You sweat over this for a few weeks until one of your interns suggests a solution to you while spilling your coffee on the photocopier. She suggests that you allow every female of the population to continue having children until they have a male child. Once they’ve given birth to a male child, then they are required by law to stop.

You ponder this for a while. Obviously, allowing everyone to have a male child will be very popular with the electorate, but can we really simply allow everyone to keep trying until they get one? Surely this would result in some kind of population explosion. Worse yet, wouldn’t it result in some kind of huge imbalance in the male-female ratio?

Actually no. The ratio remains at exactly 1:1, and the birth rate is exactly 2 – zero growth.

The mathematics is not particularly difficult. One can easily write out a table of all the probabilities and the required figures (the birth rate and the gender ratio) can be found by way of summing infinite geometric series. But how are you going to convince the electorate of this rather counter-intuitive result?¹

Let’s start by drawing a box.

A box

This box represents the first child. Let’s say it’s of unit-size, representing one child. I’ve coloured half of the box black and the other half white indicating that the probability that the child is a boy is one half, and the probability that the child is a girl is also a half.

Of course, if the child is a girl, we get to try again. That box should only be a half the width of the original because there’s only a half the chance that the event it represents will happen. The length of the box should still be one though, because given that it does happen, it still represents one child. The colour scheme is identical to the first box. We can repeat this as many times as we want, but I will stop at four.

four kids

What this is really representing, is that the probability of having exactly one boy is a half, the probability of having exactly one girl and one boy is a quarter, the probability of having exactly two girls and one boy is an eighth, and so on. The diagram however, allows you to very easily see why the gender ratio stays exactly balanced – when you sum all the probabilities, you’re still going to get the same ratio of boys to girls. In fact, using the rule suggested by the intern will preserve a gender ratio reflective of the probability of having a boy or a girl, which we have assumed to be 50:50.

The question of population growth is slightly more tricky. Below is the same diagram, except with one of the cases highlighted.

girl girl boy is highlighted

We could think through every single one of these cases and add them up. And it’s not even a very difficult thing to do. But if you’re a government minister, you’re probably too lazy even for that. So what about…

another way of stacking

All I’ve done is stack the boxes differently. If we continue stacking the progressively smaller boxes on top of each other, you will get a gap of etc., until you eventually end up with a gap at the top of which equals zero (in an asymptotic sense) and voila! You have a box that is one by two, giving your expected value of births per mother to be two.

This is a good way to control overpopulation because the birth rate should be about 2.1 for a steady population (because not all babies will live long enough to reproduce) and 2 is significantly lower than 2.1. (Things like twins don’t have much of an effect, as they only account for 2% of live births)

Just quietly, I don’t believe that implementing a policy like this would be an easy thing to do. Despite the obviousness of the mathematics, there is no doubt in my mind that many will cry foul and drum up all manner of absurd conspiracy theories accusing the government of taking away freedom and the like. That, of course, will all be a smoke screen for the real problem which is, despite the public school system, not too many people appreciate the absolute truth of mathematics, and will sooner accept a better-sounding and more intuitive dogma, handed down by a loud radio show host, than actually bother to listen to reason.

the current warming trend has more to do with us than the sun, anybody who tells you otherwise is a filthy liar

I am angry. I’m angry at the world. I’m angry at the direction it is headed, not so much because of the direction itself, but more because it was within the power of a handful of men to choose this direction, and they deliberately chose to betray the trust of future generations for their own selfish gains. Not only is this betrayal taking us on a collision course with the destruction of our own civilization, many who are able to change this course, willfully lie to us about it, and how it came to be this way.

Rather than just sit here and complain (and write some rather ambiguous poetry) I have decided to take a more positive and pro-active approach to these grievances. Firstly, I’m going to apply to various schools to undertake a PhD in economics. Secondly, I’m going to list below a few little tidbits which aren’t necessarily very widely-known, yet have a huge impact on the whole-earth-equation.

Energy Crunch

Energy is absolutely essential to the advancement of civilization. Since early antiquity, every civilization that ever did anything of significance, did so because they found a way to harness a lot of energy into a short space of time. Basically, large, advanced civilizations are all built on the back of cheap energy, whether it be in the form of domesticated animals, slaves, or chemical reactions. Take for example the advantage of iron weapons over bronze – iron takes a great deal more energy to produce than bronze does. An easier example is the use of firearms, or mounted soldiers. A more peaceful example is the use of chemical fertilizers in modern agriculture. Steam power, followed by electricity in the 19th century set the stage of the explosive growth of human civilization which we have somehow become accustomed to. More and more people require more and more energy, but where does all this energy come from?

Coal and oil provide the vast majority of the world’s energy. There may be a lot of windmills in Germany and the Netherlands, but without fossil fuels, the economies of either country would grind to a halt. Coal and Oil are both renewable resources – that is if your time frame is geological time. Both are formed by organic material (mostly plants, so basically solar power) which has become buried under certain conditions and then subject to extreme pressure and high temperature over millions of years. In the time frame of human existence, especially considering we’ve only been using the stuff for about 200 years, we can safely consider it a finite resource.

Here is the origin of one of the first (important) lies I was told by adults whom I trusted (I was quite young at the time). One is often told that new oil deposits are being found all the time, and that there is estimated to be about 800 years worth of coal left in the earth. These statements are both technically true, but they don’t take away from the fact that these resources are finite. We will eventually run out, the question is: when will that happen? The obvious implication in the whole “new discoveries” line is that we are nowhere near that limit and that there is nothing to worry about. At this point, if someone is trying to tell you this, you should quickly slap them in the face with considerable force (or a fish, if one is handy).

