”. If in a spreadsheet you have values for population, GDP, energy and CO2, then a cell with Kaya’s identity, it will not change value as you change the values for population, GDP or energy.

    But, whichever these values are (any value will do), it will always show the result as being the set CO2 value. Only changing CO2 value will change the expression value – CO2 value.

    If you write it as Population * Affluence * Energy Intensity * CO2 per energy, then it seems to make sense, in the same way as when you write that Space = Time * Average Speed (meaning you dont know which was the Space, so you know how much Average Speed was, but cannot calculate Average Speed from its components — actually you want to know it from the other terms). (For the sake of simplicity lets assume that the LHS is always the unknown.)

    If you get a spreadsheet with values for Time and Average Speed, so as to obtain Space, any change in Time or Average Speed will change Space.

    Not anymore if you replace, in the expression, Average Speed with (Space / Time) and give values for both. In that case, only changing the value for Space will… change the value for Space; changing Time will not. Meaning, if you know both total Time and total Space, you dont need to calculate the Average Speed to know the Space by multiplying Speed by Time.

    So, as presented by Willis and Wikipedia, the Kaya identity is useless as it cancels out, as when you replace Average Speed with Space/Time and render d= t * (d/t) useless.

    It all depends on what you know and what you want to know, and how much do you know about the values in the RHS. How much information do you have in the equation.

    (Of course, as is known, quantities dimensions on each side of a physics equation must be the same; but if the quantities – not dimensions – in both side are the same after simplification, the equation is useless.)

  7. Steve X says:

    The Kaya identity was first attacked by Paul Krugman when criticising Roger Pielke Jnr who frequently points out the reality that the economics of shifting to low C02 energy sources currently doesn’t work.

    Then it was attacked on WUWT.

    On twitter someone pointed out that at last the alarmists and the skeptics agree on something regarding climate change, unfortunately it is that they both don’t understand basic math.

    • Johan says:

      The reasoning of Pielke jr. is as follows. Given that (unless you’re a North Korean dictator) you cannot reduce population or GDP per capita, and also given that one cannot indefinitely decrease energy intensity (or increase energy efficiency, given the laws of thermodynamics), and also assuming – as Pielke Jr. does – that we do have to drastically cut CO2 emissions (no discussion here!), all that is left to do is switch “en masse” to low carbon energy resources (and being an environmentalist, by that I suppose he means renewables and not nuclear). So Pielke jr. keeps using KAYA to tell the whole world – switch to renewables, now! But of course, pretending that you can change CO2/E while keeping GDP at the same time constant is an illusion (which he seems to sweep under the carpet).
      And although I do not like Krugman – on the contrary – he did have somewhat of a point in telling Pielke jr. that there just might be other important driving factors other than those on the RHS of KAYA.

      • Dagfinn Reiersl says:

        No, he absolutely does not say “switch to renewables, now!” His message is that even with nuclear, there is no way the current emissions targets can be reached without crippling the economy. You can’t build that many power plants that fast. And it’s even harder with wind and solar. What he does propose (among other things) is public investment in technological innovation. “Innovation is the only game in town.” And the idea is that will be a good thing even if it turns out that humans are not causing climate change at all, since the world needs huge and increasing amounts of energy.

        The idea that we have to “act now” is based on alarmist notions that he doesn’t support. He keeps repeating messages like this one.

        Everybody should read his book, The Climate Fix. It’s brilliant.

  8. stevek says:

    Seems to me another identity is co2 = pop * co2 per person.

    So adding 1 person to world increases c02 by the amount of co2 per person.

    Kaya is just trying to break down co2 per person.

    You could also start with

    Co2 = total world energy * c02 per energy unit.

  9. Curious George says:

    I am afraid that you did not choose the best example. The most practical way to determine an average speed is to divide total miles driven by total hours driven, so the tautology does not bring anything new to the table.

    You may perhaps consider Total mileage = (fuel gallons) * (average miles per gallon), where you know the average before driving.

