# On The Kaya Identity

A reader asked about my take on the Kaya wars that are flaming at Anthony Watts’s place.

Here is the form of the Kaya Identity, which is to say, the Kaya non-equation:

$Y = X_1 \times \frac{X_2}{X_1} \times \frac{X_3}{X_2} \times \dots \times \frac{X_{n}}{X_{n-1}} \times \frac{Y}{X_n}$,

The Ys and Xis are numbers and a free choice, given the limitations of algebra (no Xi equals 0). Try it and see: for fun, let $X_i = i$. Works. A perfectly harmless manipulation.

There is no explicit word about causality in the Kaya. The Xis aren’t necessarily causing the Y or each other. If we wanted to know about what caused Y, and we believed that the Xis were in the causal path of Y, we wouldn’t set up an identity but an equation which looked like this:

$Y = f(X,\beta)$.

Where X is a vector and where the vector of (known and unknown) parameters β may be larger or smaller than X. Notice that Y does not appear on the right hand side. We are solving for Y here. One possibility (and probably a too simple one for most Y)is a linear equation:

$Y = \beta_0 + \beta_1 X_1 + \dots + \beta_1 X_n$.

This is not regression. This is a causal model: it says Y will certainly change by $\beta_i$ when $X_i$ increases by 1 unit. Regression is a probability relationship where we first assume $Y \sim N(\mu, \sigma)$ and then substitute μ for Y on the left hand side. Regressions says Y might, not that it certainly will, change.

Anyway, since the Kaya is an identity we can put anything we like in for the Xis and Y. Let’s try.

$\mbox{CO}_2 = \mbox{Puppies} \times \frac{\mbox{Cats}}{\mbox{Puppies}} \times \frac{\mbox{Meteors}}{\mbox{Cats}} \times \frac{\mbox{Cosmic rays}}{\mbox{Meteors}} \times \frac{\mbox{CO}_2}{\mbox{Cosmic rays}}$,

where each is a number existent or occurring over a year in some suitable units. Notice that I was careful to put things that we know change over time, but I needn’t have. Everything could be static (more or less), like this:

$\mbox{CO}_2 = \mbox{Mountain ranges} \times \frac{\mbox{Continents}}{\mbox{Mountain ranges}} \times \frac{\mbox{Great lakes}}{\mbox{Continents}} \times \frac{\mbox{Gold}}{\mbox{Great lakes}} \times \frac{\mbox{CO}_2}{\mbox{Gold}}$.

Some of these quantities are known for sure, and one, the amount of Gold, is not. But whether we know the value of any of these is immaterial to the Kaya. As long as none are null quantities, and Gold isn’t, we’re in business. Also note that the number of entries was up to me. I could have made the list shorter or longer as I pleased.

What do any of these things have to do with CO2? Who said the items had to have anything to do with CO2? Who said I had to use CO2? Insisting that the Xis are causative of Y and using the Kaya and not an equation is doing it, as they say, the hard way.

But we can certainly manufacture cause-like stories. Puppies eat, and their food both requires and releases CO2. Cats, too. Meteorites often have carbon in them, and boy do they disturb the atmosphere; lots of cloud nucleii strewn hither and thither during their journey. Same thing with cosmic rays. Plus, these energetic creatures effect life, and life is important for understanding carbon. This is just off the top of the head; spend some serious time and you can spin this tale out to saga length. Peer reviewed, of course.

The same thing can be done with the fixed Xs. Or with any items you care to put into the Kaya. As long as you stay away from 0, you’re in business.

Here’s what Kaya himself put (adapted to just the USA):

$\mbox{CO}_2 = \mbox{Population} \times \frac{\mbox{GDP}}{\mbox{Population}} \times \frac{\mbox{Energy}}{\mbox{GDP}} \times \frac{\mbox{CO}_2}{\mbox{Energy}}$.

This is just as valid as the examples above, though this one seems more popular with economists. But then economists are prone to tying everything to GDP, which is a number, and economists love numbers (those without uncertainty, that is), often preferring them over reality. Never mind.

For an example of the Kaya in action, see the Pielke Jr video embedded at this link starting at around 21 minutes. Pielke appears to believe that the Kaya has something to do with causation and that he and fellow economists have captured all they need know about human beings and carbon.

The good news from an analytical standpoint is that there’s nothing else. There’s no other levers that you can use out there. This is comprehensive. You may wish there was some rabbit you can pull out of a hat because the good news is also the bad news: this is all you have.

That’s false, as we know from the Puppies example (recall we can add puppies or gold or whatever to the Kaya). And these items—GDP, etc.—are far from all we know about humans and carbon, though Pielke calls it an “extremely powerful tool for policy analysis.” For instance, he says (around 20 min. mark) that, as a lever governments can pull, “Less people, all else equal, equals less emissions” (I believe the guillotine had a similar lever). Another lever the government can yank, he says, is to purposely create poverty, i.e. “Limit generation of wealth” (does that include the wealth of government, one wonders?).

These claims are not quite false, but not quite true, either. More people mean more energy is used, but there’s also a greater chance for more innovation in, say, creating more efficient energy sources. And more people also means more food, and food is a terrific carbon sequestration vehicle, to say it in economic-speak. (Incidentally, one reason that there are more people is that there is more food.)

Now there’s nothing wrong with grappling with crude ratios like Energy/GDP to have some rough, first-blush idea of the amount of energy that is now required to generate such-and-such-a-sized economy, but as for the energy required to drive a future economy, who knows? Nobody in 1990 predicted Google. The Kaya is not a forecasting tool. And since it doesn’t carry any measure of uncertainty, and since every term is mixed up causally with every other term, nobody knows how much credence to give it.