We are running out of coal and oil. In the 50s a clever fellow by the name of M. King Hubbert made a very obvious observation: That you cannot extract a finite resource at an exponentially increasing rate over an infinite amount of time. He proposed, based on well-known observations and science, that most oil wells’ extraction curves followed the same bell-shaped pattern. Oil exists in the ground in a way rather like water is absorbed in a sponge. You stick a straw into it, and it comes gushing out because it’s under pressure. After you discover it, you stick more straws in so that you can get it out more quickly. Eventually, you have to pump it out. After a while, it becomes more and more difficult to extract. Since the stuff is so valuable these days, we’ve come up with all manner of clever ways of getting those last little bits of oil out of the sponge, but the fact remains that the extraction curve tapers off because the amount of energy required to extract the oil eventually comes too close to the amount of energy contained in the oil to make it worthwhile.

The same pattern is generally true for any finite resource that has to be extracted from the ground. The other wonderful thing about this bell-curve is that the curve for number of new discoveries and the curve for actual extraction looks exactly the same and is only separated by a slight time-lag. The time lag is caused by the delay between finding a new oil field and setting everything up to extract the oil from it. The really bad news is this – it is generally agreed that oil discoveries peaked in about the year 2000. Coal is predicted to peak anywhere from 20 to 50 years later.

Oil discoveries peaked in about the year 2000

Here’s the rub. Unlike other natural resources, like a lot of rare earth metals, oil has a very low substitutability. What does that mean? Lets say that for some unknown reason, tomorrow the world ran out of butter and the means by which to produce it were somehow forgotten (c’mon, use your imagination here). Would this be a disaster? Not really. For a start, there are many substitutes for butter. Of course, it is also possible for human civilization (but not French restaurants) to continue without butter or any of its substitutes. This is not true for oil.

Our standards of living, all of our technology, all of the wonderful complexity that we take for granted is built upon cheap coal and oil. After the peak, which is happening very soon if it hasn’t happened already, the world as we known it will come crashing down rather hard. Astute readers will point out that the curve is bell-shaped and so we can use all the time we have on the downward part of the curve to figure out alternatives. Even more astute readers will have figured out that the curve is for extraction, not any measure of cost per unit of energy, which will continue to skyrocket because the cost of extraction will become ever greater.

The smart thing to do, is to come up with substitutes before the crunch, while we still have cheap energy and the high productivity that comes with it. Nuclear is a nice stop-gap option because it is high in energy and low in CO2 emissions (I’ll get to that later) but for similar reasons (Uranium is actually very rare) it can’t last long. Wind supplies about 1.5% of the world’s electricity, and ultimately may never supply more than about 20% because of intermittency problems. Solar really is the only way to go, but using current technology, we would need an area of solar panels equivalent to the size of Spain  to supply the world’s current demands (and what happens if it’s cloudy in Spain!?). Lighter and more efficient batteries should also be high on the priority list to support completely electric cars, and eventually aircraft.

There’s another good reason to switch to renewable energy apart from the fact that we are fast running out of non-renewables, and that is because we will emit a great deal less CO2 into the atmosphere and take a first and important step to addressing climate change. Perhaps we would have been better off if there wasn’t so much coal or oil in the earth, because then we would have been forced to wean ourselves off our carbon-based economy before climate change really had a chance to make our lives even more difficult.

Right now, the amount of energy that comes out of agriculture is about a third the amount of energy that goes in (think about the energy required to make the fertilizers, and the petroleum that goes into farm machinery). As climate change starts making agriculture less productive, we are going to need ever-increasing energy inputs to produce the same amount of food. (If only we could eat petrol) Running out of energy doesn’t only mean not being able to drive cars, or turn lights on… it will mean starvation for untold billions.

Government Subsidies

The trouble with modern democracies is that you need to get elected. It is quite widely known, if not widely accepted, that getting elected isn’t just a matter of having good policies, leadership qualities, or good intentions (although these things, one would hope, still factor into the electorate’s decision – I remain an optimist), but also a matter of having enough money to run a campaign against whoever else is on the ballot. It is for this reason that many politicians come from very wealthy backgrounds, or have accumulated wealth by being captains of industry. In other cases, a candidate may receive support from an established political party, who in turn receives support from whoever is kind enough to donate.

Here’s how it works – say some big company doesn’t want the government to ban product X, which is known to accidentally kill hundreds of people a year. Some politicians are like “no way, that’s absurd”, while others are like “umm… I’m kinda low on campaign funds”. The company sees an opportunity to provide an incentive for a politician to vote in their favor. So while you and I vote for someone to be in government, and our collective points of view is supposed to somehow imperfectly translate to their votes, there are some very wealthy entities out there who vote in a very different and much more effective way.

The result of this is a lot of very bad policy which favors the short term interests of a small group of people (who are, btw, already obscenely wealthy) at the expense of the long term interests of the people, and the planet.

A good example is agriculture subsidies. Just about every wealthy western democracy does this, and it’s simple – they heavily subsidize farmers to produce more food. This is politically a very easy thing to do because everyone likes the idea of food security. The result is a very large amount of food, much more than is required to feed the population. What happens to this food? A lot of things actually. I would recommend the interested reader to watch the documentary “Food Inc.” for more information on the industrialization of the food industry. When you have a surplus of something, you want to trade it. In this globalized world, what happens is all this extra (and very cheap) food gets dumped in developing countries. Countries whose farmers cannot possibly compete, not only with the huge advantages afforded by economies of scale in advanced western agriculture, but with subsidies effectively pricing these products below cost. The result of this is, at the very least, a severe hampering of an entire sector to be able to grow or function properly, and at worst, the complete destruction of the agricultural industry of a country.

This is difficult to stop for several reasons. First of all, not many people know that this happens to this extent. Secondly, if one country engages in heavily subsidizing its agriculture, then other countries (if they are able) must follow suit, or else they risk pricing their own industry out of business on the international markets. Why doesn’t everybody just agree to stop? Because in many countries, the US being the main culprit, the food industry is extremely wealthy and powerful and able to buy votes and engage in morally-questionable (I’m being charitable in my language here, the appropriate language to use in this case is “fucked up”) legal actions against anyone brave enough to stand in its way.

