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Problem

A LQG dynamic rational inattention problem (DRIP) is defined as the following tracking problem, where at any point in time the agent chooses a vector of actions $\vec{a}_t\in\mathbb{R}^m$ to track a Gaussian stochastic process $\vec{x}_t\in\mathbb{R}^n$:

\[\begin{aligned} & \min_{\{\vec{a}_t\}_{t\geq 0}} \mathbb{E}\left[\sum_{t=0}^\infty \beta^t (\vec{a}_t - \vec{x}_t'\mathbf{H})'(\vec{a}_t - \vec{x}_t'\mathbf{H}) + \omega \mathbb{I}(\vec{a}^t;\vec{x}^t|\vec{a}^{t-1})\lvert \vec{a}^{-1}\right] \\ s.t.\quad & \vec{x}_t=\mathbf{A}\vec{x}_{t-1}+\mathbf{Q}\vec{u}_t,\quad \vec{u}_t\sim \mathcal{N}(\mathbf{0},\mathbf{I}^{k\times k}) \\ & \vec{a}^{-1} \text{ given.} \end{aligned}\]

Here:

  • $\vec{a}_t\in\mathbb{R}^m$ is the vector of the agent's actions at time $t$ (a firms choosing a price, or a househld choosing consumption and labor etc.) We denote the number of actions by $m$.
  • $\vec{x}_t\in\mathbb{R}^n$ is the vector of the shocks that the agent faces at time $t$ that are exogenous to her decision, but could be endogenous to the GE model (marginal cost of a firm, real interest rates etc.) We denote the number of shocks by $n$.

Parameters

The LQG-DRI problem is characterized by the following parameters:

  • $\omega\in \mathbb{R}_+$: cost of 1 bit of information in units of the agent's payoff.
  • $\beta\in[0,1]$: rate of discounting information.
  • $\mathbf{A}\in \mathbb{R}^{n\times n}, \mathbf{Q}\in\mathbb{R}^{n\times k}$: Determine the state space representation of $\vec{x}_t$.
  • $\mathbf{H}\in \mathbb{R}^{n\times m}$: interaction of payoffs with shocks. This comes from a second order approximation to the utility function and is such that under full information $\vec{a}^*=\mathbf{H}'\vec{x}$.

Solution

The solution to the dynamic rational inattention problem is a joint stochastic process between the actions and the states: $\{(\vec{a}_t,\vec{x}_t):t\geq 0\}$. Moreover, in some economic applications, we are also interested in the law of motion for the agent's belief about $\vec{x}_t$ under the optimal information structure $\hat{x}_t=\mathbb{E}_t[\vec{x}_t]$ where the expectation is taken conditional on the agent's time $t$ information.

Theorem 2.2 and Proposition 2.3 in Afrouzi and Yang (2020) characterize this joint distribution as a function of a tuple $(\mathbf{K}_t,\mathbf{Y}_t,\mathbf{\Sigma}_{z,t})$ where

\[\begin{aligned} \vec{a}_t &= \mathbf{H}'\hat{x}_t = \mathbf{H}'\mathbf{A}\hat{x}_{t-1}+\mathbf{Y_t}'(\vec{x}_t-\mathbf{A}\hat{x}_{t-1})+\vec{z}_t \\ \hat{x}_t &= \mathbf{A}\hat{x}_{t-1}+\mathbf{K}_t\mathbf{Y}_t'(\vec{x}_t-\mathbf{A}\hat{x}_{t-1})+\mathbf{K}_t\vec{z}_t,\quad \vec{z}\sim\mathcal{N}(0,\Sigma_z) \end{aligned}\]

Here,

  • $\mathbf{K}_t\in\mathbb{R}^{n\times m}$ is the Kalman-gain matrix of the agent in under optimal information acquisition at time $t$.
  • $\mathbf{Y}_t\in\mathbb{R}^{m\times m}$ is the loading of optimal signals on the state at time $t$.
  • $\mathbf{\Sigma}_{z,t}\in\mathbb{R}^{m\times m}$ is the variance-covariance matrix of the agent's rational inattention error at time $t$.

In addition to these, we might also be interested in the agent's prior and posterior subjective uncertainty, along with the continuation value that she assigns to information:

  • $\mathbf{\Sigma}_{p,t}=\mathbb{V}ar(\vec{x}_t|\vec{a}^{t})\in\mathbb{R}^{n\times n}$.
  • $\mathbf{\Sigma}_{-1,t}=\mathbb{V}ar(\vec{x}_t|\vec{a}^{t-1})\in\mathbb{R}^{n\times n}$,
  • $\mathbf{\Omega}_t\in\mathbb{R}^{n\times n}$.

where the matrix $\mathbf{\Omega}_t$ captures the value of information (see Afrouzi and Yang (2020) for details)

Steady State of DRIPs

Use function Drip(ω,β,A,Q,H) to store and solve for the steady state of a DRIP. It takes the primitives (ω,β,A,Q,H) as arguments and returns the solution of the model within a Drip type structure that (1) stores all the primitives of the problem and (2) returns the steady state solution information structure of the model in a field called ss that is of type DRIPs.SteadyState.

See the syntax section for Drip methods for the definition, syntax and options of Drip.

Transition Dynamics of DRIPs

The Euler equation derived in Afrouzi and Yang (2020) for the also allows us to characterize the transition path of the information structure over time for an arbitrary initial prior.

The function Trip(P::Drip,s0) takes a Drip type structure along with an initial condition s0 as an input and returns a Trip structure that summarizes the transition path of the optimal information structure. The initial condition s0 can be given either as an initial prior covariance matrix or alternatively as a one time signal about the state that perturbs the steady state prior.

See the syntax section for Trip methods for the definition, syntax and options of Trip.

Impulse Response Functions

Once the model is solved, one can generate the impulse response functions of actions and beliefs using the laws of motion stated above.

We have also included a built-in function that generates these IRFs. The function irfs(P::Drip) takes a Drip structure as input and returns the irfs of the state, beliefs and actions to all structural shocks within a Path structure.

The function also returns IRFs for transition dynamics if an initial signal is specified. See the syntax section Impulse Response Functions for the definition of the Path type structure as well as for more information about irfs function.

Simulations of DRIPs

Once the model is solved, one can also generate the simulated paths for fundamental, actions and beliefs..

We have also included a built-in function that generates these IRFs. The function simulate(P::Drip) takes a Drip structure as input and returns a simulated path of the state, beliefs and actions to all structural shocks within a Path structure.

See the syntax section Methods for Simulating DRIPs for the definition of the Path type structure as well as for more information about irfs function.