And we can’t bypass the hard work of actually estimating the amount of carbon released and sequestered, both now and in the future. Yet the Kaya is mute on what causes CO2. GDP, after all, doesn’t cause CO2. That’s impossible.

The Kaya should be replaced with a probability model/equation, which can tell us how much change in GDP might be associated with a change in CO2.

That model ought to be under the same constraint as climate models. If it can’t make skillful predictions of future data, we shouldn’t believe it. Right? And how good are economists at forecasting the GDP or energy use one to two decades out?

Update I should have mentioned this above, but it might be in some problem that we know the last ratio Y/X_n (and each other ratio) but that we do not know Y. The Kaya can then be used to calculate Y. But in the case of CO_2, we do not know the last ratio.

1. Scotian says:

The following is the comment that I posted at WUWT (http://wattsupwiththat.com/2014/07/12/the-beer-identity/). I believe that it is similar to Briggs’ post.

Hard to comprehend why there are hundreds of comments on such a simple thing. It must be this burning desire to see Willis fail, no matter how trivial the point. I havenâ€™t read all the comments â€“ who has? â€“ so there is a slight possibility that the following is duplicated.

This kind of expansion is similar to a chain rule expansion in calculus. For example we might write Newtonâ€™s second law in the following form (rectilinear motion).

F(x)/m = dv/dt = (dv/dx)(dx/dt) = vdv/dx

This simplifies the equation that can now be solved for â€œvâ€ as a function of â€œxâ€ if you know F(x). The important thing to note is that you can only do this if there is only one independent variable, here time â€œtâ€. On the other hand if there is more than one independent variable, application of the chain rule involves partial derivatives and more than one term. It might look like this:

du/dt = (du/dx)(dx/dt) + (du/dy)(dy/dt)

where the derivatives on the right hand side are partial. My point in all this math is that the Kaya identity assumes that there is only one independent variable (path) and that Willis is pointing out that there are many paths or variables.

2. James says:

Scotian,

I think a lot of the ire directed at Willis was that it appeared that he was using a dimensionality check to show that the equation was meaningless. That’s how I interpreted his first post, anyway. The second one focused more on how the numbers and linearity didn’t mean anything in the real world (which is correct).

3. Mike B. says:

I’m certainly no expert in Kaya, but is it possible that there has been a time-dependent subscript left out of the equation with respect to CO2? For instance on the left side the variable is CO2(t) and on the right side it is CO2(t-1)?

4. Briggs says:

Mike B,

See Scotian’s point: short answer is yes. Also, people cause the GDP, but people also use energy, and energy is needed for the GDP. All the terms are “mixed up” or interdependent in terms of causality of Y.

5. Rich says:

On Wattsupwiththat Roger Pielke links to his paper at http://iopscience.iop.org/1748-9326/4/2/024010/pdf/1748-9326_4_2_024010.pdf

He outlines he Kaya identity as Prof Briggs has above though less transparently. Then he says, “Thus, according to the logic of these relationships, carbon accumulating in the atmosphere can be reduced only by reducing (a) population, (b) per capita GDP, or (c) carbon intensity of the economy”. But if you reduce population then you increase per capita GDP by an exactly proportional amount so carbon dioxide in the atmosphere will remain unchanged. Time dependent subscripts will not change this.

He analyses the effects of the other three factors in the same way: he varies a quantity without considering where that quantity appears in the rest of the expression and so how it will affect the overall outcome.

Maybe his analysis of the individual factors is correct but not if he insists on tying them together in an algebraic equation.

6. Mike B. says:

Rich:

Like I said, I’m unfamiliar with the formal construction of the Kaya “equation”, and was asking if the time-dependencies were inadvertently left out. If so, then Kaya is most certainly not an “identity”.

7. James says:

I got curious as to what the relationships between GDP, CO2, and energy might be like, so I went to the World Bank’s site to grab GDP, CO2, and energy consumption data.

After some plotting, the answer is what everyone already knows, which is “it depends”. Some countries have GDP/energy/CO2 that track nicely to one another. Others don’t. The USA has no GDP to CO2 correlation (per capita), but a high energy usage to CO2 correlation (per capita). That indicates what, again, we already know. We know that the USA uses a lot of energy, which makes CO2, but we are pretty productive, so GDP grows more than energy usage.

See some images from that exercise at this imgur album I put together. Again, nothing major can be demonstrated from that, except for the conclusion that those relationships in the identity are incredibly complicated (CO2/energy alone needs to be function of energy type) and that a simple linear-in-each-term function won’t cut it.

8. Doug M says:

It reminds me of DuPont’s Method.

Retrun on Equity = Gross Margins * Interest Burden * Tax Burden * Turnover * Leverage

Gross Margins=Earnings Before Interest and Taxes (EBIT) / Sales
Interest burden = Pretax income / EBIT
Tax Burden = Net Earnings / Pretax income
Turnover = Sales / Assets
Leverage = Assets / Earnings

Why don’t you cut to the chase, and just say Earnings / Equity?
I guess the idea that if DuPont acquires a smaller company, they may be able to streamline any of the sub-components, and boost the ROE.

