## Sunday, February 16, 2014

### A physicist reads the economics blogs

I just put up a two part series that motivates the equation:

$$\text{(1) }p = \frac{dD}{dS} = \frac{1}{\kappa} \; \frac{D}{S}$$

in two different ways.

I. Quantity theory and effective field theory
II. Entropy and microfoundations

The first (I) is a top down approach; take a "symmetry" of macroeconomics like the long run neutrality of money and use it to narrow down the form of macroeconomic relationships. Of course, long run neutrality is just one instance of homogeneity of degree zero and according to Leontief [1] this represents one of the fundamental assumptions of general equilibrium:
One of these fundamental assumptions - that which Mr. Keynes is ready to repudiate - defines an important universal property of all supply and demand functions by stating that the quantity of any service or any commodity demanded or supplied by a firm or an individual remains unchanged if all the prices upon which it (directly) depends increase or decrease exactly in the same proportion. In mathematical terms, this means that all supply and demand functions, with prices taken as independent variables and quantity as a dependent one, are homogeneous functions of the zero degree. In course of the following discussion, this theorem will be referred to as the "homogeneity postulate".
Equation (1) is the leading order differential equation consistent with homogeneity of degree zero. Additionally, to leading order, supply and demand systems described by equation (1) are those described by supply and demand diagrams. Diagram-based analysis advocated by e.g. Paul Krugman is exactly this leading order analysis.

The second (II) is a bottom up approach; take the efficient markets hypothesis (EMH) as a statement about information flows and use it to build a canonical microeconomic model of supply and demand.

It is nice that these two approaches result in the same form (1). However, we can ask: is it consistent to say that the price level turns out to be some predictable function of aggregate supply and aggregate demand while the EMH tells us that prices can't be predicted? Ah, but the EMH doesn't contradict supply and demand ... increasing supply, ceteris paribus, should predictably lead to lower prices on average. The EMH states that you can't beat the market in the long run, i.e. past prices are maximally uninformative, not completely uninformative. The price level and a growing company's stock both follow a long run path. There is additional uncertainty about the latter relative to the former due to the much larger number of equally probable microstates consistent with the macrostate. Statistical fluctuations should be suppressed by factors $\sim 1/\sqrt{N}$.

Systematic deviations due to bubbles, herding, or other behavioral effects don't have to obey this. In fact, things like money illusion and the involuntary unemployment Keynes was attempting to describe (that was being discussed by Leontief) are considered violations of homogeneity of degree zero -- and I would expect a program looking at homogeneity-violating terms might be a fruitful line of research [2].

[1] Leontief, W.W. The fundamental assumption of Mr. Keynes' monetary theory of unemployment Quarterly Journal of Economics 192-197, 1937.

[2] In fact, I specifically add the homogeneity violating  $\kappa = \log S / \log D$ term in Part I in order to describe the price level.