Summary

A Type Structure for LQG Dynamic Rational Inattention Problems (DRIPs)

Fields

Primitives of the DRIP

ω      : Cost of information
β      : Discount factor
A      : Transition matrix: x=Ax+Qu
Q      : Std. Dev. matrix: x=Ax+Qu
H      : Mapping of shocks to actions: v=-0.5(a'-x'H)(a-H'x)

Solution of the DRIP in the Steady State

ss     : Steady State Solution as a SteadyState type
See also: [`SteadyState`](@ref)

Drip(ω,β,A,Q,H; kwargs...) -> Drip

Solves for the steady state of a Dynamic Rational Inattention Problem (DRIP) defined by the arguments and stores the solution in a Drip type. See Afrouzi and Yang (2019) for details.

Arguments

The function takes the primitives of the Drip as arguments:

* ω      : Cost of information
* β      : Discount factor
* A      : Transition matrix: x=Ax+Qu
* Q      : Std. Dev. matrix: x=Ax+Qu
* H      : Mapping of shocks to actions: v=-0.5(a'-x'H)(a-H'x)

Optional Arguments (kwargs...)

Default values are set unless specified otherwise by user.

* fcap  [= false]    [if `true` then solves the problem with fixed capacity = ω bits]
* Ω0    [= H*H']     [initial guess for steady state information matrix]
* Σ0    [= A*A'+Q*Q'][initial guess for steady state prior]
* w     [= 1]        [updating weight on the new guess in iteration]
* tol   [= 1e-4]     [tolerance level for convergence]
* maxit [= 10000]    [maximum number of iterations]

Outputs

The function returns a Drip structure with the primitives and the solution objects:

* Y      : Weight vector for evolution of actions
* Σ_z    : Covariance matrix of the rational inattention error
* K      : Kalman gain matrix
* Σ_1    : Steady-state prior covariance matrix under the solution
* Σ_p    : Steady-state posterior covariance matrix under the solution
* Ω      : Dynamic benefit matrix

Examples

julia> P = Drip(ω,β,A,Q,H)

Summary

A type srtucture for storing the steady state solution of a Drip.

Fields

K      : Kalman gain matrix
Y      : Weight vector for evolution of actions
Σ_z    : Covariance matrix of the rational inattention error
Σ_p    : Steady-state posterior covariance matrix under the solution
Σ_1    : Steady-state prior covariance matrix under the solution
Ω      : Dynamic benefit matrix
err    : Convergence error for the solution

Summary

A Structure for the Transition dynamics of Rational Inattention Problems (TRIPs)

Fields

T       : length of TRIP
Σ_1s    : sequence of prior covariance matrices
Σ_ps    : sequence of posterior covariance matrices
Ωs      : sequence of information benefit matrices
Ds      : eigenvalues of Σ_t^(0.5)Ω_tΣ_t^(0.5) over time (marginal values of information)
err     : convergence error for shooting algorithm
con_err : convergence error for the steady state prior

     Trip(p::Drip,              # steady state of D.R.I.P.
          Σ0::Array{Float64,2}; # initial prior matrix
          T     = 100,          # optional: guess for time until convergence to steady state
          tol   = 1e-4,         # optional: tolerance for convergence
          maxit = 1000          # optional: max iterations
          ) -> Trip

Solves for the transition dynamics of the optimal information structure starting from the initial prior distribution with covariance matrix Σ0. See Afrouzi and Yang (2019) for details.

Outputs

Returns a Trip structure with the steady state and transition path of the optimal information structure.

Examples

julia> p = solve_drip(ω,β,A,Q,H)
julia> Σ0 = 0.1*p.Σ_1;
julia> Pt = solve_trip(p,Σ0);

     Trip(p::Drip,             # steady state of D.R.I.P.
          S::Signal;            # information treatment in the steady state
          T     = 100,          # optional: guess for time until convergence to steady state
          tol   = 1e-4,         # optional: tolerance for convergence
          maxit = 1000          # optional: max iterations
          ) -> Trip

Solves for the transition dynamics of the optimal information structure starting from a one time treatment with a signal S in the steady state. See Afrouzi and Yang (2019) for details.

Outputs

Returns a Trip structure with the steady state and transition path of the optimal information structure.

Examples

julia> p  = Drip(ω,β,A,Q,H)
julia> S  = Signal(L,Σ_z);
julia> pt = Trip(p,S);

Summary

A Signal Structure for Information Treatments in DRIPs. The structure encodes the signal S = L'*x+z, z₀~N(0,Σ_z).