  10. Roy Spencer says:

    Gee.. Now I feel like I’ve hijacked the arguments from Anthony’s website.

    • Bryan says:

      My first thought was that maybe that is what you were trying to do (smile).

      I told myself (twice) that I was giving up. But I think I might have a better way to explain it (than I tried before), so one more try…

      Folks, you just have to understand what the Kaya Identity is, and what it is not.

      First, what it is not.

      1) It is not an algebra problem to be solved or simplified.

      Perhaps the human skill at recognizing patterns gets us sometimes. We see all the things that can be cancelled, and we start canceling. But there is simply no reason to cancel here (other than as a check to make sure you have written it correctly).

      2) It is not meant as a way to actually calculate current CO2 emissions.

      Many people in the threads at WUWT thought that that was the point of it, so they concluded that it is only useful if “CO2 released per energy used” is calculated by some way other than simply dividing CO2 emissions by energy used, since to do that you would have to know CO2 emissions already, which of course would mean that you would not need to calculate it. But although the multiplication of the right hand side factors will give you CO2 emissions, it is not meant as a way to calculate that, so it is fine to divide CO2 emissions by energy used to find “CO2 released per energy used”. In fact, I believe that would be the most natural and best way to calculate “CO2 released per energy used”.

      3) It is not a detailed economic model, that would necessarily allow you to accurately and precisely see how a policy change would affect CO2 emissions over time.

      The reason is that (as many at WUWT pointed out) the factors on the right are interrelated. A change in one of them might affect some of the others in ways that are not accounted for in the equation. For example, an increase in GDP per person (meaning people are richer) will have many effects on their behavior and decisions, so it would be expected to affect “energy used per GDP”, for example. But the equation does give a way to do ‘back of the envelope” calculations to get an idea of how much CO2 emissions would change with a change in one of factors on the right side.

      Now, what it is:

      It is an expression of “CO2 emissions” (more specifically, energy related CO2 emissions) in terms of the main factors that drive (energy related) CO2 emissions. It is in fact a straightforward expansion of CO2 emissions into the those factors. As such it is a tool for discussion of ways to affect CO2 emissions, and a way to estimate the effects on CO2 emissions of changes to the factors on the right side.

      Dr. Spencer, thanks for weighing in on this. I hope I am pretty close with this explanation, but feel free to correct anything I have wrong.

      • Johan says:

        All true, but even IPCC knows that. I’ve given the quotes and references dozens of times. One feels like a dragon against a whole bunch of dragonslayers 🙂

        • Bryan says:

          And perhaps I should have emphasized how far off we could be if we assume that regulations aimed at reducing energy per GDP or CO2 per energy would have no effect on GDP. For example, if a regulatory regime lowered average annual GDP growth from 3.5% to 2.5%, that might not seem like so much, but after 30 years, it would make a difference of 25% in GDP, which would change GDP per person by the same percentage (unless the policies also affected population). So we would be 25% poorer than the calculation (which ignores the unintended consequences of the regulations), assumes, and that would cause an error of 25% in the estimate yielded by the equation for C02 emissions (after 30 years). Of course the lower GDP after 30 years means the resulting C02 emissions would be lower than the equation estimates, making the regulations more effective from the standpoint of lowering C02. But the fact remains that the equation’s usefulness for calculating the effect of policy changes on future CO2 emissions is extremely limited, due to unintended consequences and interrelatedness of the factors, especially when compounded over a time frame long enough to be meaningful in such discussions.

          It seems to me that it is more useful for broad discussions of options. For example, if it is believed that emissions must be reduced by a huge fraction of the current value (let’s say by 90%), one could look at the factors, and see what combinations of changes to the right side factors would accomplish that. After recognizing what would have to happen, one could evaluate the feasibility of actually achieving that, and discuss the options. In spite of large potential errors in estimated effects of policies, if you are considering a reduction of 90%, the picture you get from the Kaya Identity could be close enough to be useful, in my opinion.

