Ultimate Limits to Growth

My latest essay in Bloomberg touched on ultimate limits to energy growth (and quite possibly economic growth) due to the accumulation of waste energy in the environment. It's really just an exercise in taking basic physics into account while extrapolating trends in energy use into the future. The conclusion is that continued exponential growth in energy use -- which we've experienced over the past few centuries (and possibly much longer) -- cannot last for much more than a century or so. What about economic growth? We don't know. Economists theorize about a great "decoupling" of energy from economic productivity, but that hasn't happened so far in any country. My conclusion is that economic growth must also end, fairly soon (by which I mean, say, 100 years) -- unless we transform our economic activity to involve far less energy and in a way we have never done before.

I noticed that Noah Smith has a post criticizing physicist Tom Murphy, who I cited in my article. I love Noah's blog and read everything he writes, but I don't think this is his fairest criticism, although parts are fair. Certainly, I can see that economists might feel that their best arguments weren't put forward in the dialogue Murphy recalled between himself and an economist, the subject of Murphy's most widely read post. (Note: I didn't cite that particular post Noah refers to, but this one of Murphy's which looks at trends in energy growth alone.) But reading Noah, I'm led to believe that my understanding of the prevailing view among modern economists on economic growth is hugely mistaken. Indeed, he makes it sound as if economists generally accept that growth must end, and fairly soon (if true, I' very happy about that).

Murphy made the point that, if we extrapolate our current and past energy growth into the future, then we will actually boil the oceans in 400 years (with 2.3% energy growth; sooner with fast growth). To this Noah responds,

This is correct. And in fact, Murphy didn't even need to mention waste heat or anything like that to make his argument; he could have just said "Hey, eventually the Sun will explode, and then the whole Universe will degrade into heat, and where will your economy be then?" So what if that happens 500 million years in the future, or 10^100 years? What's the difference? One way or another, the human race is kaput!

Yes, of course. But the point is that 400 years is not very long. And we don't need the oceans boiling before we would see important temperature change (and other associated environmental changes) that would make life rather uncomfortable. I think Murphy is right that most people do not appreciate how soon in the future (soon on a timescale of human history) continued growth of energy use brings problems. This isn't a problem set some 5 billion years in the future. This is one reason I think so many people have found Murphy's posts worth reading: this seems really surprising to them.

Noah goes on:

Are economists ignoring this basic fact? Do economists' models crucially hinge on the idea that economic growth will continue forever and ever and ever? No. The "long term trend growth" that is used in growth and business cycle models is only meant to represent a trend that lasts longer than the business cycle - so, longer than a decade or two. No economist - I hope - thinks that currently living humans are making economic decisions based on what they think is going to happen in 400 years, or 2500 years, or 500 million years.

Again, I think this is just the point Murphy is trying to make -- that if these effects are looming only 400 years (or significantly less) in the future, then perhaps contemporary humans ought to be taking them into account now in making their economic decisions. Certainly it would be appropriate for our leaders to be casting an eye on this long term, and to seek advice from our best economists who might help them think clearly about how our society could manage to change in response. Does economics end with the business cycle? Nothing longer term than that? 400 years is perhaps only 20-30 business cycles away.

These are the main points I think Murphy was trying to make. I do agree with Noah that other parts of Murphy's original post are much less convincing. When he moves into proper economic territory, discussing prices, scarcity of future energy, etc., my own feeling was to take that all with a grain of salt, as quite a lot of speculation.

In any event, I'm glad Noah has brought the attention of more economists to the quite short timescale on which continued energy growth leads to problems. If this is already well understood in economics, and built into theories of growth, then great, I've learned something. In my experience a lot of people think we'll be fine and continue growth if we can only find some cheap, infinite and non-polluting energy source to power our future. That's not the case.  