Fields

• L : loading of the signal on x
• Σ_z : variance covariance matrix of the noise

     irfs(p          :: Drip,                          # Steady state of the DRIP (when treatment happens)
          S          :: Signal;                        # Signal for treatment
          T          :: Integer = 40,                    # optional: length of irfs
          reoptimize :: Bool = true,                   # if true gives the irfs with reoptimized signals, if false with steady state signals
          trip       :: Union{Trip, Nothing} = nothing # if false solves the trip, if = P::trip then takes P as the trip
          )

Returns a Path structure with the impulse response functions of the fundamental (x), beliefs (x_hat) and actions (a) to all the structural shocks under the information structure implied by a one time information treatment with S in the steady state of the DRIP P. In particular, if n is the dimension of x, m is the dimension of a and k is the number of structural shocks, then

• x has dimension n*k*T where x(i,j,:) is the impulse response function of the i'th dimension of x to the j'th structural shock.
• x_hat has dimension n*k*T where x_hat(i,j,:) is the impulse response function of the agent's average belief about the i'th dimension of x to the j'th structural shock.
• a has dimension m*k*T where a(i,j,:) is the impulse response function of the i'th action to the j'th structural shock.

          irfs(p :: Drip;        # Steady state of the DRIP
               T :: Integer = 40   # Optional: length of impulse response functions
               ) -> Path

Returns a Path structure with the impulse response functions of the fundamental (x), beliefs (x_hat) and actions (a) to all the structural shocks under the steady state information strucutre. In particular, if n is the dimension of x, m is the dimension of a and k is the number of structural shocks, then

• x has dimension n*k*T where x(i,j,:) is the impulse response function of the i'th dimension of x to the j'th structural shock.
• x_hat has dimension n*k*T where x_hat(i,j,:) is the impulse response function of the agent's average belief about the i'th dimension of x to the j'th structural shock.
• a has dimension m*k*T where a(i,j,:) is the impulse response function of the i'th action to the j'th structural shock.

      irfs(pt :: Trip;       # Transition dynamics of the DRIP
           T  :: Integer = 40  # Optional: length of impulse response functions
           ) -> Path

Returns a Path structure with the impulse response functions of the fundamental (x), beliefs (x_hat) and actions (a) to all the structural shocks under the information structure implied by P. In particular, if n is the dimension of x, m is the dimension of a and k is the number of structural shocks, then

• x has dimension n*k*T where x(i,j,:) is the impulse response function of the i'th dimension of x to the j'th structural shock.
• x_hat has dimension n*k*T where x_hat(i,j,:) is the impulse response function of the agent's average belief about the i'th dimension of x to the j'th structural shock.
• a has dimension m*k*T where a(i,j,:) is the impulse response function of the i'th action to the j'th structural shock.

Summary

A Structure for the irfs/simulations of DRIPs

Fields

T     : length of IRFs/simulation
x     : IRFs/simulation of the fundamental shocks
x_hat : IRFs/simulation of beliefs
a     : IRFs/simulation of actions

In particular, if n is the dimension of x, m is the dimension of a and k is the number of structural shocks, then

• x has dimension n*k*T where x(i,j,:) is the impulse response function of the i'th dimension of x to the j'th structural shock.
• x_hat has dimension n*k*T where x_hat(i,j,:) is the impulse response function of the agent's average belief about the i'th dimension of x to the j'th structural shock.
• a has dimension m*k*T where a(i,j,:) is the impulse response function of the i'th action to the j'th structural shock.

     simulate(p    :: Drip;
              T    :: Integer = 500,                    # Optional: length of simulation
              burn :: Integer = 100,                    # Optional: length of initial burn (in addition to T)
              N    :: Union{Int4,Nothing} = nothing,    # Optional: number of simulated agents (returns the average beliefs of a large set of agents by default)
              seed :: Union{Integer,Nothing} = nothing  # Optional: seed number for fundamental shocks
              ) -> Path

Returns a Path structure with a simulated path of the fundamental (x), beliefs (x_hat) and actions (a) under the steady state information structure. In particular, if n is the dimension of x and m is the dimension of a, then

• x has dimension n*T where x(:,t) is the simulated value of x at time t.
• x_hat has dimension n*T*N where x_hat(:,i,t) is the simulated value of x_hat of agent i at time t.
• a has dimension m*T*N where a(:,i,t) is the the simulated value of x_hat of agent i at time t.

    capacity(P::Drip;      # Drip structure
             unit = "bit"  # optional: unit of capacity (bit or nat).
             )

Returns the amount of information processes per unit of time in the steady state of the DRIP P.

infinitesum(func; tol = 1e-6,maxit = 1000,start=0)

Returns the infinite sum Σₓfunc(x) starting from x = start up to tolderance tol or max iteration maxit.