Of course, subsidies can be used for good as well as evil. For a long time, the US led the way in developing renewable energy because it provided subsidies in the form of tax breaks to those who developed it. Of course, during the 8 years of the Bush administration, this was effectively reversed and, instead, the oil industry was rewarded with many policy “favors” including, but not exclusive to, the war in Iraq. At this point, I expect many libertarians to use this as evidence that all government intervention is bad. Predictably, I would disagree by pointing out that the bad stuff isn’t really a result of the government intervening in the market, but rather, the market intervening with the government – the invisible hand is not as invisible as many laissez-faire advocates would have us believe.

How can this be stopped? With great difficulty. Huge campaign reform needs to happen in every country in the world and huge multinational corporations need to be made a lot less powerful. That’s kind of a tall order. No matter how accountable you can make governments, and how strong you can build international institutions and treaties, it will always be very difficult to hold large corporations accountable. Public companies are at least theoretically accountable to their shareholders, so if shareholders begin to care about things other than dividends and share prices, then we could see some progress (ha ha ha). Even then, I’m a shareholder in a handful of companies, and I don’t get a huge amount of input into the decisions that the companies make.

Strong international laws which are enforced, cross-border accountability, and corporate responsibility are all lofty dreams. However, the world of multinational corporations is effectively self-policing at the moment. The kind of strong, socially-responsible leadership that is needed isn’t likely to surface, especially when boards of directors will pay someone much more money to behave like a total dickhead.

The Money Supply

The way money functions in the modern economy is more than just a philosophical question. Many people have theories on how money works based on stupid anecdotes (and I should know, my father is an accountant). I should take this opportunity to debunk some of the sillier myths about how money works, and how some people think that it should work within a historical context. Then point out some of the problems with the way things are currently sailing.

A long, long time ago, we bartered goods. Barter has many limitations, of course. Who is to say how many sheep are worth two cows? Even if this value is accurately known, there would be fractions of this and that, and transactions would be horribly complicated. So we invented currency. Currency originally took the form of rare metals like gold and silver, which eventually became impractical to carry so paper money and coinage was invented to take it’s place. These forms of currency were basically receipts for gold and were simply substitutes. The world has been on and off the gold standard for a long time and in the 1970s went off it for good (the US started it).

There’s no compelling reason that we need to be on a gold standard. The value of something is simply how much you are willing to pay for it. Not having a gold standard also makes it easier for governments to generate liquidity in a short space of time, which can be a very useful thing. Of course, one of the big fears of people who advocate going back to the gold standard is inflation. To ordinary people, inflation is when things become more expensive. Slightly more numerate people will complain about inflation when the inflation of commodities exceeds that of income. Inflation is a very tricky thing to deal with because it involves dealing with numbers in a way that isn’t intuitively obvious and which requires more education than most people ever receive.

The number doesn’t matter! Money has about as much to do with economics as numbers have to do with mathematics. My interpretation of what an amount of money means, is that is represents the value of an economy.

If the money supply increases relative to the amount of goods and services that exist in an economy, then we get inflation. Everything costs more, but in reality, nothing has changed. How does this happen? A bank might lend out some money, then there is theoretically more money in the system because the money saved in a bank is counted (sort-of twice) as invested. That’s how banks make money, they lend at interest (necessarily a higher rate than they pay out to deposits). The idea is that you borrow from a bank and use that money on something that will return a higher value than what you started with, and a higher value than your loan plus interest. Then you’ve made a profit, the bank gets its money back and has a little bit more than when it started.

What does the interest rate mean? If we assume perfect competition and perfect information (optimal market conditions) then everyone operates at what is called the “zero profit line”. Then the interest rate represents the growth in productivity in the economy. If the money supply precisely follows this, then it increases in step with the increase in the value of goods and services, which is the result of this increased productivity. Of course, if everyone spent their loan-money on stuff that turned out not to increase as much in value as previously predicted, then defaulted on their loans, or if everyone who had money in a bank suddenly decided they wanted to withdraw it all, then all these interest rates must fall otherwise the banks would loose money. The banks would also make it more and more difficult to borrow money, and so there would be less liquidity in the economy, so there would be less productivity, so less people would put money in the bank… and so the spiral continues. This sort-of happened on an epic scale in mid-2008 and precipitated the financial crisis. (There’s a really good explanation of it here)

One of the perceived problems of this system is that it assumes infinite growth in productivity. You may have heard how the current evil capitalist system is obsessed with growth. I used to think that growth was immoral because, to achieve growth, you had to exploit something – be it a natural resource, the rights of another human being, or the environment. To me, growth had to be at the expense of something else. But, I was wrong – economics is not a zero-sum game. The principle of comparative advantage allows us to achieve growth simply by dividing labor. Trade was simply an extension of the same principle over larger distances and time. Technology also allows us to increase productivity for free. In principle, it is actually possible for there to be sustainable economic growth for an infinite amount of time. The trouble really is – nobody ever talks about sustainability.

What does this have to do with the money supply? Well, governments have invented a system of controlling money called “central banking”. All it means is that there is a central bank in charge of the money supply. When the government needs money that it doesn’t have, it borrows money from the central bank out of thin air. The ledger has some negative numbers on it, and there’s suddenly more money in the system. If you think this sounds very iffy, that’s because it is. It is not a fatal flaw though. It is my opinion that it is vitally important that governments are able to engage in deficit spending. Isn’t there a problem if governments can spend money that almost seems to not exist? It can be.

If the economy is doing very badly and there is high unemployment, the government has a responsibility to do something about it. For reasons of market equilibrium, it is possible for an economy to get stuck in a depressed state. What the government can do is spend this money that, in a way, doesn’t exist, and stimulate the economy. You see, the principle is the same as borrowing from a bank except on a much larger scale. If the money is spent on increasing productivity (for example, increasing employment by engaging in building large-scale infrastructure) then that increase in productivity will be reflected in an increase in tax revenues which can be used to pay back the loan.