      • Josualdo says:

        Therefore it is a “philosophical equation”, written as algebra (why?). Hence, I was fooled by pseudoalgebra and a pseudoequation. I stand enlightened now. The good part is I dont have to care about it any more.

      • Dr. Doug says:

        Bravo, Bryan!

    • wyoskeptic says:

      Sir, with all due respect, it is not a highjack, but rather an import …

      and not all imports travel was well as what a fine wine might …

      I am simply wondering if anywhere in this … uhm er mmmm, discussion, that anyone has pointed out that the Kaya meets the definition of a tautology?

      From Wikipedia:


      In logic, a tautology (from the Greek word ταυτολογία) is a formula which is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; (it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense). A formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically contingent. Such a formula can be made either true or false based on the values assigned to its propositional variables. The double turnstile notation \vDash S is used to indicate that S is a tautology. Tautology is sometimes symbolized by “Vpq”, and contradiction by “Opq”. The tee symbol \top is sometimes used to denote an arbitrary tautology, with the dual symbol \bot (falsum) representing an arbitrary contradiction.

      Tautologies are a key concept in propositional logic, where a tautology is defined as a propositional formula that is true under any possible Boolean valuation of its propositional variables. A key property of tautologies in propositional logic is that an effective method exists for testing whether a given formula is always satisfied (or, equivalently, whether its negation is unsatisfiable).

      The definition of tautology can be extended to sentences in predicate logic, which may contain quantifiers, unlike sentences of propositional logic. In propositional logic, there is no distinction between a tautology and a logically valid formula. In the context of predicate logic, many authors define a tautology to be a sentence that can be obtained by taking a tautology of propositional logic and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable). The set of such formulas is a proper subset of the set of logically valid sentences of predicate logic (which are the sentences that are true in every model).

      In other words, what the argument here is about is Philosophy, not mathematics. Aristotle would be grinning … Kepler would have a headache.

      • Josualdo says:

        Right on. Elsewhere I and others noted that, were it an equation, it would yield CO2=CO2 or, 1=1, therefore being tautological. Given that it is not algebra, no rules apply anymore.

      • Dr. Doug says:

        Yes, it was amply pointed out on WUWT that an identity in itself proves nothing. It was also pointed out that an Identity can usefully serve as an analytical and accounting framework, provided that real substance is applied to the terms (ratios) in the identity.
        Some people insisted on not getting the point, but most were convinced.

    • Johan says:

      Serves you right for opening a can of worms 🙂

  11. Jere Krischel says:

    Units are not variables.

    Say it again:

    Units are not variables.

    The “Kaya Identity” is actually:

    V CO2 = W Population X (GDP/Population) Y(Energy/GDP) Z(CO2/Energy)

    Note the units properly cancel, so that the result on the RHS is the same units as the LHS.

    Note that there is nothing about this that is an identity without units, it looks like:

    V = W * X * Y * Z

    V varies for any change in W, X, Y or Z.

    You cannot construct this identity algebraically:

    V = V

    V = W * X * Y * Z ???

    • bk51 says:

      Exactly. I will say it again with you, units (of measure) are not variables.

      V is the dependent variable, its unit of measure is gigatons of CO2.

      W is a variable, its unit of measure is persons.

      X is a variable, its unit of measure is $$ per person.

      And so on.

      V = W * X * Y * Z

  12. geran says:

    How about calling it the “Kaya Absurdity” instead?

    Because it really is absurd. But, it is “climate science” and that is pretty absurd.

    Climate science assigns a “forcing” to atmospheric CO2, which is NOT a heat source. Climate science calculates a 33K “warming” by mis-application of the Stefan-Boltzmann equation.

    No need to go on.

    The best way to illustrate absurdity, often, is with absurdity. So, here goes:

    UAH hires experts in many fields. I am glad to offer my services. My field is “saving the planet”, and my fee is based on the well-known “Saving The Planet Identity” (STPI).

