## Monday, June 20, 2016

### Metzler diagrams from information equilibrium

Paul Krugman has a post today where he organizes some DSGE model results in a simplified Mundell-Fleming model represented as a Metzler diagram. Let me show you how this can be represented as an information equilibrium (IE) model.

We have interest rates $r_{1}, r_{2}$ in two countries coupled through an exchange rate $e$. Define the interest rate $r_{i}$ to be in information equilibrium with the price of money $M_{i}$ in the respective country (with money demand $D_{i}$) -- this sets up four IE relationships:

\begin{align} r_{1}& \rightleftarrows p_{1}\\ p_{1} : D_{1}& \rightleftarrows M_{1}\\ r_{2}& \rightleftarrows p_{2}\\ p_{2} : D_{2}& \rightleftarrows M_{2} \end{align}

This leads to the formulas (see the paper)

$$\text{(1) }\; r_{i} = \left( k_{i} \frac{D_{i}}{M_{i}}\right)^{c_{i}}$$

Additionally, exchange rates are basically given as a ratio of the price of money in one country to another:

$$e \equiv \frac{p_{1}}{p_{2}} = \alpha \frac{M_{1}^{k_{1}-1}}{M_{2}^{k_{2}-1}}$$

And now we can plot the formula (1) versus $M_{1}^{k_{1}-1}$ (blue) and $M_{2}^{1-k_{2}}$ (yellow) at constant $D_{i}$ (partial equilibrium: assuming demand changes slowly compared to moneytary policy changes). This gives us the Metzler diagram from Krugman's post and everything that goes along with it:

Also, for $k \approx 1$ (liquidity trap conditions), these curves flatten out:

1. Super cool! This is the first time you have explicitly stated these relationships for exchange rates. As an amateur investor, this seems to be a way to ascertain market inefficiencies in exchange rates. If one can independently determine k1 and k2 for a set of countries, one can then determine the "correct" information transfer exchange rate. One can then take advantage of temporary market mispricing of the exchange rates (perhaps?).

1. Hi Todd,

I think I lost this comment in the craziness that has been my life the past couple weeks. Sorry about that!

I'd say the problem with this is that 1) I don't think this is well established enough to do real trading with money you don't expect to lose and 2) even if it were established, exchange rates are so volatile that the random fluctuations around this "equilibrium" could cause sufficient losses to make it not worth it.

Also, I explicitly stated the relationships for exchange rates in the link with the hypertext "exchange rates" above. I just hadn't put the interest rate and exchange rate models together in a Metzler diagram.

In any case, it could use a lot more empirical work. I just wanted to make the connection between IE models and some traditional economics.