9. Just seems like we’re converting miles/hour into furlongs/fortnight.

10. Johan says:

I do not like the KAYA identity / equation for a number of reasons, but just a few remarks. 1) KAYA was the result of a discussion trying to identify the main “driving forces” of CO2 emissions. Anyone who in such a discussion would suggest cats, puppies etc. would probably not be taken too seriously, as you are well aware of. 2) Kaya, after whom the identity is named, is a professor (now emeritus) of engineering (his collaborators at the time were also engineers). BTW, KAYA is very similar to the IPAT identity, concocted by Ehrlich and Holdren, a biologist and a physicist. Ironically, being an engineer, I surmise that prof. Kaya couldn’t just come up with a list, so he wrote down an equation: F = P * g * e * f , where F, P , g ,e and f are variables; derived from the true identity F = P * (G / P) * (E / G) * (F / E), wher F, P, G and E are simply numbers.
The identity itself is of little use – you can look up historical values for F, P, G and E, and it will duly tell you that F = F (such is the nature of an identity).
The equation is used for policy analys (by the likes of Pielke jr., who I believe is not an economist but an environmental scientist, whatever that is), and all that you can reasonably get out of the equation is: “ceteris paribus, if you divide or multiply one of the variables on the right hand side by a certain factor, you have to divide or multiply the variable on the left hand side by the same factor” (such being the nature of equations).
The KAYA equation is a very crude decomposition method. Economists, real economists that is, use slightly more sophisticated decomposition methods, such as the logarithmic mean Divisia Index. Those methods have been known to economists since the 1920s (Irvin Fisher and CÂ°).
Although slightly more sophisticated, they are still pretty useless, for the same reasons KAYA is useless.
But my main point being, do not blame economists for the mistakes of engineers, biologists, physicists and “environmental scientists”. There are plenty of things left to blame economists for.

11. Brandon Gates says:

Johan,

Anyone who in such a discussion would suggest cats, puppies etc. would probably not be taken too seriously, as you are well aware of.

Hope springs eternal.

12. Milton Hathaway says:

I couldn’t really follow the logic in the assigned reading material. Based on what I did manage to read, it reminded me of the Simonâ€“Ehrlich wager .

It also reminded me of an engineer I used to work with. He was a mechanical engineer, and his kind was outnumbered a half-dozen-to-one by his electrical engineer coworkers. He found humor in the EE jargon, and he loved to walk up to a group of EE’s deep in a hall discussion about some sticky problem, appear to be listening intently for a few seconds, then ask “why don’t you multiplex it?” or some such, then walk away. The EE’s would look at each other, one would say “Hey, that might work”, and the conversation would change to an intense discussion of the applicability of multiplexing.

13. John Moore says:

Perhaps folks are overthinking this.

If it’s being used as an accurate predictive tool, then it is being misused.

However, it should be quite useful as an explanatory tool to counter a lot of the foolish arguments going around about reducing carbon emissions. These arguments are put forward by people who (some on purpose, some out of ignorance) are not looking at the problem with any sense of numeracy.

This identity can be helpful in teaching those who do so out of ignorance, because it provides an *illustrative* way of showing interdependencies that too often are conveniently left out.

It can also be used as an illustrative model, demonstrating statements like “if we want to cut CO2 emissions by 50%, we have to either cut the population by 50%, cut the standard of living by 50%, or increase carbon efficiency by 100%.” While we know that this statement is not perfect, it does give a ballpark idea of what’s going on. And, while those of us who think clearly about the issue already know this is true (in a very approximate sense), at least some of those who are thinking less clearly can be taught.

And, as the creators were engineers, and I’m an engineer, a ballpark idea for us is a heck of a lot better than no idea at all.

14. Thank you William! I just used this to “prove” to my family that we’d all be financially more secure if they’d just all give me some of their income!

🙂
MJM

15. The real subject here, masked by a thin veil of carbon dioxide and Kaya, is pollution and how it’s effecting the climate. Well, we now have a satellite that’s going to to add some real figures to the multiple formula involved. Though I doubt any amount of evidence will convince you that pollution is a bad thing, it should at least clear the Kaya off the veil.

JMJ

16. Sheri says:

JMJ: It seems unlikely we can ever get you to stop stuffing that straw man. Makes you look foolish and incapable of learning.

Okay, dumb blonde here. What is the usefullness of an “identity”. It really tells us nothing and cannot predict. Can one of you math types explain to me why someone invented the “identity” in the first place. I’m kind of track with Johan, who if I understand him correctly, sees little use in the entire thing.

17. Sheri, let me explain irony to you in one easy lesson.

“JMJ: It seems unlikely we can ever get you to stop stuffing that straw man. Makes you look foolish and incapable of learning.

Okay, dumb blonde here. What is the usefullness of an â€œidentityâ€. It really tells us nothing and cannot predict. Can one of you math types explain to me why someone invented the â€œidentityâ€ in the first place. Iâ€™m kind of track with Johan, who if I understand him correctly, sees little use in the entire thing.”

And that, my friend, is irony.

The whole point is that the Identity is pointless, it is as relevant as a name, as Briggs believes is the argument for pollution causing climate change.

JMJ

JMJ

18. JH says:

Scotian,

I am not sure what the partial derivatives and the number of independent variables have anything to do with the Kaya Identity in your comment, though I understand the simple calculus just fine. But here is what I know about multiplicative models.

The percentage change in CO2 at period t is defined as

CO2(t)-CO2(t-1)] / CO2(t-1),

which can be well approximated by

log[CO2(t)] – log[CO2(t-1)]

when the change is small.

If one takes logarithms of both sides of the multiplicative Kaya Identify with four components, say,

CO2=C1*C2*C3*C4,

ln CO2=ln C1 + ln C2 + ln C3 + ln C4.

And after taking derivative with respect to t, we have

d(ln CO2)/dt = d(ln C1)/dt + d(ln C2)/dt + D(ln C3)/dt + d(ln C4)/dt

That is, the sum of the percent changes in each of the components closely approximates percent change in carbon emissions between any two years.

