A sideways look at economics
This week, Roger Penrose (along with Reinhard Genzel and Andrea Ghez) received the Nobel Prize in Physics, for his work on black holes among other areas. But it’s his work on the philosophy of mind that appeals most to me, especially in his brilliant book The Emperor’s New Mind. Economics has always shamelessly stolen techniques developed in other fields — and some of Penrose’s ground-breaking work has found its way into the field of economics too. The area I focus on here is the distinction between computable and non-computable problems. People who actually know physics and mathematics: look away now. I speak as an unapologetic interested outsider, seeking resonances with the world of economics rather than formal linkages (which would require a far higher level of knowledge than I possess).
There is a class of economic models known as ‘computable general equilibrium’ (CGE) models. The qualifier ‘computable’ is important. It points towards a tension between computability and general equilibrium, suggesting that there are some aspects of, or types of, general equilibrium in economic systems that cannot be computed, and that the model restricts itself, as it must, to those aspects/types that can be computed. In fact, there is a strong argument that general equilibrium properly defined is never ‘computable’ (see for example this paper by Benjamin H. Mitra-Kahn).
General equilibrium occurs when all markets have cleared in the sense that all adjustments of prices and quantities corresponding to the set of shocks in the system, are complete. There are many problems here, including two biggies. First, how is the set of equilibrium prices achieved? Do we jump straight to the market-clearing vector of prices? (And then, how do we know what that is?) Or do we iterate gradually towards it? (And maybe never actually get there?) And, second, what variables are allowed to adjust to those new prices?
In the real world, nothing is exogenous — everything adjusts in response to everything else, and often simultaneously. But computer modelers must treat some things as exogenous (or at least as occurring in some already known chronological sequence) in order for the model to solve. Which implies, if it’s computable, it’s not really general equilibrium.
The term: ‘computable’ here means ‘can be solved by an algorithm’ — by a set of instructions or a code that proceeds for an indefinite period of time around an indefinitely numerous set of loops until a ‘solution’ has been achieved. Usually, in economic models, the solution is defined by compliance with a set of convergence criteria, such as that the differences in outcomes for an economic variable determined within the system at a given point in time between successive loops of the algorithm must be less than a certain target.
If one or more in a system of differential equations contains contemporaneous (or forward expectation) terms for another variable on the right-hand side, it becomes a system of simultaneous equations. CGE models are typically made up of systems of equations like these. Such systems are computable under certain conditions.
Some simultaneous equations can be solved algebraically, by substitution or subtraction/addition — you’ll recall from school algebra (probably with the same sense of dread that I feel) examples like:
3x + y = 7 …..(1)
3x – y = 5 …..(2)
And you’ll remember that you can solve (if a solution exists) by addition/subtraction, by substitution, and by trial and error. Trial and error is what convergence routines do with economic models.
Imagine writing an algorithm to do trial and error here. The algo can’t just intuit the solution the way you can. Instead it has to proceed along a predefined decision tree: a set of instructions.
Routines to solve computer models of the economy are the same — they are essentially a set of instructions that lead to a solution.
There can only be a computable solution at all on condition that some variables are treated as exogenous, determined outside the system. The problem is, we know that restriction is not realistic — but we can’t relax it and get a computer to solve the model. The problem of how, in the real world (where all economic variables are endogenous), the economy achieves general equilibrium, is not computable.
This has big implications! We might know everything there is to know about the current state of the world, all the preferences of every individual, all the endowments, all the shocks: everything. And still we would not be able to compute — to see in advance via some algorithm — what the economic equilibrium conditioned on that information set would look like. We can only let the world unfold and take us there. There is no other way.
The parallels between this idea (though couched in terms of the non-computability of human behaviour rather than in terms of economic equilibria, which are functions of human behaviour inter alia) and the eternal questions in the philosophy of mind relating to free will and determinism are what Penrose explores in the book cited above. Minds, for Penrose, are not like computers, in the sense that human behaviour cannot be ‘solved’ algorithmically. You can’t write a set of instructions that will reproduce human behaviour, so you can’t get computers to be humans, since what computers do — all they do — is follow sets of instructions.