    GAS = GGI * EB/GGI * $/EB

    Where–GAS = Geran’s Annual Salary
    GGI = Geran’s Great Ideas
    EB = Economic Benefit
    $ = Dollars (USD)

    For my first year on the staff of UAH, I will have 11,647 great ideas. The economic benefit per great idea will be 8043. The $ per EB, as everyone knows, is 14,787.

    Therefore, using the STPI, my first year’s salary, payable in advance, will be $1,385,199,152,130.

    (I have rounded to the nearest 10 dollars, as I am not greedy.)

    Dr. Roy, please submit this bill to the relevant office at UAH. I will accept payment in cash only. Let me know when it is ready, and I will pick it up.

    I know you are very busy, so expect 5% of the total I receive, for your efforts. (I told you I was not greedy.)

    It’s all about climate science and saving the planet.

  13. ray says:

    It is a tautology.

    Tautologies are NOT used to make empirical statments.

    Tautologies ARE used to change one (possibly complex) empirical (“contingent”) statement into another empirical statement. The truth or falsity of the second statement is completely bound up with the truth or falsity of the first, so long as the tautology is proper. Most of the “gobble-de-gook maths” in science books is merely establishing such tautologies, so that different problems can be solved using the same basic theory.

    So, with this puerile Kaya tautology:

    : IF

    GDP-per-person-living is fixed, by some social considerations (empirical statement)
    Energy intensity is fixed by some techical considerations (empirical statement)
    The proportion of energy use which involves CO2 emission is fixed by some technical considerations (emprical statement)


    An increase in population will cause a proportionate increase in CO2 emissions (empirical statement or forecast).

    And if the assumptions are WRONG (they are contingent statements, after all) then the conclusion will be WRONG.

    • sunsh says:

      “…the conclusion will be wrong.”

      And the policy adopted in panic on the precautionary principle will be wrong.

  14. ray says:

    And ,of course, in Economics, one always allows for the things one leaves out by saying “other things being equal.”

  15. Laura S. says:

    These identities are useless when the terms are inversely proportional.

    For instance a famous equation in economics is MV=PY. Milton Friedman famously recommended a constant growth rate in “M” to stabilize the growth rate in PY. The issue with such a rule is that “V” is not a stable constant and indeed under some circumstances “V” moves counter to movements in M. So changing M does not wag PY. This lead to some effort over many years to find a definition of “M” for which “V” was stable. These efforts ultimately failed…

    Now going back to the Kaya identity. Suppose you create an identity:

    emissions = grams C02/goat * number of goats.

    The trouble is changing the number of goats will inversely change grams / goat and emissions will stay the same. This is because the number of goats does not control the emissions.

    So it is with the Kaya identify. Just because the units work out physically does not mean the equation has physical meaning because some of the terms of proportionality are not universe constants/not independent.

  16. paqyfelyc says:

    Miles = [hours]x[miles/hour] is useful … or not.
    You may use it if you have some control of the element ; that is, say, if you have some indepedante choice of speed or time, then you can estimate distance.
    Now, if because of some technical constrain, the two things are not independants (e.g. : you have a limited gasoline tank, and the speedier, the less time you’ll drive…), you’d better switch to a better equation with more proper variables, for instance
    Miles = [gallons]x[miles/gallons]

    Kaya’s is not good because it uses variables that you cannot control and are not independant. (1) population , (2) GDP per person (affluence term) and (3) energy used per GDP (energy intensity) just run together (obviouly a change in any of the 3 may change the other two) and cannot be easily controlled, so they just are worst than useless in such a equation.
    The only thing that may be have some relevance is the (4) the amount of CO2 released per energy used. Not even sure, though, because there are many source of energy with very different “amount of CO2 released per energy output”, so we should probably use a more complex form :
    CO2= sigma ( amount of CO2 released per energy output for each technology x energy output of this technology)
    CO2= energy output x sigma ( amount of CO2 released per energy output for each technology x share of this technology in energy output)

  17. jimc says:

    Not having seen the WUWT comments, I proclaim in complete ignorance:
    Such equations are designed to show/estimate/guess/simplify relationship/dependence among variables.
    Double this and that will double. Double this and halve that and the other will remain the same. Etc.