19. JH says:

What is the point of such decomposition in Kaya Identity? Regardless of what important components are used in the identify, I imagine if one calculates and plots the historical tend in each of the Kaya Identity components, one can find a reference point for evaluating current and future policy development on each of the components, and for discovering the driving forces/components behind the past change in CO2 emission.

No, nothing is perfect. A method can always be improved. One probably can break down each components! So I would think, better decomposition methods with various relevant components have been proposed.

20. Shawnhet says:

“Thatâ€™s false, as we know from the Puppies example (recall we can add puppies or gold or whatever to the Kaya).”

Really? I would be interested in seeing how specifically changing the puppy population affects the CO2 levels in a country. If we euthanized all dogs, could we logically expect that our CO2 emissions would drop to zero? I would say no which would pretty much invalidate your puppies expression. OTOH, if we moved any of the factors in Kaya to zero, it would be pretty reasonable to expect CO2 emissions to drop to zero (I’m not saying it is desirable).

Cheers, 🙂

21. Nullius in Verba says:

“Okay, dumb blonde here. What is the usefullness of an â€œidentityâ€. It really tells us nothing and cannot predict. Can one of you math types explain to me why someone invented the â€œidentityâ€ in the first place.”

Suppose you’re asked to calculate the height of a brick wall. Well height of wall = height of brick times number of rows of bricks in the wall. It’s an identity: W = B * (B/W). It’s just telling you how to calculate the quantity you want from the information you’ve got.

You can decompose a quantity in many arbitrary ways – but for an explanation the trick needed is to decompose it into independent variables. Changing the number of bricks doesn’t change their height. Using bigger bricks doesn’t change their number. Both are well-defined constants, independent of one another. You can manipulate each one independently without affecting any of the others. This is the critical property needed, and what decides whether an identity is useful or not. What you’re really doing is decomposing the effect into independent factors, each associated with one policy lever that can be adjusted independently of the others, and that is a useful thing to be doing.

So the issue is really whether the terms in the Kaya identity are independent, not that it is an identity. And they’re not. While some policy levers might affect one term more than the others, there are many others that affect several at once. Nor are the terms well-defined constants. Energy/CO2 is different for trees versus cities, for example, so you’re getting weighted averages of different stuff, rather than a fixed property of common stuff. At the least, it needs to be treated with a lot of caution.

22. JH says:

Nullius in Verba,

“You can decompose a quantity in many arbitrary ways â€“ but for an explanation the trick needed is to decompose it into independent variables. Changing the number of bricks doesnâ€™t change their height. Using bigger bricks doesnâ€™t change their number. Both are well-defined constants, independent of one another. “

Yes, one can add any terms on the right hand side of the identity equation. An explanation is only legitimate when a plausible causal mechanism is available. Why does it have to be decomposed into independent variables for an explanation? What do you mean by â€œindependent variablesâ€? Does it refer to two variables that are statistically independent or something else?

It is evident that B and B/W are not statistically independent in your example. Nor are GDP/population and Energy/GDP in the Kaya identity. Both dependent on GDP, hence they are not statistically independent.

The Kaya identity can be used to draw useful information on each of terms in the decomposition in for various countries. However, one needs to know that those terms are generally not independent of each other, simply by how those terms are defined. I wouldn’t be surprised if some researchers have proposed identity equations with interaction terms, which can provide better explanations.

23. JH says:

Dear Briggs, just for you –

â€œIf youâ€™re only skeptical, then no new ideas make it through to you. You never learn anything. You become a crotchety misanthrope convinced that nonsense is ruling the world. (There is, of course, much data to support you.) Since major discoveries at the borderlines of science are rare, experiences will tend to confirm your grumpiness. But every now and then a new idea turns out to be on the mark, valid and wonderful. If youâ€™re too resolutely and uncompromisingly skeptical, youâ€™re going to miss (or resent) the transforming discoveries in science, and either way, you will be obstructing understanding and progress. Mere skepticism is not enough.â€

~ Carl Sagan

24. Johan says:

JH

Just one example to give you an idea of the many problems (energy) economists are faced with. In determining whether economic growth is the cause of improved energy efficiency, or whether energy efficiency is a cause of the growth in economic output, some economists use a modified Solowâ€“Swan (neoclassical) model for economic growth. Well, all you have to do is change the underlying aggregate production function from – say a Leontief production function to – say a Cobb-Douglas production function, to get totally different results. But of course, is does not matter what type of production function you use, it has long been known that these functions are devoid of any economic content. They are only used for their algebraic properties.

Legend has it that Stanislaw Ulam once challenged Paul Samuelson to name one theory in all of the social sciences which is both true and nontrivial. It took Samuelson several years to come up with Ricardo’s theory of comparative advantage. But even that has been debunked.

25. Sheri says:

JMJ: Okay, that’s what I thought. The “identity” is worthless. Or that seems to be what you are saying.
As for pollution causing climate change, that’s a ridiculous claim that cannot be backed up by any science. It may affect climate, but climate changes with or without pollution. Then, when a naturally occurring molecule, CO2 was declared “deadly” and “evil”, well, there went any credibility. So, my dear JMJ, how much more straw do you have left?

JH: A couple more Carl Sagan quotes
“If it can be destroyed by the truth, it deserves to be destroyed by the truth.”
â€œOne of the saddest lessons of history is this: If weâ€™ve been bamboozled long enough, we tend to reject any evidence of the bamboozle. Weâ€™re no longer interested in finding out the truth. The bamboozle has captured us. Itâ€™s simply too painful to acknowledge, even to ourselves, that weâ€™ve been taken. Once you give a charlatan power over you, you almost never get it back.â€

N in V: If I go back to elementary math, the Kaya seems to reduce to CO2=CO2 which is true, but useless. How does an identity speak to anything in the real world? I can fill in anything in place of GDP, energy, etc and still get the same answer. CO2=CO2 I’m not understanding the point here. In fact, Kaya would be more useful as a list of suspected components of CO2 increase and ranked according to what is probably the biggest contributor. Or maybe a diagram. Unless it speaks to actually changes and amounts, how does it help (which would then imply causality)? I’m not getting it. Sighâ€¦â€¦.(Your brick wall example doesn’t reduce to wall height=wall height, as does Kaya.)