Of course, governments are not always so wise in their spending. It is a well-advertised fact that the US has a huge deficit. Unfortunately, their recent spending patterns aren’t great. If you spend on things like infrastructure, you increase productivity and get your money back. However, if you spend money on very expensive wars, then these are expenditures that aren’t likely to ever pay you back. As my dad once said of me, it is a poor “return on investment”. What does this mean in the “whole earth equation”? It means that it is more politically difficult for governments (especially the US) to spend money on things that they really need to spend money on (yes, I am implying that the wars are not strictly necessary to the betterment of humankind).

Climate Change

I’ve left this until last because it is probably the most important. We’re screwing up the climate. A great deal of misinformation is being passed around by some very loud voices and most of it is so amazingly wrong that it’s almost funny. The main arguments about man-made climate change are detailed in the previous installments of the earth debate (box on the right -> ). Even if you don’t believe that we are responsible for the mess we’re making for the climate, there are plenty of good reasons to take precautions. Like the precautionary principle.

Think about it this way. There are four possibilities – (1) we’re right about climate change, and we do something about it (2) we’re right about climate change, and we don’t do anything about it (3) we’re wrong about climate change, and we do something about it (4) we’re wrong about climate change, and we don’t do anything about it.

Possibilities (1) and (4) are good outcomes, and we can put them aside. Possibilities (2) and (3) are possible mistakes that we might make.

If we’re wrong about climate change and we take action on climate change, where does that leave us? Well, less reliant on oil, and more technologically advanced with regard to renewable energy. Both of these are more or less inevitable anyway. The cost? Well… in truth, the only real losers would be large oil companies. Also, as you have just read, there are many other reasons than climate change to stop using fossil fuels.

If we’re right about climate change, and we don’t do anything about it, then what? We will make the planet increasingly difficult to inhabit for humans. We will make it increasingly energy-intensive for us to get the same amount of productivity out of our agriculture. We will continue to fight over a smaller and smaller amount of resources and eventually destroy our civilization, and possibly species.

So… there’s a fairly obvious rational course of action. I realize that most of the people who read my website agree with me on most, or all, of these points. But perhaps they have a friend (or not a friend) who is sitting on the fence, or whose rational decision making ability is being obscured by the mountains of information (and disinformation) that is being pushed by a largely amoral and irresponsible news media.

Eventually, we will have to rely on technology to save us. Even if we stopped emitting CO2 tomorrow, there’s a lot of damage that needs to be undone to stabilize the earth’s climate. The climate is a very complicated system, and the safest course of action is not to mess with it. We desperately need to invest in mitigation as well as adaptation. Many misinformed people believe that mitigation is useless and adaptation is all we need… however, as detailed above, if all we do is adaptation, we’re still facing an epic energy crunch. We need to demand access to good information, call out the liars for what they are, and take desperately needed action to save ourselves from… ourselves. We have no time to waste.

“All I have is a voice
to undo the folded lie
the romantic lie in the brain
of the sensual man in the street
and the lie of authority
whose buildings grope the sky
there is no such thing as the state
and no man exists alone
hunger allows no choice
to the citizen or the police
we must love one another
or die” ~ W. H. Auden

With the world population at 6.7 billion, it is difficult to question the insight of this 19th century philosopher-economist. But even Malthus couldn’t have predicted the environment of the early 21st century. Looking at population figures from the history of human history, one indeed observes exponential growth. The total population of the world in 1500 was about 500 million. In roughly 1800, the world’s population was 1 billion, it didn’t double again for another 130 years, in 1930. 44 years later, in 1974, we had hit 4 billion. It only took 25 more years to add another 2 billion. Those readers who have a “slight acquaintance with numbers” will realize that these numbers are fairly consistent with models for exponential growth. So are we ultimately doomed to having our numbers controlled only by disease, famine, or warfare?

In a word – no.

To understand this, we need to examine a concept called the “logistical growth model”. When observing the population of just about anything except humans, scientists noticed that initially, when resources were plentiful, the growth rate is exponential. Eventually, resources become scarce, and the growth slows down and eventually stops. This can be modeled by the differential equation:

basically means “the rate of population growth over time”, r is a constant which defines the growth rate and K is the “carrying capacity” or the maximum population that can be sustained. Let’s think this through – at the start, when N (the population) is small relative to the carrying capacity, is a small number. This means that is a value close to 1. So the rN dominates. The solution to is simply – exponential growth. When N approaches the carrying capacity K, becomes a value close to 1, so goes to 0 making the whole thing go to zero – growth stops. Perhaps a graph would be instructive at this point:

Standard Logistic Curve

I’ve been very lazy here, this is the graph of (because it was easy to plot). The actual solution to the equation above is (where is the initial population) which would put different numbers on the axes but can easily be shown to have the same shape. The carrying capacity can be limited by any number of things such as the amount of available food, the amount of available space, and so on.  Curiously, while the overall trend of global population doesn’t appear to follow this curve (or at least seems to only be following this curve as far as the initial stages are concerned) there are some countries that do follow this curve. Moreover, these countries are the developed countries which are nowhere near pushing the boundaries of their carrying capacity (measured in food/space consumption vs production). When we think about this, it doesn’t seem to make sense – the growth levels off because of a scarcity of resources, yet it is in the developing countries where the resources are scarce, while in developed countries resources are plentiful. Why this apparent paradox?