Why expected value is a mistake

I want to take a closer look at the very interesting work of Ole Peters I mentioned in my last post. He argues that the ensemble averages typically used in economics and finance to compute "expected" returns are, in many cases, inappropriate to making decisions in the real world; in particular, they severely underestimate risks. Peters begins with a simple gamble:

Let’s say I offer you the following gamble: You roll a dice, and if you throw a six, I will give you one hundred times your total wealth. Anything else, and you have to give me all that you own, including your retirement savings and your favorite pair of socks. I should point out that I am fantastically rich, and you needn’t worry about my ability to pay up, even in these challenging times. Should you do it? ... The rational answer seems to be “yes”—the expected return on your investment is 1,583 1/3% in the time it takes to throw a dice. But what’s your gut feeling?

As he notes, almost no real person would take this bet. You have 5 chances out of 6 of being left destitute, one of being made very much wealthier. Somehow, most of us weight outcomes differently than the simple and supposedly "rational" perspective of maximizing expected return. Why is this? Are we making an error? Or is there some wisdom in this?

Peters' gamble is a variation on the famous St Petersburg "paradox" proposed originally by Nicolas Bernoulli, and later discussed by his brother Daniel. There the question is to determine how much a rational individual should be willing to pay to play a lottery based on a coin flip. In the lottery, if the first flip is heads, you win $1. If the first is tails, you flip again. If the coin now comes up heads, you win $2, otherwise you flip again, and so on. The lottery pays out 2^n (^ meaning exponent) dollars if the head comes up on the nth roll. An easy calculation shows that the expected payout of the lottery is infinite -- given by a sum that does not converge (1*1/2 + 2*(1/2)^2 + 4*(1/2)^3 + ...) = (1/2 + 1/2 + 1/2 + ...). The "paradox" is again why real people do not find this lottery infinitely appealing and generally offer less than $10 or so to play.

This is a paradox, of course, only if you have some reason to think that people should act according to the precepts of maximizing expected return. Are there any such reasons? I don't know enough history of economics and decision theory to say if there are -- perhaps it can be shown that such behavior is rational in some specific sense, i.e. in accordance to some set of axioms? But if so, what the paradox really seems to establish, then, is the limited relevance of such rules to living in a real world (that such rules capture an ineffective version of rationality). Peters' resolution of the paradox shows why (at least for my money!).

His basic idea is that we live in time, and act in time, and have absolutely no choice in the matter. Hence, the most natural way to consider the likely payoff coming from any gamble is to imagine playing the gamble many times in a row (rather than many times simultaneously, as in the ensemble average). Do this indefinitely and you should encounter all the possible outcomes, both good and bad. Mathematically, this way of thinking leads Peters to consider the time average of the growth rate (log return) of the wealth of a player who begins with wealth W and plays the gamble over N periods, in the limit as N goes to infinity. In his paper he goes through a simple calculation and finds the formula for this growth rate:

               
The third line here is explicitly for the St Petersburg lottery, while the second line holds more generally for any gamble with probability p_i of giving a return r_i (with the sum extending over all possible outcomes).

This immediately gives more sensible guidance on the St Petersburg paradox, as this expected growth rate is positive for cost c sufficiently low, and negative when c becomes too high. Most importantly, how much you ought to be willing to pay depends on your initial wealth w, as this determines how much you can afford to lose before going broke. Notice that this aspect doesn't figure in the ensemble average in any way. It's an initial condition that actually makes the gamble different for players of different wealth. Coincidentally, this result is identical to a solution to the paradox originally proposed by Daniel Bernoulli, who simply postulated a logarithmic utility and supposed that people try to maximize utility, not raw wealth. This idea reflects the fact that further riches tend to matter relatively less to people with more money. In contrast, Peters result emerges without any such arbitrary utility assumptions (plausible though they may be). It is simply the realistic expected growth rate for a person playing this game many times, starting with wealth W. Putting numbers in shows that the payoff becomes positive for a millionaire for a cost c less than around $10. Someone with only $1000 shouldn't be willing to pay more than about $6.

It's also useful to go back and work things out for the simpler dice game. One thing to note about the formula is that the average growth rate is NEGATIVE INFINITE for any gamble in which a person stands to lose their entire wealth in one go, no matter how unlikely the outcome. This is true of the dice gamble as laid out before. I was wondering whether this really made any sense, but after some further exploration I now think it does. The secret is to again consider that the person playing has wealth W and that the cost of "losing" isn't the entire wealth, but some cost c. A simple calculation then shows that the time average growth rate for the dice game takes the form shown in the figure below, showing the growth rate versus the c/w, the cost as a fraction of the players' wealth.

