Wednesday, August 17, 2016

DSGE, part 2


I am continuing to build a standard DSGE model (specifically, the simple three equation New Keynesian DSGE model) using information equilibrium (and maximum entropy). In part 1, I summarized the references and built a "Taylor rule". In this installment, I will use the Euler equation to derive the "IS curve". I'll assume rational expectations for simplicity at first (one can drop the $E$'s), but will add some discussion at the end.

Let's start with the information equilibrium relationship between (real) output and (real) consumption $Y \rightleftarrows C$. This tells us that

$$
Y \sim C^{1/\sigma}
$$

or in log-linear form $y = \frac{1}{\sigma} \; c$. I took the information transfer index to be $1/\sigma$ so that we end up something that might be recognizable by economists. Now let's import the maximum entropy condition relating two periods of consumption at time $t$ and $t+1$ from this post:

$$
C_{t+1} = C_{t} (1 + r_{t})
$$

or in log-linear form $c_{t+1} = c_{t} + r_{t}$. Substituting output $y$, defining the real interest rate in terms of the nominal interest rate $i$ and expected inflation $r_{t} \equiv i_{t} - \pi_{t+1}$, and rearranging we obtain:

$$
y_{t} = -\frac{1}{\sigma}\left( i_{t} - \pi_{t+1} \right) + y_{t+1}
$$

And there we have the NK IS curve. We can add in the expectation operators if you'd like:

$$
y_{t} = -\frac{1}{\sigma}\left( i_{t} - E_{t}\pi_{t+1} \right) + E_{t}y_{t+1}
$$

And this is where the information equilibrium version of the IS curve has a different interpretation. The information equilibrium model can be viewed as a transfer of information from the future to the present. We can interpret the "expected" value as the ideal information transfer value, and deviations from that as non-ideal information transfer. The value added by this interpretation is that instead of rational expectations where the deviation from the expected value has some zero-mean distribution, we generally have e.g. prices that will be bounded from above by the ideal information equilibrium solution. Here's an example using interest rates:


We could think of the $E$ operators as a warning: this variable may come in below expectations due to coordinations (financial panic, recession). Therefore, we should think of the information equilibrium NK DSGE model as a bound on a dynamic system, not necessarily the real result. With this in mind, it is no wonder DSGE models would work well for the great moderation but fail during a massive coordination event.

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