26. Scotian says:

JH,
“I am not sure what the partial derivatives and the number of independent variables have anything to do with the Kaya Identity”

The Kaya identity is a crude type of chain rule expansion that assumes a single independent variable. It is not clear what the variable is but a defence of the identity should identify it and show that all the other terms depend only on it. If there is more than one independent variable, as is almost certainly the case, the chain rule expansion has many terms (paths of CO2 production). NiV gets it although there is a typo in his equation. You must know what independent variables are.

Shawnhet,
“OTOH, if we moved any of the factors in Kaya to zero, it would be pretty reasonable to expect CO2 emissions to drop to zero”

Are you claiming that there was no CO2 before humans entered the scene? Maybe you are interpreting the identity to refer to only changes in CO2 due to industrial society as measured by the ratios given, but that would beg the question. That is why the identity should be explicitly written as rates of change (sinks as well as sources) giving multiple paths tested with empirical data. As it is we have a political and not a scientific or even mathematical identity.

27. Johan says:

Sheri
How does an identity speak to anything in the real world?

Letâ€™s take the following two countries, Austria and Belarus. In the year 2011 Austria emitted 67182180 Metric Tons of CO2; and Belarus 67157810 Metric tons of CO2. Thatâ€™s almost the same. We call F = CO2 emissions in Metric Tons.

One can easily look up their population (P in Millions); GDP (G in in Billion U.S. Dollars), and their Total Primary Energy Consumption (E in Quadrillion Btu). From that it is easy to calculate GDP per capita (g in thousand U.S. Dollars per capita); energy intensity (e in Btu per U.S. Dollars) and CO2 emissions per unit of energy (f in Metric Tons of CO2 per billion Btu].

For Austria P = 8.217280; g = 40.894340; e = 4386.944790 and f = 45.572266, with F = P*g*e*f = 67182180 Metric Tons CO2, as expected (disregarding rounding errors).

For Belarus P = 9.661510; g= 4.609136; e = 26821.632430 and f 56.227235, with F = P*g*e*f = 67157810 Metric Tons of CO2 (again disregarding rounding errors).

So what can we learn from this? Those two countries do not differ much in terms of CO2 emissions. Belarus is slightly more populated (by a factor of 1.18); and it has higher CO2 emissions per unit of total primary energy consumption, but not all that much (a factor 1.23). But, Austria has a GDP per capita almost 9 times higher than that of Belarus; whereas the energy intensity of Belarus is more than 6 times higher than that of Austria.

How does KAYA help e.g. Belarus in determining their energy and climate change policies? Well, they may try to increase GDP per capita, and hope that energy intensity as a consequence will decrease. Or perhaps they should lower energy intensity, and hope that as a consequence GDP per capita will increase. Or perhaps they should do both. Or perhaps, they should stay poor and nonetheless try to lower energy intensity, who knows? KAYA really does not tell you what to do. It only tells you why two different countries, so similar in some respects, can be very different in other respects.

PS: I hope I got the units correct, as Iâ€™m not used to working with Btu.

28. JH says:

Scotian,
My question is then how the Kaya identity is a crude type of chain rule expansion that assumes a single independent variable? What is the independent variable exactly?

The Kaya identity simply involves the basic operation of cancellation.

Yes, there is a typo in NIVâ€™s example. Yes, he is also right that those terms are not independent. However, is it possible to decompose CO2 emission or anything else into variables that are independent of each other? Or, in statistics, is it possible to observe only explanatory variables that are truly independent of each other?

Johan,
I have no doubt that economists… and climate scientists are faced with many difficulties when trying to sort through complicated situations. But, that some readers here outrightly trash researchersâ€™ ideas without thorough understanding is really not admirable behavior.

Sheri,
Yes, indeed, â€œIf it can be destroyed by the truth, it deserves to be destroyed by the truth.â€ However, as Suntzu said, one cannot win if he doesn’t understand and know his opponents first. If Briggs and you indeed know the truth and understand the climate research, you gotta show it.

29. Scotian says:

JH,
“What is the independent variable exactly?”

The criticism relates to the fact that there isn’t a single independent variable, but there should be for the identity to make sense.

“The Kaya identity simply involves the basic operation of cancellation.”

If so then the procedure is nonsensical as Briggs and others have claimed.

“But, that some readers here outrightly trash researchersâ€™ ideas without thorough understanding is really not admirable behavior.”

What an odd claim. I assume then that you will enlighten us.

30. Sheri says:

JH: You clearly stated in a past post that you would not debate climate science with me (http://wmbriggs.com/blog/?p=12914). No point to showing that which you do not want to see. Briggs has presented evidence that models do not predict accurately, which means the models are flawed. Flawed models=theory without proof when using models as proof of theory. There could be other proof, but models are what are thrown out by virtually all global warming advocates.

31. Shawnhet says:

Scotian: “Maybe you are interpreting the identity to refer to only changes in CO2 due to industrial society as measured by the ratios given, but that would beg the question.”

The Kaya is intended only as you say to refer to industrially derived changes in CO2 not the natural ones, of course. I’m not sure what question you think I am missing but if the question is can any set of factors give just as good an idea of causation as Kaya does, then that answer is plainly false. The Puppy-Kaya would state, for instance, that it is impossible for a puppy-free society to have any CO2 emissions whatsoever.