Put simply, the paradox is a result of our ability to use our brains to plan our lives. That is not to say that people living in developing countries lack this ability, but because they are poorer, they don’t have the luxury of being able to exercise this ability.  When a country is wealthy, the life expectancy of its citizens generally increases. When life expectancy is good, parents don’t feel the need to have lots of children because there is a much higher chance that they will survive to adulthood. Wealthier countries also tend to be more urbanized, resulting not only in space constraints, but also in a diminished need for “farm hands” – a typical role for the offspring of families living in rural areas. In addition, better education and empowerment of women also contributes to a lower birth rate. If we were just like all the other animals, we would do what most other animals do when they’re bored – kill each other and make babies – both very easy paths to short term gain (and long term ruin), but thanks to our brains we can see past that (at least most of the time).

However, it is precisely these brains that may lead to our own undoing. Obviously, Malthus underestimated our ability to control our own population well before we hit our carrying capacity ceiling. He also underestimated our ability to dramatically increase our food production (although we’re not doing it in a very sustainable way – more on that later). Despite many predictions, globally we still produce far more food than we consume, although food distribution follows income distribution and that is obviously lacking. Of course, just because we are able to control our population before we hit our carrying capacity ceiling doesn’t mean that we will. The most optimistic projections estimate that the world population will level off at about 9 billion people, other predictions have that number as high as 12 billion, which doesn’t look like that much more on a graph, but is actually a very significant difference. While some argue (and I disagree) that we are already past carrying capacity, there is certainly a strong case that 9 billion will be pushing it, and 12 billion will almost certainly be over it. If this really is the case though, if the logistical model is to be believed, shouldn’t we already be seeing a dramatic leveling-off of populations across the board and not just in developed countries?

The reason that this hasn’t happened is because there is significant lag between the depletion of resources and the response resulting from it. An example I like to use to illustrate this is fish because it is easy to understand and the time-frame in which things happen in the fish example is easy enough for the average person to grasp.

Let’s just say for the sake of illustration that there are only two things that matter in our hypothetical earth – humans and fish. Humans eat fish. The amount of fish eaten is directly proportional to the number of humans. If the human demand for fish exceeds the amount of fish that there is, the human population will decline. Obviously, if the human population declines enough, the demand for fish will decrease and the fish will be able to replenish their numbers. For simplicity’s sake, we will assume that the fish population follows the general pattern of the logistic function discussed previously. The human population also follows this pattern but the carrying capacity is not a fixed value, but is determined by the number of fish. What happens?

Well, it depends. What does it depend on? It depends on the growth rate, or more accurately, the ratio of the growth rate of humans to fish. If the growth rate of the fish is very large compared to humans, the number of humans climbs steadily as the number of fish declines slowly and they eventually reach equilibrium.

Humans vs Fish round 1

The number of fish, and the number of fish that each individual human eats will obviously change all the numbers, but the shape of the graph would remain the same. What is basically happening, mathematically is , that means rate of growth of fish is the same as the rate of growth of humans (and this “rate” of growth happens to be zero). The reason it is zero is simple – if the number of humans was growing, the number of fish would decline, the growth of the number of fish is limited by carrying capacity for fish, and the growth of the number of humans is limited by the number of fish. Easy. If the growth rates for fish and humans are much closer, one of two things can happen.

Humans vs Fish round 2

The populations initially react very quickly to each other, but they eventually even out and reach equilibrium. The key thing to remember here is that we are solving the same equations, the only thing that has changed is the constant that tells us how quickly each of the populations can grow. If we tip the ratio just slightly so that ratio that humans can grow relative to the fish is a tiny bit higher, then the following happens:

Humans vs Fish round 3

It doesn’t take a genius to figure out what’s going on here. I don’t want to sound too alarmist, but this is precisely what is happening. Our rate of population growth may be slowing ever so slightly, but our rate of consumption (which is what actually matters here) is still increasing at an alarming rate.

Now I don’t want people to start thinking that I have objections to eating fish. I love fish. But we are simply using our resources much faster than they can be replenished and it will eventually bite us in the arse. Fish are a particularly good example because the feedback loop is very quick, but we are actually doing the same thing with agriculture as well (the energy that goes into producing food is now much greater than what we get out of it, more on this later).

How do we get the maximum out of our fish stocks? If we look at our logistic curve, we see the answer. For any given amount of fish stock, we need to consume at exactly the rate at which those stocks can be replenished. Now it remains to figure out when that rate is the highest. We could go to the trouble of solving the differential equation to find the maximum (and it isn’t very difficult), but looking at the graph is much easier. By symmetry, the maximum rate of growth occurs at a half of the carrying capacity. So… everyone just agrees to fish his or her fraction of whatever that sustainable number is, and we’ll all be fine right? Not quite.

Most people are familiar with the prisoner’s dilemma. Where the police have caught two suspects but haven’t enough evidence to convict. They are separated and each offered a deal whereby if they betray the other suspect, and the other suspect remains silent, then they can go free. The payoff matrix is shown below:

Prisoners dilemma payoff matrix

At first glance, the obvious thing to do would be for both to cooperate. But if you take the view of any of the individuals, you discover that the equilibrium (indeed, the Nash Equilibrium) is to defect. This is because, no matter what the action of the other suspect, you are always better off if you defect. In game-theory-speak we say that cooperating is strictly dominated by defecting.

What does this have anything to do with fish? Everyone can cooperate and limit their catch of fish, which is generally recognized as being the optimal solution. But if a fisherman decides he wants to fish more… if nobody else does, then that makes it easier for him to fish more fish, if others do and he doesn’t, then he loses out because the fish are more difficult to fish because there are fewer of them, and if everyone decides to disregard the limit, then everyone loses. Since the damage caused by any one individual is shared by everyone, the incentive for that individual not to cause that damage is very easily outweighed by any benefits that the individual might gain – the classic tragedy of the commons.

In part 5: Energy crunch, subsidies, and money supply

There is a good reason that our “pale blue dot” is blue, and that is because over 70% of the Earth’s surface is covered with water. It is no surprise that life began in the oceans, and water is still essential to life today. On average, a human can survive no longer than 3 days without it. Water is also very useful because it moderates our climate, and it is in this context that it will be discussed in Part 3.

