Posts tagged with welfare economics

## Nonlinear Superdimensions?

I’m not the first person to say "ceteris paribus is a lie". What this aphorism means is that if you make a c.p. assumption in order to think something through, then the conclusion you reach may be irrelevant to the real world.

Worse, because people don’t understand models, someone might take your careful “A implies B” statement to mean “Both A and B are the case”. For example rather than Edgeworth boxes implying that trade be always mutually beneficial, people might take you to mean that

1. exchange is characterised by Edgeworth boxes
2. private transactions are always Pareto optimal

which is not at all what the theory’s saying. The theory is just connecting assumptions to conclusion: yes, if this were true, then that would surely follow. Which is great because some people don’t actually think such things through.

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Anyway. Ceteris paribus assumptions make thinking easier, but they hamstring whatever you find out—so that it may be useless, or (hopefully not) worse than useless: misleading.

But maybe it’s possible to keep the crutch of c.p. and make it less foolish.

There are some situations where it’s impossible to do what I’m going to suggest—like where space overlaps itself. But in Euclidean spaces it is possible.

Econometricians are already familiar with principal components analysis. You make one “composite dimension” which is composed of a fixed combination of existing dimensions.

`composite = .4 × X₁   +   .2 × X₂   +   1.7 × X₃`

This is what I’m calling a “super-dimension”.

You hold all other things constant so you can think logically about a situation that has the geometry of a single straight line. By creating a composite dimension maybe one could still use the handy ceteris-paribus assumption but roll more of real-life into the model too.

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For example let’s say as wealth ↑, trips to the emergency room ↓. Then you could form a composite dimension with a positive coefficient on wealth, negative on emergency room visits, and talk about both at once with everything else held constant. One step forward relative to talking only about only wealth ↑↓.

But wait — maybe these are only linearly related around a small neighbourhood of some point. Well, we could still create a composite “super-dimension” by varying the coefficients. This could either come in the form of pre-transforming wealth to be log of wealth, or something else — like a threshold effect where we use two or three linear pieces (eg, rich enough with slope=0, way too poor with slope=0, and middle with a linear decrease). In general, whereas linear means `+k+k+k+k+k+k+…`, nonlinear can be interpreted as `+1.2k+k+.9k+.8k+k+1.1k+1.3k+1.2k+1.4k+…`. So instead of constructing a composite dimension with fixed coefficients before ignoring everything else, perhaps one could vary the coefficients along with the space.

That’s all. This may not be a new idea.

If you buy a loaf of bread from the supermarket both you and the supermarket (its shareholders, its employees, its bread suppliers) are made to some degree better off. How do I know? Because the supermarket offered the bread voluntarily and you accepted the offer voluntarily. Both of you must have been made better off, a little or a lot—or else you two wouldn’t have done the deal.

Economists have long been in love with this simple argument. They have since the eighteenth century taken the argument a crucial and dramatic step further: that is, they have deduced something from it, namely, Free trade is neat.

If each deal between you & the supermarket, and the supermarket & Smith, and Smith & Jones, and so forth is betterment-producing (a little or a lot: we’re not talking quantities here), then (note the “then”: we’re talking deduction here) free trade between the entire body of French people and the entire body of English people is betterment-producing. Therefore (note the “therefore”) free trade between any two groups is neat.

The economist notes that if all trades are voluntary they all have some gain. So free trade in all its forms is neat. For example, a law restricting who can get into the pharmacy business is a bad idea, not neat at all, because free trade is good, so non-free trade is bad. Protection of French workers is bad, because free trade is good. And so forth, to literally thousands of policy conclusions.

Deirdre McCloskey, Secret Sins of Economics

A wonderful essay. I’ll just add what I think are some common answers to common objections:

## A One-Line Viewpoint on Trading

Given a time-series of one security’s price-train `P[t]`, a low-frequency trader’s job (forgetting trading costs) is to find a step function `S[t]` to convolve against price changes `P``′``[t]`

$\large \dpi{200} \bg_white \int^{\tau}_0 dP_t \ast S_t \ dt \quad = \quad \text{profit}$

with the proviso that the other side to the trade exists.

`S[t]` represents the bet size long or short the security in question. The trader’s profit at any point in time `τ` is then given by the above definite integral.

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• I haven’t seen anyone talk this way about the problem, perhaps because I don’t read enough or because it’s not a useful idea. But … it was a cool thought, representing a `>0` amount of cogitation.
• This came to mind while reading a discussion of “Monkey Style Trading” on NuclearPhynance. My guess is that monkey style is a Brownian ratchet and as such should do no useful work.
• If I were doing a paper investigating the public-welfare consequences of trading, this is how I’d think about the problem.

Each hedge fund / central bank / significant player is reduced to a conditional response strategy, chosen from the set of all step functions uniformly less than a liquidity constraint. This endogenously coughs up the trading volume which really should be fed back into the conditional strategies.
• Does this viewpoint lead to new risk metrics?
• Should be mechanical to expand to multiple securities. Would anything interesting come from that?

I wouldn’t usually think that multiplication of functions has anything to do with trading. Maybe some theorems can do a bit of heavy lifting here; maybe not.

It at least feels like an antidote to two wrongful axiomatic habits. For economists who look for real value, logic, and Information Transmission, it says The market does whatever it wants, and the best response is a response to whatever that is. For financial engineering graduates who spent too long chanting the mantra`μ dt + σ dBt`" this is just another way of emphasising: you can’t control anything except your bet size.

UPDATE: Thanks to an anonymous commenter for a correction.