Dynamic Equilibrium

23Nov07

I keep wondering if this is stupid and wrong or brilliant and subtle. I will try to elaborate and deepen the idea in another post later. I have a feeling this may be something useful.

Bear with me for an outlandish thought experiment. Imagine an empty water tank with a special tap. The tap is special because it would follow what we will call an “exponential law”. This is how it works:

•  It would deliver water to the tank in infinitely short but regular, periodic bursts

•  The amount of water in each burst would be exactly proportional to the remaining empty capacity during the immediately preceding period

A little bit of thinking will make you see why this is an “exponential” law – the first burst would deliver the most amount of water, the subsequent ones would deliver ever lower amounts. The tank would never be full, but would only approach full capacity while the bursts would get nearly but not quite zero. This state would still be dynamic, although, after a long, long time, it would look static. Aha, say the engineers among the readers – this is nothing but a single-pole negative feedback system!

Now imagine a second tap with similar properties, but with shorter time periods between bursts – as if it could process faster than the first tap how much water to deliver in each subsequent burst. The total instantaneous water level in the tank would now be decided by the net result, which would be a sum of a fast exponent and a slow exponent. Overall, the tank would reach its “nearly full” state faster than with only one tap.

Now stretch your imagination further. Each tap has an operator, whose employers have a magical device. They can find out the exact contributions from each tap in the total water at the end and proportionately define the pay of each operator. Obviously, rational operators would have an incentive to increase the size of their bursts (meaning raise the multiplication factor between their burst size and the instantaneous empty tank capacity) while still following the exponential law. Similarly, they would try to reduce their processing times and try to make their bursts at shorter intervals. In following their self-interest, they would inevitably improve the speed at which the tank fills up. This process would only intensify with the addition of more taps.

Financial economists among the readers will be quick to notice that this is really an analogy for arbitrage – competition to maximize payoffs arising from exploiting the water level gap. In financial markets, when going from one dynamic equilibrium to another, agents compete to maximize their gains from arbitrage and have to be quick enough to exploit their advantage before the word is out. A proliferation of such agents intensifies competition and shortens the “processing delays” so much that it seems like we always maintain a steady state, which is actually dynamic but looks static – called efficient markets by economists.

Businessmen, on the other hand, will see in this example a model for competition in real life. Competition looks static, but in reality, each competitor is constantly working on improving his arbitrage gains and agility in the face of change. A shock to the existing dynamic equilibrium means either a sudden “doubling of the tank size” – say addition of a new market – or a technological innovation where a new upstart tap operator comes in with a radically “faster exponent”.

The time elapsed between the sudden change and the new dynamic equilibrium is one of upheaval. Overall, it is essential to minimize this time to maintain an efficient market system. For this to take place, tap operators need freedom to maximize their water bursts and minimize their processing times – in other words, benefit from and seek out arbitrage opportunities. Also, potential new tap operators need freedom to set up taps and keep incumbent operators on their toes.

Control engineers might mistakenly (but understandably) think that I am making a case for designing systems that are not slew-rate limited. Finance buffs and economists would understand that I am really making a case for freedom of financial innovation and business innovation. This is a case for the Hayekian and Schumpeterian view of “competition as a verb rather than as a noun”.

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