Throughout my travels, I have discovered that the people who are most likely to believe in the urgency of addressing climate change, are those who live in environments which are very sensitive to the effects of the change. For example those who live on the boundaries of climate zones, or near large glaciers – very sensitive to the global temperature. One of these groups is, of course, small island nations who have noticed their territory slowly shrinking as the sea level rises. Older readers of this website may recall a rather terrible Kevin Costner film called Waterworld which was set in a future Earth where the sea level had risen enough to cover almost all of the Earth’s land area. Is such a scenario really possible?

Archimedes’ principle states: Any object, wholly or partly immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.

So, if you have a block of ice floating in some water, when it melts, the water level neither rises nor falls (think about that for a while… I have a funny feeling that that statement is going to generate a few comments). What does this mean in terms of sea-levels though? This means that when arctic sea ice melts, as it has been more and more prone to do lately, the sea level doesn’t change. However, it is important to note that most of the world’s ice isn’t contained in arctic sea ice. The antarctic leads the way, followed by Greenland, then the Andes. All three of these bodies of ice are supported on land. That means that if that ice melts, when it eventually gets to the sea, the sea level will rise.

That all seems very simple, but it isn’t quite the whole story. Water is very heavy. Anyone who has ever lifted a large, full bucket of water will know this (and to those of you who haven’t, the activity comes highly recommended). All matter (“stuff” in layman’s terms) is attracted to all other matter by gravity, but the force is immeasurably small unless the mass of the stuff happens to be very heavy. The Earth is very heavy (it is; have you ever tried lifting it?) and the gravity associated with its mass is what keeps our feet on the ground. All this ice that we keep talking about is also quite heavy. If all of the ice in Antarctica were to melt, that loss of weight would be significant enough to affect the distribution of gravity around the world, enough so that the sea level around Antarctica would actually go DOWN (the mass of all the ice in Antarctica is significant enough to pull the water in the ocean towards it). Elsewhere, where glaciers are melting, the resulting loss of mass on top of the land ends up making the land rise slightly (like a mattress does after you’ve rolled out of bed).

So if the sea level, at least relative to the land, is going DOWN near the poles where all the ice is melting, then where does all this extra water go? The answer is: near the equator. Sadly, it is in tropical areas, where most of the low-lying island nations lie, where sea levels will rise the most.

Sadly, these will be the first to go

So rising temperatures will melt the ice and this will raise sea levels. Moreso near the equator than near the poles. I wish that was all there was to it, but there’s more. Anyone who’s thought about how a thermometer works knows that when a liquid is heated, it will expand. The water in the Earth’s oceans is no exception to this rule, and thermal expansion is also contributing to sea level rise.

But wait, there’s more. Remember all that talk about CO2? Well, we’re very lucky that we have so much ocean, because water, like all liquids, can absorb gas. It’s not unlike dissolving sugar into a cup of tea actually. In fact, for a while it was not known why the atmospheric concentration of CO2 was not as high as our output of CO2 would suggest that it should have been. We eventually figured out that the oceans were absorbing a lot of the stuff. Of course, anyone who has tried to add an infinite amount of sugar to a cup of tea knows that you can’t do it forever (believe me, I’ve tried). (Edit: I have assumed that the cup of tea is finite, which I believe to be a reasonable assumption. If you happen to have one of those rare cases of an infinite cup of tea, then it is possible to add an infinite amount of sugar to it. The oceans are likewise assumed to be finite) The same is true of the oceans. As if that wasn’t enough to make you depressed about the world, consider the fact that warmer liquids cannot hold as much gas in solution as colder ones.

So… we release CO2, some of it goes into the ocean and some of it warms the Earth a little bit. As the Earth warms, the oceans lose a bit of their ability to hold CO2, so more CO2 gets released, causing the world to warm a bit more. This is called “positive feedback”, where all the feedback mechanisms reinforce each other. The same sort of thing happens when you put a microphone too close to the speakers… often with similar results – the whole system crashes – something’s gotta give.

The trouble with feedback, is that it sometimes takes a long time for the feedback to make itself known. The amount of carbon in the oceans is affected by the solubility pump, the biological pump, and the carbonate pump. Solubility we’ve already discussed, the biological pump is simply underwater plants photosynthesizing in the same way that land-based plants do. The carbonate pump is shellfish turning carbon into shells. In a cruel twist of fate, many of these shellfish now have much thicker shells as a result of the increased availability of carbon and, as a result, sometimes die because they get trapped inside, or because the shells are too heavy.

A curious side-effect of the carbon chemistry in the oceans is that the oceans are becoming more acidic. This has many bad effects. Coral reefs around the world are already showing signs of literally dissolving away because of the increased acidity. Without these coral reefs, many fish species will die out because they loose their breeding grounds, placing a huge strain on fisheries that are already on the verge of collapse. The increased acidity also speeds up certain types of erosion. The erosion of carbonate rocks may also liberate additional CO2.

So we’re in a bit of a situation. We’ve got CO2 going into the atmosphere which is pushing the thermal equilibrium of the planet toward the warm end of the scale. Moreover, while some of it is being absorbed into the oceans, the warmer temperatures will eventually cause even MORE CO2 to be released from the oceans. Meanwhile, the chemical consequences of the carbon in the oceans is possibly causing more potentially catastrophic cascades of bad stuff to happen – the increased acidity of the oceans, the destruction of coral reefs, the mass extinction of many species of fish.

A Coelocanth

In Part 4: Population dynamics, Nash equilibrium, and the tragedy of the commons.

In part 1, it was established that warming was taking place, and that the consequences of warming were not good for humans. I wanted to get those points out of the way first because they are the easiest, and are widely accepted as fact, even by most so-called climate change skeptics. The response so far has been fairly quiet, with no real objections raised, but merely more questionable material from the skeptic’s camp being brought to my attention.