What’s the point of economic models, given that if they are computable then they cannot be describing general equilibrium, since general equilibrium is in principle a non-computable problem? It’s the same as with human behaviour in general: why do we even bother trying to think ahead? Shouldn’t we just let things unfold, since we cannot compute how they will end? A phrase that I often use in discussions with Fathom staff is: the gods laugh loudest at those who make plans. But we make plans nevertheless. Making plans, and then changing those plans in the face of unforeseen events: that is what humans do. That process is one reason why you can’t see, ahead of time, where our behaviour will lead us — its unfolding involves continual making of and amending of plans. The same goes for the economy. We know the forecasts or scenarios that we develop will prove to be inaccurate, but we make them all the same, and try to learn when we get things wrong.
One thing we have learnt, over the years, is not to pretend that the future is deterministic: rather it is probabilistic. We should think about the future path for the economy as a probability distribution, not a point forecast. The same, of course, applies to individual behaviour, as Penrose eloquently discusses.
 Penrose, R. (1989), The emperor’s new mind: Concerning computers, minds, and the laws of physics, Oxford University Press.
 General equilibrium models describe what those equilibria look like — even though in the real world we are never ‘at’ a general equilibrium because there are always shocks occurring. Only on an extreme assumption that all markets clear instantaneously could we ever say we were at a general equilibrium — and in that case, we would always be there. Dynamic general equilibrium models take the additional step of capturing the rate at which the economy will converge towards its general equilibrium: the dynamic path that corresponds to a given set of shocks.
 Something like: initialise with random estimates of x and y that satisfy equation (1), then look at equation (2) to see how close those values get you to the answer. Say I start with x=1 and y=4 (which satisfies equation (1). Then I input those to equation (2) and find that those values leave me at -1 instead of +5: a long way away from satisfying the equation. In the next step, I might choose arbitrarily a lower value for x=0.5 implying y=5.5 (which satisfies equation (1)). But then I find I am even further away from satisfying equation (2). So, I probably proceeded in the wrong direction from the first estimate for the value of x. How about x=1.5 and y=2.5? That gets me closer, so it looks like the right direction. You can imagine the rest. This iterative process, in this example, will stop when the two equations are perfectly satisfied. In a large system of simultaneous equations, though, often the number of iterations that would be needed to get you to the perfect solution might run into the millions. Usually, convergence routines stop when you have reached a stable solution, one that varies very little (by less than some predefined, arbitrarily small quantity) from one iteration to the next.
 Some economic models do not ‘solve’, in the sense that they never converge on a stable solution no matter how many iterations you allow. Usually this means there is a problem in the way the model is structured — either it is not closed (meaning there is at least one variable that is neither determined within the system nor imposed exogenously), or it has explosive roots (which has nothing to do with root canal surgery, you’ll be relieved to hear, although the equivalent kind of intervention is often needed to correct the structure of the model in these cases), or it drives towards solutions that require logs or roots of negative numbers. Sometimes those problems are knowable in advance, without trying to solve the model, but sometimes they are not.
 You can get computers to mimic humans — at the time Penrose wrote The Emperor’s New Mind, no computer had yet passed what’s known as the Turing test. (That a human interrogating a machine should not be able to detect, from its answers alone, that the machine is not a human.) But he anticipated that day would come shortly, and indeed it has — many AI devices would now ace the Turing test, aided especially by the proliferation and massive power of search engines like Google. But for Penrose, passing the Turing test is not sufficient to attribute mental states to the machine. A machine could pass the Turing test, for example, without ‘understanding’ any of the questions it was asked, in the way that we commonly use the word ‘understanding’. Machines can mimic human behaviour, as in some respects a parrot can, without experiencing any of the underlying mental states that make us human.
 I’m not sure of the origin of this epithet, though I am sure it’s not me. Woody Allen has said: “If you want to make God laugh, tell him about your plans”.