My student Miguel Carrión Álvarez got his Ph.D. in 2004. This is when I was still working on loop quantum gravity. So, he decided to work on a rigorous loop quantization of the electromagnetic field. I like his thesis a lot:
• Miguel Carrión Álvarez, Loop Quantization versus Fock Quantization of p-Form Electromagnetism on Static Spacetimes.
However, he decided to leave mathematical physics when he got his degree… and he switched to finance.
There’s a lot of math in common between quantum field theory and mathematical finance. When you take quantum fluctuations in quantum fields, and replace time by imaginary time, you get random fluctuations in the stock market!
Or at least in some models of the stock market. One difference between quantum field theory and mathematical finance is that the former is famous for predicting certain quantities with many decimal digits of accuracy, while the latter is famous for predicting certain quantities with no digits of accuracy at all! I’m talking about the recent financial crisis.
Miguel and I share an interest in the failures of neoclassical economics. My interest comes from the hope — quite possibly a futile hope — that correcting some mistakes in economic theory could help show us a way out of some problems civilization now finds itself in. In fact, the Azimuth Project has its origins in my old economics diary.
Right now we’re having a little conversation about mathematical economics. Maybe you’d like to join in!
I’ll be teaching parts of two courses on mathematical finance and financial risk management in an ‘Mathematical Engineering’ MSc course at the Universidad Complutense here in Madrid.
Cool! Has the way people teach these subjects changed any since the economic crisis? I would hope so…
I don’t think it has.
First of all, these courses are mostly technical, as part of a Master’s programme intended to teach people what people do in practice. I don’t think criticizing the foundations is part of the program.
But you may have noticed (for instance, if you follow Krugman in the NYT) that the economic establishment has been very resistant to recognizing that the crisis is an empirical refutation of neoclassical economics. This crisis doesn’t fit within the conceptual framework of NCE, but that’s not a problem as they can just call it an “external shock” and continue pretending that the economy will trend to equilibrium from its new perturbed position. Related jokes of mine include that the recession part of the economic cycle is considered an outlier.
And this is not to speak of mathematical finance, where the situation is even worse. Academics still think that the best way to manage a new risk is to quantify it, define an index, and then create derivatives markets to trade it. In other words, turn all risk into market price risk and then push it all to the least solvent participants on the periphery of the financial system.
I think there is good progress being made by non-mainstream economists. Notably Steve Keen – see here:
• ARGeezer, Steve Keen’s dynamic model of the economy, European Tribune, 23 September 2009.
One of the problems with economics that Steve Keen complains about is that economists generally don’t know much about dynamical systems. I doubt they know what the Lotka-Volterra equation is, let alone understanding it (if you discretize it, the predator-prey model displays chaos like the logistic equation). I also doubt economists know about chaos in the logistic equation:
even if they know about logistic growth models which may not be generally the case either. Their model of the economy seems to be basically the Clausius-Clapeyron equation.
I believe the proper way to look at macroeconomics is as a flow network and as such ideas from category theory may be useful at least to organize one’s thinking.
The economy is a network whose nodes are “agents” and links are “economic relations”. Economic relations are flows, of goods and services in one direction and of money in the other direction (opposite categories via arrow reversal?).
Each node also has a balance sheet: assets and liabilities, and it’s the liabilities that are the key here, because they are mostly monetary. But they are also intertemporal. When you buy on credit, you get a physical asset today in exchange for the promise of cash at a later date. Presumably, the discounted present value of that future cash covers the value of the asset today, and that’s what you book as an asset or a liability. But the asset/liability accrues interest over time so its value changes on the balance sheet, and every so often you still need to make an actual transfer of cash along the an edge of the network. And when IOUs become tradeable (you issue me a bearer-IOU which I can then give to someone else who trusts your credit) they become money, too. And the relative variations in the prices of all these different kinds of money, their liquidity, etc, are key in understanding a recession like the one we’re in, or the growth (ehm, bubble) phase of a business cycle.
I don’t have a theory of this, but I keep thinking in these terms and one thing that seems to come out of it is a sort of “fluctuation-dissipation” relation between market fluctuations in prices and trade volumes, the creation of money-as-debt, and inflation. Because nodes abhor insolvency, fluctuations in cash flows lead to the creation of IOUs, which inflate the money mass. With the analogy inflation ~ dissipation, you get a sense that the more intense the market-induced cash flow fluctuations, the higher the inflation rate of the money mass, through the creation of money-as-tradeable-credit.
But “Dynamic Stochastic General Equilibrium” is not going to cut it.
Anyway, maybe I could do a guest post on your blog about these things… It’s all very inchoate as you can see.
I also think the parallels between these economic flow networks and ecosystems and climates as matter/energy flow networks are important and such dynamical systems are a relatively unexplored area of mathematical physics – it’s just too difficult to say anything general about them!
Miguel allowed me to post this exchange, noting that he could fill in gaps or moderate excessive claims in the comments. It would be nice if together we could all figure out how to take his thoughts and make them a bit more precise.
• Quantitative ecology, Azimuth Project.
I’m also dying to talk about flow networks in ecology, so it’s nice to hear that Miguel has been thinking about them in economics.
But here’s a basic question: in what sense do economists model the economy using the Clausius-Clapeyron equation? Is the idea that we can take this equation and use it to model economic equilibrium, somehow? How, exactly?