Cheers, 🙂

32. Scotian says:

Shawnhet,
“The Puppy-Kaya would state, for instance, that it is impossible for a puppy-free society to have any CO2 emissions whatsoever.”

Are you saying that 0/0 = 0? 🙂

33. JH says:

Scotian,

The idea of Kaya identity is indeed based on a basic operation of cancellation. No chain rule or derivatives of any sort unless you want to claim that the cancellation is some sort of chain rule.

If so then the procedure is nonsensical as Briggs and others have claimed.

Why would that imply the procedure is nonsensical? Simplicity doesnâ€™t imply nonsensical. It also doesnâ€™t imply its application is of no use. For example, see Johanâ€™s comments. No need for me to repeat good comments. Another example, Monte Carlo simulation methods use the simple Riemann sum, which is a simple idea but turns out to be very useful.

34. Sheri says:

Okay, further research indicates an “identity” may be useful in trigonometry and other mathematical processes, but I cannot see what use the Kaya “identity” has because unlike trig, have how much energy is used, population, GDP been shown to in 100% of the cases to actually definitively affect the carbon output? Does the type of energy matterâ€”say nuclear versus lignite? From what I’ve read, it does. So this is actually looking at causation and is not just a definition (sort of) of a mathematical relationship.

35. Scotian says:

JH,
“For example, see Johanâ€™s comments. No need for me to repeat good comments.”

You might explain why you think that they mean anything. The same numerology can be done with Briggs’ puppy equation and even Johan admits that the “KAYA really does not tell you what to do.”

36. JH says:

Scotian,

I thought Johan’s comment was quite clear in explanaing why Kaya identity can be meaningful, which doesn’t mean it is perfect. I have also pointed them out in my previous comments based on what I can see as to how KI can possibly be useful.

Right, Kaya identity itself does not tell you what to do, but Johan demonstrated how it can be applied. As I said, an explanation is only legitimate when a plausible causal mechanism is available. Yes, Briggs has used the cancellation operation to derive his puppy identity. If Briggs can make any sense out of his puppy equation, he should show how it can be done via data; see Johan and Brandon’s comments. Well, we know that at least you would have taken him seriously if he actually did so.

37. Shawnhet says:

Scotian
23 JULY 2014 AT 1:43 PM
“Are you saying that 0/0 = 0? :-)”

Heh. How is it possible for the amount of CO2 emitted to be undefinable? Isn’t it simpler to just accept that the Puppy-Kaya makes no sense (and is not thusly equivalent to regular Kaya)?

38. Scotian says:

Shawnhet,
“Heh. How is it possible for the amount of CO2 emitted to be undefinable?”

That was your choice not mine. You could have done the same thing with the original. The main problem is as I explained above.

39. Scotian says:

JH,
“As I said, an explanation is only legitimate when a plausible causal mechanism is available.”

Exactly, there must be an unique independent variable as I outlined above. Unfortunately this has not been shown nor is it likely.

“If Briggs can make any sense out of his puppy equation, he should show how it can be done via data”

The numbers seem to impress you but because of the nature of the cancelation the numbers will always work out, even for puppies. There has to be an underlying theoretical justification, i.e. a chain rule.

“Well, we know that at least you would have taken him seriously if he actually did so.”

Do you understand the nature of the reductio ad absurdum argument used?
I am not sure that Johan is on your side when he says “Although slightly more sophisticated, they are still pretty useless, for the same reasons KAYA is useless.”

40. James Gibbons says:

To fully try to understand the Kaya Identity, I went to the U of Chicago web site and downloaded the FORTRAN code for their model calculator. They don’t provide the historical data, but it is easy to harvest those numbers from their plots. Once I went over this and did some example calculations, I felt that I fully understood it, and what I found out is that there is a lot of fuzziness to it.

For example, start with Johan’s discussion at 23 JULY 2014 AT 10:59 AM:

To calculate a volume of CO2 produced by a group of people or nation we must multiply several values, P, g, e and f. But one must also pay close attention to the units, how they are used and what they really mean.

Let’s start with the right most term, f, or Tons of CO2 per GigaBtu. This is one of the fuzzy values. To know this value we need to know all the types of energy used, the quantities of each used and then calculate up an average that we can use as a single number. Burning natural gas to generate electrical power will not create as much CO2 as burning coal, assuming the conversion of heat (Btu) from both processes produces the same amount of electrical energy. Using wind turbines produces much less CO2, and so on for all the different sources of energy. One only hopes that you can get this right, and how do you figure this for the future should we ever develop fusion power? But we will ignore the details and simply say we have units of Tons CO2 per GigaBtu.

We next need the e value or Btu per dollars. Again, another fuzzy value. We need the energy mix again and for future values, how do you cost out that new fusion power plant? Forget all the details and the units are simply Btu per dollars. This is really how much energy you can buy per dollar, not the energy intensity. Intensity implies how fast you burn through something (hey, look at that intense house fire over there) which is totally unrelated to these units.

The next term is the g value or 1K dollars per person. This is the most fuzzy of all the values because it must be the average dollars you give each person to buy the fuel that produced the energy. It is not GDP! It is only a part of the GDP. We don’t spend all our money on fuel or even the cost of production of fuel.

Now multiply all these values by the population in millions: P * g * e * f

And convert to units:

(1,000,000 * person) * ($* 1,000 / person) * (Btu /$) * (Ton / (1,000,000,000 * Btu))

And we get Tons which is what we were looking for.