Here you see that the payoff is positive, and the gamble worth taking, if the cost is less than about 60% of the player's wealth. If more than that, the time average growth rate is negative. And, if becomes strongly more negative as c/w approaches 1, with the original game recovered for c/w=1. Again, everything makes more sense when a person's initial wealth is taken into account. This initial condition really matters and the question of the likely payoff of a gamble depends strongly on it, as lower wealth means higher chance of going bankrupt quicker and then being out of the game entirely. The possibility of losing all your wealth on one turn, no matter how unlikely, becomes decisive because this becomes certain in the long run.

Again, this way of thinking likely has significance far beyond this paradox. It's really pointing out that ensemble averages are very misleading as guides to decision making, especially when the quantities in question, potential gains and losses, become larger. If they remain small compared to the overall wealth of a person (or a portfolio), then the ensemble and time averages turn out to be the same, giving a formula in which initial wealth doesn't matter. But when potential gains/losses become large, then the initial condition really does matter and the ensemble average is dangerous. These points are made very well in this Towers Watson article I mentioned in an earlier post.

Which brings me to one final point. Ivan in comments suggested that perhaps Peters has changed the initial problem by looking at the time average rather than the ensemble average, and so has not actually resolved the St Petersburg paradox. I'm not yet entirely sure what I think about this. The paradox, if I'm right, is why people don't act in accordance with the precepts of expected return calculated using the ensemble average. To my mind, Peters' perspective resolves this entirely as it shows that this ensemble average simply gives very poor advice on many occasions. In particular, it makes it seem that a person's initial wealth should have no bearing on the question. If you face gambles, and face them repeatedly as we all do throughout life in one form or another, then thinking of facing them sequentially, as we do, makes sense. But that's not, as I say, my final view..... this is one of those things that gets deeper and deeper the more you mull it over....

Ergodicity -- the Biggest Mistake in Economics?

I'm increasingly convinced that Ole Peters has identified the nub of an utterly essential problem in the framework of contemporary (i.e. last 50 years) economics. In a series of recent papers (here, here, here), he has argued with impressive clarity that the usual ensemble averages used to compute "expected" returns in finance are, in many cases, simply inappropriate to making decisions in the real world. Take a risky gamble, and the usual average over different outcomes mixes potential worlds in which we go broke with others in we get rich, and, importantly, takes the often irreversible consequences of these outcomes (bankruptcy, for example) out of the picture. If you make hugely risky investments, this average gives you full credit for all the wonderful possible outcomes, weighted appropriately for their likelihood, which of course seems sensible. What it doesn't do is account for the very real fact that the bad outcomes may effectively wipe you out entirely and take you out of the game, making it impossible to play again -- in which case you will never get to experience those eventual big payoffs.

Maybe the best thing to read about this is this wonderful paper by people from the financial firm Towers Watson (credit: I learned of this from Rick Bookstaber's blog). The potential implications of this are really huge, as Peters' perspective suggests that the standard way of assessing risk versus reward in financial economics is wrong and systematically underestimates risks (and not merely because it ignores fat tails). The paper above, the first paper of Peters I mentioned above, and this interview with Peters are among the most interesting things I've read this year.

I'm going to do an in depth post on this stuff soon, but I must admit that I need to study it in detail a little more. I'm convinced that Peters insight -- which brilliantly resolves the centuries old "St Petersburg paradox" of probability theory proposed originally by Bernoulli -- also has a lot to do with the work of Doyne Farmer and John Geanakoplos on economic discounting, which I've written about before. Both suggest that our basic thinking about probability in time series suffers from some terrible misconceptions, and generally makes us underestimate risks. More coming on this soon.

Why diversification doesn't work

You're standing in your canoe, on a beautiful Canadian lake, taking photos of the wildlife, occasionally fishing. Why standing, not sitting? Well, you've read about those disturbing studies that show how sitting is really bad for your long term health; how every hour of television viewing, for example, takes about 20 minutes off your life expectancy, and why the same is probably true for sitting at the computer, sitting reading a book, whatever. So you're standing and that's OK because you're balanced and stable, with your weight distributed uniformly.

