## Saturday, August 30, 2014

### Walras' law, information theory edition

Nick Rowe has a new post up and it inspired me to take up his challenge (entering as a non-economist). Rowe is probably one of the best economist bloggers out there if you want to get more technical than the typical post from Scott Sumner or Paul Krugman. His question is this:
Q. Assume an economy where there are (say) 7 markets. Suppose 6 of those markets are in equilibrium (with quantity demanded equal to quantity supplied). Is it necessarily true that the 7th market must also be in equilibrium (with quantity demanded equal to quantity supplied)?
I've looked at Walras' law before (e.g. this post). I'm going to answer this using information theory with progressively more complexities, but I'll start with some notation.

Define $I(D_{k})$ to be the source (demand) information in the $k^{\text{th}}$ market and $I(S_{k})$ to be the received information (supply). Define aggregate source information (aggregate demand, AD) and aggregate received information (aggregate supply, AS) as

$$I(AD) = I(\sum_{k} D_{k}) \;\;\text{and}\;\; I(AS) = I(\sum_{k} S_{k})$$

If the information in each market is independent, this becomes:

$$I(AD) = \sum_{k} I(D_{k}) \;\;\text{and}\;\; I(AS) = \sum_{k} I(S_{k})$$

And lastly, define excess information in the $k^{\text{th}}$ market as

$$\Delta I_{k} \equiv I(D_{k}) - I(S_{k})$$

Rowe's question becomes

$$\text{If } \Delta I_{k = 1 .. 6} = 0 \text{ then what is } \Delta I_{7} \text{ ?}$$

First is the "Walras' law is correct" version  ...

We assume that the information in each market is independent and that $I(AD) = I(AS)$, so that

$$0 = I(AD) - I(AS) = \sum_{k} I(D_{k}) - \sum_{k} I(S_{k}) = \sum_{k} \Delta I_{k}$$

rearranging the terms

$$0 = \Delta I_{7} + \sum_{k = 1}^{6} \Delta I_{k} = \Delta I_{7} + 0$$

Therefore, $\Delta I_{7} = 0$.

Now the thing is that all we can really say is that $I(AS) \leq I(AD)$ (the market doesn't necessarily transfer all the information), so that brings us to the non-ideal information transfer version  ...

We assume that the information in each market is independent and that $I(AS) \leq I(AD)$, so that

$$0 \leq I(AD) - I(AS) = \sum_{k} I(D_{k}) - \sum_{k} I(S_{k}) = \sum_{k} \Delta I_{k}$$

rearranging the terms

$$0 \leq \Delta I_{7} + \sum_{k = 1}^{6} \Delta I_{k} = \Delta I_{7} + 0$$

Therefore, $\Delta I_{7} \geq 0$.

That means Walras' law doesn't pin down that last market, and says that there can be excess demand. But it's even worse than that, which brings us to the non-independent (i.e. mutual) information version  ...

As I keep mentioning, we're assuming the information in each market is independent, i.e.

$$I(D_{j} + D_{k}) = I(D_{j}) + I(D_{k})$$

But this isn't necessarily true and in general (e.g. Shannon joint entropy)

$$\text{(1) } I(D_{j} + D_{k}) \leq I(D_{j}) + I(D_{k})$$

This says for practical purposes that some of the information in the source in one market may be the same as the information in the source in another, hence they do not necessarily add to yield more information. So that all we really know is that

$$I(\sum_{k = 1}^{6} D_{k}) \geq I(\sum_{k = 1}^{6} S_{k})$$

based on the fact that you can't get more information out than you put in. This means that knowing the six markets clear doesn't necessarily even tell us about the aggregate demand of the 6 markets (ignoring the seventh).

Nick Rowe basically arrives at this last version -- he says there can be excess demands/supplies of money in each of the six markets so Walras' law can't really tell us anything about the seventh. The information theory argument presented here does not require money, which is consistent with Rowe. He says that the same result could hold in a barter economy because some good could effectively operate as money and there would be excess demands for various barter goods in each of the individual markets. Rowe says that:
Walras' Law is true and useful for the economy as a whole only if there is only one market in the whole economy, where all goods are traded for all goods.
This appears to be saying that if you can't decompose $AD = D_{1} + D_{2} + \cdots$ (or the decomposition is trivial), then you get Walras' law back -- and it's true. If you can't decompose the markets, then there are no "joint entropies" that can be formed from their decomposition, so there is no information loss in equation (1) above. This doesn't rule out non-ideal information transfer in version  above, but assuming markets work, saying you can't decompose the markets (or the decomposition is trivial) gets you back to version  where Walras' law holds.