Now suppose you say I’m all mixed up because g is really total national GDP like Kaya and his friends say it is, and e is really intensity and f is efficiency. Well I say BS. In any accurate physical calculation, not only must the units make sense and balance out, but they must also be fully understood in terms of what they measure. All the groups of units (except possibly population) in the Kaya Identity are fuzzy, and only when you perform a specific calculation for a specific example do you find out what they really mean.

I think Kaya simply started adding groups of units to create this identity, because in any valid physical calculation the units must balance, but he then got to thinking this group is GDP, this one is intensity (whatever that means) and this one is how efficiently we generate the energy relative to CO2 produced. But in reality, it is very hard to nail these numbers down and what we are told was GDP is only a part of real total GDP.

41. Shawnhet says:

Scotian
23 JULY 2014 AT 7:57 PM
“That was your choice not mine. You could have done the same thing with the original. The main problem is as I explained above.”

My point is that doing the same thing to Kaya does not give the same sorts of results as doing it to the Puppy example.

If the GDP falls to 0, we can be pretty confident emissions will approach 0 as well. OTOH, if Puppies fall to 0, what can we say about CO2 emissions – I would say next to nothing. It is entirely possible that any and all changes in Puppies have no effect whatsoever to the CO2 emissions. This argues pretty strongly against any sort of causal connection between Puppies and CO2, something which cannot be said of GDP.

42. Johan says:

I’m not going to delve deeper into these matters, because I would have to write down some (fairly) sophisticated equations, and am not sure that’s even possible in the comments.

It is funny though that IPCC is very much aware of the shortcomings of the KAYA (and the very similar IPAT) “identity”. Some quotes:

While the Kaya identity above can be used to organize discussion of the primary driving forces of CO2 emissions and, by extension, emissions of other GHGs, there are important caveats. 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.

Although, at face value, the IPAT and Kaya identities suggest that CO2 emissions grow linearly with population increases, this depends on the real (or modeled) interactions between demographics and economic growth (see Section 3.2) as well as on those between technology, economic structure, and affluence (Section 3.3).

Using KAYA as a “model” to build scenarios with “What-if analysis” is IMHO an unequivocal “abuse”. Unfortunately, some researchers keep doing that.
But KAYA does have some limited use in first-order cross-section and time-series analysis, i.e. comparing the emissions of different countries / regions in the same year (see my example) or trying to figure out why the emissions in one country /region have evolved the way they did. Again, I repeat, only “first-order”. Obviously, a lot more analysis would still be required.

James Gibbons
“…this one is intensity (whatever that means)”
I know engineers prefer energy efficiency, defined as (total useful output) divided by (total energy input from all sources). Energy intensity is simply defined as the reciprocal of energy efficiency.

43. Johan says:

James Gibbons
Forgot to add … the reason that in the aggregate (total useful output) is expressed in monetary values should be quite obvious. How else are you going to add e.g. “gallons of beer production” and “numbers of cars produced”?
Of course, once you start disaggregating, you can begin using physical outputs. But even so, strictly speaking, you cannot simply add “tons of Basic Oxygen Furnace steel” to “Electric Arc Furnace steel”, and call the sum total “steel”.
In the end, everything gets very, very complicated (and believe it or not, at least some economists are aware of that).

44. JH says:

Scotian,

Exactly, there must be an unique independent variable as I outlined above. Unfortunately this has not been shown nor is it likely.

I donâ€™t see how you go from my statement to this conclusion. Why would you think there must be a unique variable (letâ€™s drop the word independent for now)? What is the unique variable for?

Do I understand reductio ad absurdum? Here is what I know about the method of reductio ad absurdum based on my academic training. Of course, one can easily google it. The idea is to show that the negation of a proposition is inconsistent with other things we already know, which is basically the proof of contradiction. For example, it is used to prove â€œthere is no greatest even integer.â€

I would say that Briggs has attempted to ridicule the KI, but no reductio ad absurdum argument used here.

Nope, itâ€™s not that I am impressed with the numbers or who is on my side. I have read many research articles in statistics (not the kind that Briggs like to criticize) throughout my 23-year academic career, almost all authors would first clearly explain the method under discussion and then point out the merits and shortcoming to be improved. In addition to mathematical proofs, if necessary, authors would often demonstrate their proposed method by simulation studies or real data evidence. I guess it is too much to uphold such a standard here.

Well, we probably all have other things to do. Criticizing Briggs’s statistical knowledge isn’t one of things I have to do today, though I am glad that he finally interprets the linear coefficient correctly. So, have a good day!

45. Sheri says:

Scotian: Excellent response to JMJ!

Shawnhet: I thought identities did not involved causality but rather were simply statements about a mathematical relationship.

46. Shawnhet says:

Sheri
24 JULY 2014 AT 8:30 AM
” I thought identities did not involved causality but rather were simply statements about a mathematical relationship.”

IMO, they may or may not say something about causality – the Kaya as stated does correctly identify (however imperfectly) some of the causal factors of CO2 emissions.

Scotian
24 JULY 2014 AT 9:31 AM
“Actually you get another 0/0.”

And the 0/0 tells you that your expression has broken down. When this happens, it is time to take a step back and think about things *logically*. With no GDP, there is no industry, with no industry there can be no industrially emitted CO2. QED.

47. Sheri says:

If Kaya “identifies sources of CO2” (without natural ones included, of course), then one must justify each number and how much each contributes. The Kaya says nothing about this and no matter what numbers you plug in, you still get CO2=CO2 which cannot address causation. When you introduce causality, the level of proof is much different that an identity, which is true no matter what numbers you plug in. Generally, causation changes when you plug in numbers.