  18. JohnKl says:

    Hi Roy,

    After consideration, I suppose there’s nothing necessarily wrong with the equation other than it doesn’t indicate very much. As I mentioned above there’s not much right with it. One other consideration that should be brought into any equation regarding environmental impact is land area. After all in economics the four factors of production land, labor, capital and management all prove key to any rational analysis and or decision. For example, country x may have large population but relatively small land area like Germany or a somewhat larger population on an enormous land mass like Russia. Of necessity the Germanic population must husband resources much more carefully than Russia. The environmental impact of any given citizen will likely differ greatly. In any case many other factors should be included in any environmental impact equation that purports to relate any useful information at all.

    Thanks Roy and have a great day!

  19. Ruth Dixon says:

    Roy said “To get total global CO2 emissions with the Kaya Identity, you multiply together (1) population , (2) GDP per person (affluence term), (3) energy used per GDP (energy intensity) and (4) the amount of CO2 released per energy used.”

    Exactly – and it works for individual countries too, which all vary in terms of the factors 1-4. It is worth looking at real-world data for these factors (which as Johan says are not necessarily independent) as the results are illuminating. I show some plots in a couple of recent posts on my blog.

    My second post shows how the KI can help to detect spurious claims of CO2 reductions.

  20. Tim Jenvey says:

    As several folks have stated “hours” is totally different than “hour”.

    In the Kaya thingy it uses co2 and co2. Not co2’s.

    Just can’t see what all the fuss is about. The equation/identity/or whatever as stated is totally ridiculous.

  21. filo says:

    TimJenvey says:

    “Just can’t see what all the fuss is about.”

    It is a WEASEL tautology. That is, one which is true by implicit or slyly hidden definition. It is puerility to the nth degree,which unfortunately CAN be effective in creating a fuss about nothing.

    Also, calling anything by a catch-phrase or tag (“The Kaya Equation”), using a sort of matey familiarity, gives a spurious profundity to it. Another example was calling the speculation of Mr Higgs, “The Higgs Boson”; and then even better, “The God Particle” – acts of evil genius which spawned billions in funding for the physicists who send things round in circles.

    Compare what the philosopher David Hume had to say about wasting time on bad ideas and arguments:

    “Does a man of sense run after every silly tale of hobgoblins or fairies, and canvas particularly the evidence? I never knew anyone, that examined and deliberated about nonsense, who did not believe it before the end of his enquiries.”

  22. ray says:

    “…who did not believe it before [sic] the end of his enquiries.”

    But,it is to be hoped, not at the END of his enquiries.

    To some extent, you have to believe provisionally in anything silly to understand at least “where it is coming from.” Of course, if the provisional belief lasts all your time on earth…

  23. Scott Bennett says:

    Hey! I am Scott Wilmot Benett and I have never actually written an Identity with Land area as one of the terms!! I have suggested that the Kaya, lacks the qualification of land area though!

    Dr Roy, I can’t believe that you could mix up units with the variables of a formula!

    Units are never explicitly handled by a formula. The result must balance dimensionally or it is badly formed.

    You have mistakenly illustrated dimensional analysis for a formula.

    Miles = [hours]x[miles/hour], all of these are units not magnitudes.

    They should be replaced by variables to create a properly formed formula:

    Miles (Total miles traveled) = k = ?
    hours (Total hours ) = h = 2
    speed (Displacement/Elapsed time or Miles/hour) = s = 100

    k= s x h

    k = 100 x 2

    k= 200

    Total MILES traveled = 200

    Your answer is expected to be in Miles, not miles per hour and so it is,because dimensionally, the hours cancel!

    And it is absolutely appropriate to add values to the variables of an identity because this is in fact the proof!