Of course, anyone with even a few minutes of experience in a canoe knows this isn't as safe as it seems. What really matters isn't how well-balanced you are when the canoe rests peacefully, but what happens when a few waves come along, kicked up by rednecks passing in a souped-up bass trawler (I lived in rural Virginia for several years, so I know the experience). As you shift your stance to stay upright, and the boat shifts, that balanced distribution vanishes and you can easily tip. Stability demands balance in the midst of the boat's dynamics, not only in the static peace beforehand.

As it turns out, this same lesson applies to investment portfolios -- a new paper in Nature Scientific Reports shows just how important this insight may be.

Famously, of course, Harry Markowitz introduced the idea of diversification into investing back in the 1950s (at least he formalized the idea, which was probably around long before). Using information on the mathematical correlations between the returns of the different stocks in a portfolio, you can choose a weighted portfolio to minimize the overall portfolio of volatility for any expected return. This is maybe the most basic of all results in mathematical finance.

But it doesn't work; it suffers from the same problem as the balanced man in the canoe. This is clear from any number of studies over the past decade which show that the correlations between stocks change when markets move up or down. If the market suddenly plunges downward, you would hope that your well-diversified portfolio, invested as it is in stocks that tend to move unlike one another, would be OK. But when markets move significantly down (or up), it turns out, the correlations are no longer what they were. Trending markets induce strong correlations among stocks that aren't there beforehand, and aren't obvious from long-term averages. So the risks to a portfolio are actually much larger than the simple diversification analysis suggests -- just as the risk of a canoe tipping is much more than it seems to a man standing balanced on a peaceful lake.

The new paper by physicist Tobias Preis and colleagues makes this point with probably the largest data set used so far, looking at the stocks in the DJIA over about 70 years. It's a fairly simple analysis (modulo some nitty gritty details). Roughly, they look at the correlations between different stocks in the DJIA and see how these correlations depend on the recent average return of the DJIA. Are the correlations stable? Or do they go up as the market begins to move? The figure below showing the average correlation coefficient versus the return indicates that the result is clearly the latter: a trending market, in either direction, induces significant correlations among the DJIA stocks.

One of the interesting things here is that this link holds on many different timescales, from 10 days up through two months. The worrying thing for an investor, of course, is that these correlations make the risks of large losses significantly larger than they would appear to be on the basis of long-term correlations alone. As the authors conclude:

... a “diversification breakdown” tends to occur when stable correlations are most needed for portfolio protection. Our findings, which are qualitatively consistent with earlier findings^{42, 44} but quantitatively different, could be used to anticipate changes in mean correlation of portfolios when financial markets are suffering significant losses. This would enable a more accurate assessment of the risk of losses.

 As any canoeist knows, dynamics really matter.

The future of economics?

Ali Wyne at the blog "big think" asked eight notable young (under 40) economists about the future of their profession and key topics for future research. Their responses make for interesting but not really surprising reading. The research frontier, in their eyes, faces its key challenges in 1) understanding the nature of economic development and growth, and how the world's poor can be brought out of poverty, 2) learning how our growing understanding of human behavioral psychology can be used to replace the inadequate framework of rationality in economics, 3) gaining a much better perspective on macroeconomics, including bubble and herding phenomena, 4) building a new theoretical perspective to handle the vast influence of new information technology on human economic decisions and 5) learning to deal with massive data.

All in all, these seem like worthwhile goals and I'm encouraged that at least two of the economists make semi-explicit their view that economics dearly needs to explore new kinds of models going beyond the equilibrium framework.

I'm also struck, however, by something a little more depressing, which is the rather narrow, conservative scope expressed in their comments. Perhaps this is to be expected from young economists hoping to find stable jobs for coming decades, but not one of them even mentions the need for a deeper understanding of the nature and long term consequences of economic growth. Such growth is -- still -- simply assumed to be an absolute good to be pursued always and as rapidly as it can be. Given the alarming picture painted by studies such as this one -- in Nature a few months ago, it reviewed how human economic growth has significantly altered virtually all global biological and geophysical processes -- you might think that young economists would be scrambling to develop ideas about human society in a post-growth world, or at least one in which growth has to be strongly constrained and managed.