I did note that there seems to be 2 different versions of the Kaya Identity:
the one Briggs uses which looks like a cancellation exercise and then this–Carbon Dioxide Emissions = population* per capita GDP * energy intensity * carbon intensity which is what Johan appears to be using.
The one here is not capable of predicting causation since all values yield the same answer. The other equation could be used to calculate “causes” of CO2 if you can verify all the parameters.

48. Scotian says:

Shawnhet, Paraphrasing we have:

And the 0/0 tells you that your expression has broken down. When this happens, it is time to take a step back and think about things *logically*. With no puppies, there has been a breakdown of human will, and with no will there can be no industry, and with no industry there can be no industrially emitted CO2. QED.

In both cases we have gone beyond the identity and this outside speculation can not be used in support of Kaya in either form. In other words, although the breakdown of an equation at its poles doesn’t disprove its applicability in general, it certainly can not be used to support it.

49. Shawnhet says:

Scotian
24 JULY 2014 AT 12:59 PM
“And the 0/0 tells you that your expression has broken down. When this happens, it is time to take a step back and think about things *logically*. With no puppies, there has been a breakdown of human will, and with no will there can be no industry, and with no industry there can be no industrially emitted CO2. QED.

In both cases we have gone beyond the identity and this outside speculation can not be used in support of Kaya in either form. In other words, although the breakdown of an equation at its poles doesnâ€™t disprove its applicability in general, it certainly can not be used to support it.”

Your paraphrase is silly. People without puppies still have human will.

Whether or not the Kaya expression works, it still deals with plausible causative factors behind the emission of CO2 – imagining that GDP is 0 plausibly leads us to a correct-seeming conclusion about what CO2 would be. Reality check here: if the GDP fell to 0, what do you think the CO2 emissions would be?

50. Scotian says:

Shawnhet,

“Reality check here: if the GDP fell to 0, what do you think the CO2 emissions would be?”

Difficult to say since the GDP is an economic construct. But in any case the Kaya identity doesn’t tell us.

51. Sheri says:

If the GDP fell to zero, the emissions would not fall to zero. Anthroprogenic CO2 emissions have existed since humans found fire. Unless the entire remaining humans ceased to use fire in any way, the emissions would be smaller (depending on how many humans there areâ€”England was pretty smoky at times in its history) but they certainly would only go to zero if humans forget fire and/or die out.

52. Shawnhet says:

Sheri and Scotian,

Respectfully, you are both missing the forest for the trees here. The question is: does the Kaya Identity deal with causative factors for CO2 emissions? Setting GDP to 0 in our hypothetical shows that for GDP at least it does. The fact that there may be still be some massively smaller amount of CO2 emitted if GDP fell to 0 or the fact that Kaya breaks down when one of its factors is set to 0 doesn’t alter that at all. We don’t need Kaya to be quite confident that CO2 would fall *precipitously* with a 0 GDP. We can’t say the same about Puppies.

53. Sheri says:

No, I am not missing the forest for trees. I agree that puppies don’t affect CO2 (or at least not much, though one could argue that the factories that make dog food, dog clothes, dog houses, dog jewelryâ€”you get the ideaâ€”do). That was not the point. Done as an identity, with cancellation, the fact is you get the same answer using puppies or GDP. That appears to be what identities areâ€”equal no matter what you plug in. That’s where I don’t see any use for the identity.

Now, if you use the other version
Carbon Dioxide Emissions = population* per capita GDP * energy intensity * carbon intensity
then of course it speaks to emissions but it doesn’t seem to qualify as an “identity”. At that point, you have to justify the components and their equal weighting before you can actually say that the equation represents reality or even a useful model (useful for something other than propaganda, of course). On the surface, it appears to be somewhat useful. It does not account for the use of fuels in poorer countries that are very carbon intensive but the country has a very low GDP, so far as I can tell.

54. Shawnhet says:

Sheri
25 JULY 2014 AT 11:30 AM

“Done as an identity, with cancellation, the fact is you get the same answer using puppies or GDP.”

You mathematically get the same answer using puppies or GDP you cannot get so *realistically*. Can you see the distinction? In the real world, we cannot just decide to have the same or higher levels of GDP with the same level of CO2 emissions – our ability to do that is strongly limited.

OTOH, we can decide *in the real world* to have whatever level of Puppies to C02 we want – if we want to have 10 times as many puppies or 10 times as few at the same(more or less) level of CO2, we can do this. IOW, the ratios in Kaya reflect real world phenomena that affect CO2 and the Puppy expression doesn’t. It is as simple as that.

55. Sheri says:

I understand that in the real world, puppies are different from GDP. But in the Kaya identity, any value gives the same answer CO2=CO2. Which is why if you are looking at the “real world”, the equation “Carbon Dioxide Emissions = population* per capita GDP * energy intensity * carbon intensity” makes sense. However, that is not an identity as mathematically defined, so far as I can tell. So the Kaya “identity” really doesn’t do us much good in the real world. The two versions are very different and serve very different purposes. (Identities seem to be more a definition in mathâ€”especially trigonometry, etc. They do not seem to apply to CO2, economics, and the like.)

56. Shawnhet says:

Sherri, perhaps you’re right about Kaya not being an actual identity whatever that means to you. So long as we agree that factors Kaya includes are causative to the level of CO2 emissions, I don’t see anything worth talking about.

Cheers, 🙂

57. JH says:

Sheri,
Per capita GDP = GDP / population
Energy intensity = energy / GDP
Carbon intensity = CO2 emissions / energy

58. Sheri says:

Shawnhet: No problem.

JH: Argggghhhh! Now I see that some sites I checked left out crucial information and that both versions are identities and therefore not useful. Back to the drawing board. Thanks for the information!