DSGE, part 5 (summary)

I've just finished deriving a version of the three-equation New Keynesian DSGE model from a series of information equilibrium relationships and a maximum entropy condition. We have

$$
\begin{align}
\Pi & \rightleftarrows N \;\text{ with IT index } \alpha\\
X & \rightleftarrows C \;\text{ with IT index }1/\sigma\\
R & \rightleftarrows \Pi_{t+1} \;\text{ with IT index }\lambda_{\pi}\\
R & \rightleftarrows X_{t+1} \;\text{ with IT index }\lambda_{x}
\end{align}
$$

along with a maximum entropy condition on the intertemporal consumption $\{ C_{t} \}$ subject to a budget constraint:

$$
C_{t+1} = R_{t} C_{t}
$$

We can represent these graphically

These stand for information equilibrium relationships between the price level $\Pi$ and nominal output $N$, real output gap $X$ and consumption $C$, nominal interest rate $R$ and the price level, and the nominal interest rate and the output gap $X$. These yield

$$
\begin{align}
r_{t} & = \lambda_{\pi} \; E_{I} \pi_{t+1} + \lambda_{x} \; E_{I} x_{t+1} + c\\
x_{t} & = -\frac{1}{\sigma} \left( r_{t} - E_{I} \pi_{t+1}\right) + E_{I} x_{t+1} + \nu_{t}\\
\pi_{t} & = E_{I} \pi_{t+1} + \frac{\alpha}{1-\alpha}x_{t} + \mu_{t}
\end{align}
$$

with information equilibrium rational (i.e. model-consistent) expectations $E_{I}$ and "stochastic innovation" terms $\nu$ and $\mu$ (the latter has a bias towards closing the output gap due to the maximum entropy state being "the most likely" — i.e. the IE version has a different distribution for its random variables). With the exception of a lack of a coefficient for the first term on the RHS of the last equation, this is essentially the three equation New Keynesian DSGE model: Taylor rule, IS curve, and Philips curve (respectively).

One thing I'd like to emphasize is that although this model exists as a set of information equilibrium relationships, they are not the best set of relationships. For example, the typical model I use here (here are some others) that relates some of the same variables is

$$
\begin{align}
\Pi : N & \rightleftarrows M0 \;\text{ with IT index } k\\
r_{M} & \rightleftarrows p_{M} \;\text{ with IT index } c_{1}\\
p_{M} : N & \rightleftarrows M \;\text{ with IT index } c_{2}\\
\Pi : N & \rightleftarrows L \;\text{ with IT index } c_{3}\\
\end{align}
$$

where M0 is the monetary base without reserves and $M =$ M0 or MB (the monetary base with reserves) and $r_{M0}$ is the long term interest rate (e.g. 10-year treasuries) and $r_{MB}$ is the short term interest rate (e.g 3-month treasuries). Additionally, the stochastic innovation term in the first relationship is directly related to changes in the employment level $L$. In part 1 of this series, I related this model to the Taylor rule; the last IE relationship is effectively Okun's law (in terms of hours worked here or added with capital to the Solow model here — making this model a kind of weird hybrid of a RBC model deriving from Solow and a monetary/quantity theory of money model).

Here is the completed series for reference:

DSGE, part 1 [Taylor rule] 

DSGE, part 2 [IS curve] 

DSGE, part 3 (stochastic interlude) [relates $E_{I}$ and stochastic terms] 

DSGE, part 4 [Phillips curve]

DSGE, part 5 [the current post]