Economic policy and a simple dynamic system:

One of the simplest dynamic systems is the harmonic oscillator. Big word for a yo-yo or a weight on the end of a rubber band or spring. The motion (dynamics) of such a system is described by a second order differential equation.

m d^2x/dt^2 + b dx/dt + wx = F(t)

m is mass. b is drag, w is the elasticity of your rubber band or spring. F(t) is what you do with the other end of the spring, or your finger in the case of a yo-yo.

Let’s assume for the moment that F(t) = 0, a constant. What this equation describes is the bouncy bouncy motion of the weight at the end of a spring. If the drag term is big enough, there isn’t much bouncy bouncy, think of the suspension of a car but with dampers, or shocks as the Americans call them. By arranging the right ratios of m, b and k, you get either a smooth ride a la Lexus, or bouncy bouncy like a Land Rover Defender. Forget about the F(t) for now.

Economic growth is cyclical and can be modelled as an oscillation like we described above. If all the long term policies are right, rule of law, demographics, industry diversification, etc etc, then there is less chance of bouncy bouncy. If an economy has concentrations of risk, imbalances, poor corporate governance, then there is more chance of bouncy bouncy. In fact if you solve the equation for the path of the economy, the general solution is such that the set of solutions for which there is no bounciness, is very small, almost infinitesimal compared to the set of solutions for which there is a lot of bounciness. We have a technical term of this, namely that the probability of no bounciness is almost surely zero. Yes, almost surely is a technical term.

Lets get back to F(t). This term is like economic policy, both fiscal and monetary. Its how the government or central planner can ‘guide’ the economy and try to smooth out the bounciness of growth. The central planner basically tries to obtain the solution to the Left Hand Side of the equation, figure out how bouncy things will be and then use F(t) to try to smooth things out. The risk here is that if you time things wrong, then F(t) can make things even more bouncy. This is bad. Also, things are path dependent. Once you start your F(t), managing the system down the road is dependent on what you did before.

If the central planner has perfect information, i.e. knows everything there is to know about the economy, then it can obtain a solution to the Left Hand Side and design an F(t) to damp the oscillations. Alas, life is not like that and the central planner either doesn’t have perfect information, or makes mistakes, is plain dumb, or has been trading their PA a bit too actively. Using the wrong F(t) can lead to big bouncy bouncy. Which is bad.

Technically, the solution to the second order differential equation is:

x(t) = A exp(pt)+ B exp(qt)

If p or q are real numbers, you have an exponential blow up (bad) or decay (good). If p and q are complex, and because they are the roots of a second order polynomial equation they are very very likely to end up being complex, you have an oscillation. The chance of all the stars lining up so that F(t) is countercyclical is almost surely zero. So good luck to all those central bankers out there trying to manage the economic cycle.

Disclaimer: I am not suggesting that the economy is in fact described by a second order differential equation but the analogy is useful in highlighting the difficulty that central planners face when trying to guide a dynamic process such as economic growth. In fact the economy is far more complex that this two dimensional example. If central bank policy has to be made factoring in economic agents which adjust their behaviour to the policies then the analysis gets hugely more complex. A feedback mechanism is introduced which leads to more variability of outcomes in response to the policy. (Feedback is a word used to describe the horrible screeching sound when an electric guitar is held close an amp. The sound from the amp causes resonance in the strings which gets picked up by the pickups and routed to the amp and well... you get the idea.) Now imagine a world where central bank policy impacts exchange rates and interest rates which impact the decisions of other central banks which impact all other central banks and the scope for a right royal mess becomes apparent.