    The IPCC call the Kaya a multiplicative identity and that is what it is!

    The issue of speed is very interesting from an abstract point of view though, because you can drive your car right through the Newtonian reality of length/time straight into the relativity of space/time where you can’t separate either. And that is apparent even in the simple problem of displacement/elapsed time because instantaneous velocity is the same formula as that for average velocity or speed. The difference is, that the former is given as the limit of the ratio where time is very small but not zero!!

    • Bryan says:

      Concerning units vs. variables — I agree that it is a valid distinction. Dimensional analysis is used to make sure the units cancel properly and yield a result with the correct units. This can in fact be applied to the Kaya Identity. However, with the Kaya Identity, you can also cancel the actual variables, and this is indeed a different thing than just canceling the units.

      I look at it this way: Lets say you start with CO2 emissions (we are talking about energy related CO2 emissions, but it is too long to type that every time, so just call it CO2 emissions), and want to expand it into the factors that drive it. Since CO2 emissions come from using energy, you could reasonably expand it into:

      CO2 emissions = energy x CO2 emissions / energy

      You can read this as “energy x CO2 emissions per energy”, but either way, the “CO2 emissions” on the right really is the same as the “CO2 emissions” on the left. Also, “energy” is the same both places it appears. So indeed energy cancels, which gives CO2 emissions = CO2 emissions, showing that it is true.

      I think the reason so many people think it is useless is that they are thinking that it is supposed to help you calculate current CO2 emissions, which would be silly since CO2 emissions is on the right side, and if you already know it, why do you need to calculate it? But of course it is not our intent to find a way to calculate current CO2 emissions. We are just trying to expand CO2 emissions into the factors that drive it, so we can discuss ways to influence it, and estimate the effect on future CO2 emissions of policy changes that affect those drivers. The ratio of CO2 emissions per energy (calculated by dividing CO2 emissions by energy) is one of those drivers. The total amount of energy used (“energy”, in our equation) is another. Our expansion into those 2 factors is straightforward. As examples of ways that we could influence those drivers: To lower “CO2 emissions / energy”, we could enact policies that encourage nuclear energy. To lower “energy”, we could provide tax breaks for conservation efforts.

      But of course, recognizing that energy use comes from producing GDP, we can similarly expand energy into GDP x energy / GDP. Making that substitution, we get:

      CO2 emissions = GDP x energy / GDP x CO2 emissions / energy

      And GDP is produced by the population, so it can be expanded into population x GDP / population. Making the substitution:

      CO2 emissions = population x GDP / population x energy / GDP x CO2 emissions / energy

      We just arrived at the Kaya identity by starting with “CO2 emissions” and repeatedly expanding in reasonable ways. Each of the factors can be influenced by public policy. [Many at WUWT pointed to nefarious ways of influencing the first factor (population). But it could also be influenced by something as simple as immigration policy. Also, it might be influenced by policies that increase GDP, since the birth rate tends to go down as poorer people become richer] So we have a tool to discuss ways of lowering CO2, and estimating the results on CO2 emissions of policy options.

      As many (including me) have pointed out, its usefulness is limited, but since it is simply a straightforward expansion of CO2 emissions into factors that influence it, there is nothing wrong with it mathematically.

  24. geran says:

    Scott, you win the hilarity award.

    1) You don’t know if you are “Bennett”, or “Benett”.

    2) Northing else you wrote makes sense, but you pretend it does.

    Now, on to Comedy Central..

    • Scott Bennett says:

      Dear Deranged,

      Nice bit of gutless trolling!

      Did you write it yourself?

      Thanks for the heads-up on the typo!

      I’m sorry for your lack of comprehension.


      Scott Wilmot Bennett

      • geran says:

        “Deranged” is not here right now, he went out to support AGW, and got killed by a sudden Arctic blast. Poor chap.

        Anyway, I’m glad you finally learned to spell your name.

        Double Cheers,


  25. dp says:

    OMG! It’s started all over again!

Leave a Reply