That seems to be a step too far. The idea of growth forever, unconstrained by any physical laws or biological realities, still seems to be a core belief even of the next generation of economists.

Stability through simplicity

I gave a talk last week at Oppenheimer Funds in NYC. I met some great people there, really creative and open minded. I spoke on the general theme of this blog -- natural instabilities in finance and economics and ideas we need to understand them.

One question asked afterwards was "what can we do as far as regulations to cope with these kinds of instabilities?" As I recall, my answer was pretty lame. I tried to suggest (vaguely) that the answer probably lies not with highly complex regulations, but with simpler ones, but I didn't say much more. I've been thinking about that point since, and thought it might be worth writing a few things down.

Essentially, the first lesson I think we should draw, once we acknowledge the existence of pervasive instabilities in finance, is the need to deal with persisting uncertainty. We will never understand the terrain so well that we can reduce the future to a set of known possibilities to which we can assign specific probabilities (which standard economics assumes we can). We can and should work hard to explore the space of what might happen, and so gain some forewarning of dangers, but we will still encounter surprises and we should expect to do so. So our approach to regulation ought to be centered on that premise -- that we face a world of uncertainty.

I should have read Andrew Haldane's wonderful essay The Dog and the Frisbee before giving my talk, but I only got around to that this morning. He makes some hugely important points on this very topic. The essay is one extended argument for why financial regulation is now too complex, and why our best hope at achieving financial stability in the future probably lies in a vast simplification of the regulatory apparatus and system of rules.

He begins from the observation that, in many settings where decision making involves weighing up many conflicting factors, simple rules often out-perform more complex ones:

Among physicians diagnosing heart attacks, simple decision trees beat a complex model. Among detectives locating serial criminals, simple locational rules trump complex psychological profiling. Among investors picking stocks, simple passive strategies outperform complex active ones. And among shopkeepers understanding spending patterns, repeat purchase data out-predict complex models.

The general message here is that the more complex the environment, the greater the perils of complex control. The optimal response to a complex environment is often not a fully state-contingent rule. Rather, it is to simplify and streamline (Gigerenzer (2010)). In complex environments, decision rules based on one, or a few, good reasons can trump sophisticated alternatives. Less may be more.

In complex environments, tallying strategies have been found to be superior to risk-weighted alternatives. Take avalanche prediction. Avalanches are difficult to predict, as they are drawn from a fat-tailed (Power Law) distribution. Yet simple tallying of a small number of avalanche indicators has been found capable of predicting over 90% of historical accidents. It has also been found to be superior to more complex decision methods (McCammon and Hägeli (2007)).

Haldane goes on at length to consider this in the context of financial regulation, where the legal framework has really exploded in complexity over the past few decades. The pursuit of ever-more complex models to assess risks, whether used by regulators or by banks and financial institutions independently, has led to a proliferation of models that create an overwhelming fog of complexity and can blind us all to obvious risks:

During the 1990s, the bluntness of the risk judgements embodied in Basel I came increasingly to be questioned – and arbitraged. Basel I was perceived as lacking risk-sensitivity, at least by comparison with the new wave of credit and market risk models emerging at the time. Change came in 1996 with the Market Risk Amendment. This introduced the concept of the regulatory trading book and, for the first time, allowed banks to use internal models to calculate regulatory capital against market risk. ...With hindsight, a regulatory rubicon had been crossed. This was not so much the use of risk models as the blurring of the distinction between commercial and regulatory risk judgements. The acceptance of banks’ own models meant the baton had been passed. The regulatory backstop had been lifted, replaced by a complex, commercial judgement. The Basel regime became, if not self-regulating, then self-calibrating.

The ink was barely dry on Basel II when the financial crisis struck. This exposed gaping holes in the agreement. In the period since, the response has been to fill the largest of these gaps, with large upwards revisions to the calibration of the Basel framework. Agreement on this revised framework, Basel III, was reached in 2010. In line with historical trends the documents making up Basel III added up to 616 pages, almost double Basel II. ... The length of the Basel rulebook, if anything, understates its complexity. The move to internal models, and from broad asset classes to individual loan exposures, has resulted in a ballooning in the number of estimated risk weights. For a large, complex bank, this has meant a rise in the number of calculations required from single figures a generation ago to several million today (Haldane (2011)).