There are generally two main things which are wrong with arguments coming from skeptics and they are as follows: bad argument structure, and bad science. Argument structure is really something that we should all be able to spot, although sometimes this is difficult, especially if the idiot skeptic is a good writer. Bad science is even more difficult for the layperson to spot because it is difficult for a person without formal training in the field of climate science to tell, becuase they lack the requisite background knowledge. In this part, I will attempt to give some of this background knowledge in the form of explanation of CO2 and the greenhouse effect, which will hopefully provide the link between human activity (CO2 emissions) and the warming of the Earth.

WARNING: The following contains mathematical equations. Don’t be discouraged, if you are, for whatever reason, afraid of mathematical equations (and who isn’t a little bit afraid), then feel free to skip over them. It should not significantly detract from the learning experience.

First, an obvious statement – we get all our energy from the Sun. There are actually minor exceptions, like nuclear power and geothermal power, but the vast majority of all the energy we deal with came from the Sun at some point. The effective temperature of the sun is just short of 6000 Kelvin. The radiation from the sun spreads out according to the inverse-square law and a tiny fraction of it hits the Earth. The Earth reflects some of this radiation, and the radiation that is absorbed is eventually re-emitted. This happens because the energy coming in must be the same as the energy going out (otherwise the Earth would get hotter and hotter, or cooler and cooler).

Planck’s law tells us about the distribution of the wavelength emitted by a blackbody. This is important because it tells us how much of each wavelength is emitted by the Earth. Certain gases in the atmosphere absorb (and subsequently re-emit) radiation at different wavelengths. First, let’s look at the emissive curves of the Earth and the Sun.

(note the logarithmic scales and poor resolution)

The crucial thing to take away from this, is that the peak emissions wavelength for the Earth is mostly in the infrared while the Sun’s is just past the visual and towards the ultraviolet. This translates to the fact that most of the energy coming in to the Earth is coming in at a different wavelength to that going out.

So we need to calculate the energy flow coming from the surface of the Earth. Turns out that there is this law called the Stefan-Boltzmann law which states that this radiative flux is directly proportional to the fourth power of the temperature. The equation is very simple, it looks like this: flux. ( is the Stefan-Boltzmann constant, and is about ).

So… it seems we have enough to figure out what average temperature the Earth ought to be. Turns out, with this information alone, we get a figure of about 254 Kelvin, or -18 Celsius, which clearly isn’t true. (If you’re interested, the equation is where S is the solar flux – 1360 , and A is the albedo (the reflective index of the Earth) which is about 0.3, I shall leave the derivation of this to the interested reader). Some of you may have already guessed the omission from our calculations is the atmosphere.

What the atmosphere basically does, for the sake of our calculation, is absorb outgoing radiation, then re-radiate it. How can this possibly make a difference? Well, the outgoing radiation is coming from one direction while the re-radiated energy goes in all directions, some of it coming back to the Earth. Why doesn’t incoming radiation also get absorbed? You may ask. This is because the wavelengths at which incoming radiation comes in are different from those of the outgoing radiation. Most gases have what’s called an absorption profile which is a fancy way of saying that it will absorb energy at certain wavelengths while at other wavelengths, it will be effectively invisible. I have illustrated what is happening in the diagram below.

The temperature is simply: . To find we use and , and a simple substitution yields a temperature at the surface of 302.7 Kelvin which is about 30 degrees Celsius.

There are two major greenhouse gases in the atmosphere, and they are CO2 and H2O. H2O, due to the water cycle is self-regulating and the amount of it at any one time in the atmosphere doesn’t change a great deal. CO2, on the other hand, stays in the atmosphere for years and the processes for taking it out and putting it into the atmosphere are very slow, that is until humans came along.

no, that is not a typo, we are PAST the safe level

So, pre-industrial CO2 levels were about 270 parts per million (ppm), and at the moment, they are at about 385 ppm. FYI scientists believe that the maximum “safe” level is about 350 ppm (no, that is not a typo, we are PAST the safe level). For simplicity’s sake, let’s double the amount of greenhouse gases in the atmosphere, and see what happens. In terms of our diagram, it is equivalent to adding another “slab” of atmosphere.

Intuitively, it looks like it will get warmer, but just how much warmer?

Our first equation – the one whose solution we are interested in, would look like this: . But to find the value of , we will need and . These equations might look very scary, but the substitutions are, in fact, very simple (and to think I got extra credit on an assignment for doing this). comes to about 335 Kelvin, or 62 degrees Celsius, which is alarmingly high.

Obviously, this isn’t true, because, as I mentioned before, CO2 is one of many greenhouse gases and is the second most important one behind H2O (because there’s a bit more H2O in the atmosphere). But the point is made that adding more CO2 is going to shift the temperature equilibrium of the planet towards a warmer state.

The trouble with a lot of the critics of the theory of anthropogenic climate change, is that they get so caught up in the politics and the back-and-forth that inevitably goes on in research (which most of these pundits hardly understand) that they miss some of the very basic science that underpins these theories. Obviously, this is not the whole story, but this is the most important part of it. This is the distilled essence of the answer to the question of “how is CO2 linked to warmer temperatures?”. So the next time someone tells you that they heard somewhere that there is no link between CO2 and climate change, you can tell them that they’re full of shit, and point them in the direction of this page.

In part 3: Oceans, feedback, and other reasons that increased CO2 is bad

(note: Paul Krugman said more-or-less the same thing as I am about to do so in his NY Times Column this morning)

The trouble with the Earth, is that it is our home. Because it is our home, it is exceedingly difficult to treat any policy decisions to do with the earth in a completely impartial way. We are inescapably drawn into emotional debates about this and that, and the raw data – the truth, if you would go so far as to call it that, is prone to being manipulated and misrepresented for all manner of reasons. Also, because this pale blue dot is the only home we’ve ever known, the issue is important and affects us all, making a detached unemotional analysis impossible.