So is Nick's post essentially re-deriving the sub-additivity of joint entropy?

1. O/T: David Glasner had a question for you (which I'd like to hear your answer to as well):

"Jason, Thanks for the link and the explanation. How do you know which direction information is moving?"

1. 2. Thanks. I guess I didn't scroll down enough!

2. Jason, what do you mean by this exactly?: "...or the decomposition is trivial"

Is a case where this is trivial the case of exactly one market?

1. Exactly one market or the decomposition is an arbitrary division of exactly one market (all the apples sold to people named Pete, all the apples sold to people named Patricia, and all the apples sold to everyone else).

3. Jason, assuming a monetary economy (as per Nick's description) could a situation with ΔIk > 0 correspond to both:

1. An excess supply of money in the kth market
2. An excess demand for money in the kth market

1. Yes, I think so. If you go back to this expectations post and look at the KL divergence, you can imagine an excess supply of money as one possible distribution of apples and money over consumers and firms, call it P1 and an excess demand for money as a different distribtion P2, then if the market clearing distribution is P, then both

D(P||P1) and D(P||P2) are going to show information loss relative to P (the loss is minimized when P1 = P2 = P).

4. Off topic, but I'd be interested to know if you've tried (or thought of) how to model/test MMT (modern monetary theory) using an information transfer framework. I don't know how to even begin conceptualizing that, given that I can't find a nailed down starting point/set of axioms for MMT which would be amenable to this type of analysis!

I'm partial to the MMT take on central banks, as their starting point is looking at how things actually work vs. say idealized MM formulations..

1. I'm not entirely sure what MMT is either, but here are a few statements of MMT I copied from the introduction of the Wikipedia entry:

http://en.wikipedia.org/wiki/Modern_Monetary_Theory

Let's see what the ITM has to say about it ...

1. According to modern monetary theory, "governments with the power to issue their own currency are always solvent, and can afford to buy anything for sale in their domestic unit of account even though they may face inflationary and political constraints".

The Information Transfer Model (ITM) should be able to model the resulting hyperinflation ... but the ideas of government solvency or political constraints are generally political in nature and outside the ITM's purview.

2. In MMT, money enters circulation through government spending. Taxation and its legal tender power to discharge debt establish the fiat money as currency, giving it value by creating demand for it in the form of a private tax obligation that must be met using the government's currency.

The quantity of government debt for a country with its own currency seems largely irrelevant to inflation in the ITM. Deficit-financed government spending can affect inflation if dP/dM0 ~ 0 (liquidity trap conditions), but can be offset by monetary policy when dP/dM0 > 0. The relevant form of money (currency M0) enters the economy through requests by banks to the Fed who sends the requests to the Treasury for printing. M0 and NGDP have a strong relationship -- and causality can go both ways: NGDP causes more currency to be printed and more currency causes NGDP to grow. In the former case government spending can cause NGDP to go up, causing M0 to go up -- both of which can cause the price level to go up.

[That was kind of a long-winded way of saying: "Sometimes government deficit spending can boost the economy and cause currency to enter the economy and affect inflation. Sometimes government deficit spending can directly affect inflation. But mostly, it's currency."]

The source of value of a currency is outside the ITM's purview.

3. Because the government can issue its own currency at will, MMT maintains that the level of taxation relative to government spending (the government's deficit spending or budget surplus) is in reality a policy tool that regulates inflation and unemployment, and not a means of funding the government's activities per se.

The level of taxation seems largely irrelevant to inflation (see 2 above about government deficit spending, though). The overall level of NGDP impacts employment, so government spending can probably reduce unemployment. Unemployment itself seems to a market failure in the ITM (i.e. the ITM doesn't describe the unemployment spikes associated with recessions) -- and it's potentially caused by human psychology (the spikes seem like panicky over-reactions).