Taking all of this together, the parameter space of a large bank’s banking and trading books could easily run to several millions. These parameters are typically estimated from limited past samples. For example, a typical credit risk model might comprise 20-30 years of sample data – barely a crisis cycle. A market risk model might comprise less than five years of data – far less than a crisis cycle.

Viewed over an historical sweep, this pattern is even more striking. Contrast the legislative responses in the US to the two largest financial crises of the past century – the Great Depression and the Great Recession. The single most important legislative response to the Great Depression was the Glass-Steagall Act of 1933. Indeed, this may have been the single most influential piece of financial legislation of the 20th century. Yet it ran to a mere 37 pages. The legislative response to this time’s crisis, culminating in the Dodd-Frank Act of 2010, could not have been more different. On its own, the Act runs to 848 pages – more than 20 Glass-Steagalls. That is just the starting point. For implementation, Dodd-Frank requires an additional almost 400 pieces of detailed rule-making by a variety of US regulatory agencies.

Haldane ends the essay with some exploration of how we might reverse this trend, and manage to simplify regulations. I won't go into detail other than to second his suggestion that reducing the complexity of the financial system itself ought to be a principle target of such simplified regulations. But even before that, changing the mindset of financial economics is the first task. That mindset still remains fixated on the endless pursuit of optimal strategies by long calculations over risk weighted alternatives, when in reality we rarely know the risks or even the alternatives with much accuracy.

An important consequence of thinking in this world of known risks and optimal solutions is that we end up accepting some dubious arguments that we can achieve the best of all possible worlds with the right pricing mechanism. With bad assumptions, in other words, we fall into the trap of believing the economists' standard models, when they really have little to do with the real world. If we break free of this illusion, and face up to living in a world with real uncertainty, we may return to an era in which old-style regulations and prohibitions against certain activities make perfect sense, and dreams of perfect pricing mechanisms become evident as the fantasies they are:

Over the past 30 years or so, the regulatory direction of travel has been towards pricing risk in the financial system, rather than prohibiting or restricting it. In the language of Weitzman, regulators have pursued price over quantity-based regulation (Weitzman (1974)). That makes sense when optimising in a risky world.

It may make less sense when optimising in an uncertain world. Quantity-based restrictions may be more robust to mis-calibration. Simple, quantity-based restrictions are the equivalent of a regulatory commandment: “Thou shalt not”. These are likely to be less fallible than: “Thou shalt provided the internal model is correct”. That is one reason why Glass-Steagall lasted for 60 years longer than Basel II. Quantity-based regulatory solutions have gained currency during the course of the crisis. In the US, the Volcker rule is a quantity-based regulatory commandment: “Thou shalt not engage in proprietary trading”. In the UK, the Independent (“Vickers”) Commission on Banking has also proposed structural, quantity-based reforms: “Thou shalt not co-mingle retail deposit-taking and investment banking”.

Yet even these notionally simple, structural proposals run some risk of backdoor complexity. For example, the consultation document accompanying Volcker already runs to 298 pages. Were these proposals to become mired in detail, they risk sinking, like the Tower of Basel, into the swamp. This is not because these proposals go too far but because they may not go far enough. These reform efforts have too many commas, semi-colons and sub-clauses. They would benefit from a few more full stops.

wisdom of crowds

My next Bloomberg column comes out this weekend is now out here. I wanted to give readers some further detail on the experiments I wrote about in the column, experiments designed to test how social influence affects the Wisdom of Crowds phenomenon. I actually wrote about the experiments in this post last year, and that post gives quite a lot of detail.

I think it is the most illuminating set of experiments I have seen on this phenomenon. Most important in the current environment, it's pretty clear I think that one can't look to the wisdom of crowds as a mechanism to enforce any kind of "wisdom" on the part of the financial markets. (This doesn't mean they're always wrong either, of course.)