The debate itself also has an interesting history. All manner of people have championed all manner of causes under the banner of “saving the planet”, some of these causes being very different from each other. Recently the urgency of the matter has reached new heights with the introduction of “man made climate change” as a concept, and, as is typical when a very serious and important issue presents itself before humankind, the divisive stakes of hate have been raised, obscuring the facts, and ultimately diluting and delaying the process.

The distinguishing feature of the various different groups which interact in this debate is their motives and their means. Most have seemingly noble motives, and by seemingly I am implying that it is seen as acceptable to pretend to have altruistic motives in order to get their way. At the end of the day, public opinion matter a lot, and the public’s perception of one’s motives can count for a great deal when the common people are deciding on who to “align” themselves with. I don’t use the word “align” lightly – the debate has taken on almost religious proportions with the accompanying hypocrisy flying thick and fast from both sides.

Take for example the big oil companies. I would not say that any individual working for an oil company is evil per se (although, having met a few, I sometimes question this belief). But they don’t want to loose their jobs, the executives don’t want to loose their bonuses, and suppliers don’t want to loose their political influence. However, collectively these forces sum to acts which truly are evil – the deaths of innocent people and irreparable damage to ecosystems. But even they have to bow to public opinion, and are spending hundreds of millions of dollars on advertising campaigns to make common folk believe that they care about the environment. Don’t be fooled, actions speak louder than words, and the sum of their actions speaks very clearly – they care only for profit.

Closer to home, Federal Senator Steve Fielding of Australia is the lynchpin in many important government decisions including ones to do with the climate. He is thoroughly unqualified and misinformed, his arguments and position papers betraying a complete non-understanding of the issues he is affecting legislation on. I believe his motives are noble, and that is admirable, but owing to a complete incapacity to think he consistently makes very bad decisions. His mistakes are the Senate’s which are, in turn, Australia’s. He has no business being in government.

The real problem with the Earth Debate (I’m deliberately being broad here, I will be more specific shortly) is that, since the problem concerns everybody, everybody should have a say. The trouble is that the problem is highly complex, requiring the understanding of many different fields of the natural sciences. The terms of the debate, however, are largely determined by the news media who have made something of a dog’s breakfast of explaining everything properly. Politicians, large corporations, environmental groups, and others have scrambled to selectively display information, sometimes unknowingly but often with deliberate intent, to mislead and misinform the general public about what could be one of the most important challenges of our time.

So where do I stand in all of this? I believe that we are grossly mistreating our planet. I believe in man-made climate change. I believe that we are slowly poisoning the biosphere with chemicals. I would identify with being fairly liberal in most political thought, and in economics some would probably call me neo-keynesian. First, and foremost though, I am a mathematician. Numbers will make me turn on a dime in terms of my beliefs, and have done so on a few occasions. However, this also makes it very difficult to fool me with dodgy numbers, and I will be one of the first to make my objections known if anyone tries it (and it happens a lot).

Let’s begin with the biggest catch-phrase of the so-called skeptic’s camp. That the “climate is always changing”. Obviously, this is true, but it is not the whole truth. The climate is always changing, and has done so for all of the earth’s history. In the past it has been much hotter, and there has even been much more CO2 in the atmosphere. The crucial difference in very recent history is that it is exactly that – very recent history. The speed at which the temperature is changing is many orders of magnitude faster than what the earth is used to.

So what? You may ask. Well, let’s think about what a rise in global temperature really means. If it was a degree hotter tomorrow than it was today, you would  barely notice. Let’s think about this a different way, if you woke up tomorrow and all the farmland, desert, and forests in the world had shifted their boundaries several kilometers, would you notice? The climate is a very sensitive and finely-balanced system, indeed one of the first large-scale statistical analyses of climate was an attempt to understand the Indian monsoon. Even a small temperature shift can disrupt the climate system in a very chaotic and unpredictable way.

The Earth's climate zones would shift drastically with even a small change in global temperature.

So what? You may continue to ask. Put simply, when the climate changes slowly and gradually over hundreds of thousands of years, ecosystems have time to adapt, to change, to shift, and so on. But when change happens this quickly, these very delicately balanced systems can no longer keep up and we have mass extinctions.

There may still be some who, at this point, continue with “so what?”. As far as the Earth is concerned, that may be a valid thing to say. We are in no danger of destroying life as we know it, and perhaps the slogan of “save the planet” is somewhat misleading because the planet is not actually about to be destroyed. However, on the point of mass extinctions, it would be worth remembering that only a handful of species of plants and animals are used to feed us. Humans may not be one of the species to suffer extinction, but the extinction of any one of these species of crops or animals would drastically reduce the Earth’s ability to support humans.

Fair enough, some may say. But there are those who doubt that the Earth is warming at all. Aside from such extreme examples as “rapid warming of the earth’s climate is GOOD for us”, this is one of the more unbelievable claims of some climate skeptics. Indeed, it should be noted that many climate change skeptics believe the Earth to be warming but merely contend the point of human activity being the cause. The Earth has been slowly warming since the last glaciation and that warming has been especially rapid in the last thirty years. The evidence of this warming is well documented in temperature readings of the atmosphere and the oceans, as well as in ecology, in the shift of habitat in movement patterns for climate-sensitive life forms.

Patterns of overall warming aren’t always entirely obvious, and some may quip that last winter was unusually cold. This is merely a variation in a very squiggly line, which happens to have a very definite overall upward trend. Perhaps last winter was cold in the US, but in Australia (and at the same time, I might add) many regions experienced the hottest summers on record. Indeed, on the point of whether or not warming is occurring, the body of evidence is so especially compelling, I don’t feel that I would be able to convince you if you have already taken a view to the contrary in spite of all this evidence, and you can go back to watching Fox News and listening to Rush Limbaugh now.

In part 2: The science behind CO2 and the greenhouse effect