## Methods for Steady State of DRIPs

DRIPs.DripType

Summary

A [Mutable] 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

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
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DRIPs.solve_dripMethod
solve_drip(ω,β,A,Q,H;
fcap  = false,
Ω0    = H*H',
Σ0    = A*A'+Q*Q',
w     = 1,
tol   = 1e-4,
maxit = 10000) -> Drip

Solves for the steady state of a Dynamic Rational Inattention Problem (DRIP) defined by the arguments. 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

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 = solve_drip(ω,β,A,Q,H)
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DRIPs.solve_dripMethod
solve_drip(P::Drip;...) -> Drip

Same as above but infers ω,β,A,Q and H from P and returns a Drip structure with the primitives and the solution.

Examples

julia> P = Drip(ω,β,A,Q,H)
julia> P = solve_drip(P)
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## Methods for Transition dynamics of DRIPs

DRIPs.SignalType

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
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DRIPs.TripType

Summary

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

Fields

P    : a DRIP structure with its primitives and steady state solution
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 err
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DRIPs.solve_tripMethod
     solve_trip(Ss::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> Ss = solve_drip(ω,β,A,Q,H)
julia> Σ0 = 0.1*Ss.Σ_1;
julia> Pt = solve_trip(Ss,Σ0);
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DRIPs.solve_tripMethod
     solve_trip(Ss::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> Ss = solve_drip(ω,β,A,Q,H)
julia> S  = Signal(L,Σ_z);
julia> Pt = solve_trip(Ss,S);
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## Methods for Impulse Response Functions

DRIPs.DripirfsType

Summary

A Structure for the impulse response functions of DRIPs

Fields

T     : length of IRFs
x     : IRFs of the fundamental shocks
x_hat : IRFs of beliefs
a     : IRFs 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.
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DRIPs.dripirfsMethod
     dripirfs(Ss::Drip,          # Steady state of the DRIP (when treatment happens)
S::Signal;         # Signal for treatment at time 0
T = 40,            # optional: length of irfs
reoptimize = true, # optional: if true gives the irfs with reoptimized signals, if false with steady state signals
trip       = false # optional: if false solves for the optimal trip, if = P::trip then takes P as the transition dynamics after treatment
) -> Dripirfs

Returns a Dripirfs 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.
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DRIPs.dripirfsMethod
     dripirfs(P::Drip;  # Steady state of the DRIP
T = 40    # Optional: length of impulse response functions
) -> Dripirfs

Returns a Dripirfs 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.
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DRIPs.dripirfsMethod
     dripirfs(P::Trip;  # Transition dynamics of the DRIP
T = 40    # Optional: length of impulse response functions
) -> Dripirfs

Returns a Dripirfs 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.
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## Methods for Simulating DRIPs

DRIPs.dripsimsMethod
     dripsims(P::Drip;      # Steady state of the DRIP
T    = 500,   # Optional: length of simulation
N    = 1,     # Optional: number of simulated agents
burn = 100,   # Optional: length of initial burn (in addition to T)
seed = true   # Optional: seed number for fundamental shocks
) -> Dripirfs

Returns a Dripirfs 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*N*T where x_hat(:,i,t) is the simulated value of x_hat of agent i at time t.
• a has dimension m*N*T where a(:,i,t) is the the simulated value of x_hat of agent i at time t.
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## Aux. Functions

DRIPs.capacityMethod
    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.

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DRIPs.infinitesumMethod
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.

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