4.16 Optimal policy

Dynare has tools to compute optimal policies for various types of objectives. You can either solve for optimal policy under commitment with ramsey_policy, for optimal policy under discretion with discretionary_policy or for optimal simple rule with osr.

Command: osr [VARIABLE_NAME…];

Command: osr (OPTIONS…) [VARIABLE_NAME…];

Description

This command computes optimal simple policy rules for linear-quadratic problems of the form:

such that:

where:

• denotes the unconditional expectations operator;
• are parameters to be optimized. They must be elements of the matrices , , , i.e. be specified as parameters in the params-command and be entered in the model-block;
• are the endogenous variables, specified in the var-command, whose (co)-variance enters the loss function;
• are the exogenous stochastic shocks, specified in the var_exo-command;
• is the weighting matrix;

The linear quadratic problem consists of choosing a subset of model parameters to minimize the weighted (co)-variance of a specified subset of endogenous variables, subject to a linear law of motion implied by the first order conditions of the model. A few things are worth mentioning. First, denotes the selected endogenous variables’ deviations from their steady state, i.e. in case they are not already mean 0 the variables entering the loss function are automatically demeaned so that the centered second moments are minimized. Second, osr only solves linear quadratic problems of the type resulting from combining the specified quadratic loss function with a first order approximation to the model’s equilibrium conditions. The reason is that the first order state-space representation is used to compute the unconditional (co)-variances. Hence, osr will automatically select order=1. Third, because the objective involves minimizing a weighted sum of unconditional second moments, those second moments must be finite. In particular, unit roots in are not allowed.

The subset of the model parameters over which the optimal simple rule is to be optimized, , must be listed with osr_params.

The weighting matrix used for the quadratic objective function is specified in the optim_weights-block. By attaching weights to endogenous variables, the subset of endogenous variables entering the objective function, , is implicitly specified.

The linear quadratic problem is solved using the numerical optimizer csminwel of Chris Sims.

Options

The osr command will subsequently run stoch_simul and accepts the same options, including restricting the endogenous variables by listing them after the command, as stoch_simul (see section Computing the stochastic solution) plus

maxit = INTEGER Determines the maximum number of iterations

used in the non-linear solver. Default: 1000

tolf = DOUBLE Convergence criterion for termination based on

the function value. Iteration will cease when it proves impossible to improve the function value by more than tolf. Default: 1e-7

The value of the objective is stored in the variable oo_.osr.objective_function, which is described below.

After running osr the parameters entering the simple rule will be set to their optimal value so that subsequent runs of stoch_simul will be conducted at these values.

Command: osr_params PARAMETER_NAME…;

This command declares parameters to be optimized by osr.

Block: optim_weights ;

This block specifies quadratic objectives for optimal policy problems

More precisely, this block specifies the nonzero elements of the weight matrix used in the quadratic form of the objective function in osr.

An element of the diagonal of the weight matrix is given by a line of the form:

VARIABLE_NAME EXPRESSION;

An off-the-diagonal element of the weight matrix is given by a line of the form:

VARIABLE_NAME, VARIABLE_NAME EXPRESSION;

Example

var y inflation r; 
varexo y_ inf_;

parameters delta sigma alpha kappa gammarr gammax0 gammac0 gamma_y_ gamma_inf_;

delta =  0.44;
kappa =  0.18;
alpha =  0.48;
sigma = -0.06;

gammarr = 0;
gammax0 = 0.2;
gammac0 = 1.5;
gamma_y_ = 8;
gamma_inf_ = 3;

model(linear); 
y  = delta * y(-1)  + (1-delta)*y(+1)+sigma *(r - inflation(+1)) + y_;
inflation  =   alpha * inflation(-1) + (1-alpha) * inflation(+1) + kappa*y + inf_;
r = gammax0*y(-1)+gammac0*inflation(-1)+gamma_y_*y_+gamma_inf_*inf_; 
end;

shocks; 
var y_; stderr 0.63; 
var inf_; stderr 0.4; 
end;

optim_weights; 
inflation 1; 
y 1; 
y, inflation 0.5; 
end;

osr_params gammax0 gammac0 gamma_y_ gamma_inf_; 
osr y; 

MATLAB/Octave variable: oo_.osr.objective_function

After an execution of the osr command, this variable contains the value of the objective under optimal policy.

Command: ramsey_model (OPTIONS…);

Description

This command computes the First Order Conditions for maximizing the policy maker objective function subject to the constraints provided by the equilibrium path of the economy.

The planner objective must be declared with the planner_objective command.

This command only creates the expanded model, it doesn’t perform any computations. It needs to be followed by other instructions to actually perfrom desired computations. Note that it is the only way to perform perfect foresight simulation of the Ramsey policy problem.

See section Auxiliary variables, for an explanation of how Lagrange multipliers are automatically created.

Options

This command accepts the following options:

planner_discount = EXPRESSION

Declares the discount factor of the central planner. Default: 1.0

instruments = (VARIABLE_NAME,…)

Declares instrument variables for the computation of the steady state under optimal policy. Requires a steady_state_model block or a …_steadystate.m file. See below.

Steady state

Dynare takes advantage of the fact that the Lagrange multipliers appear linearly in the equations of the steady state of the model under optimal policy. Nevertheless, it is in general very difficult to compute the steady state with simply a numerical guess in initval for the endogenous variables.

It greatly facilitates the computation, if the user provides an analytical solution for the steady state (in steady_state_model block or in a …_steadystate.m file). In this case, it is necessary to provide a steady state solution CONDITIONAL on the value of the instruments in the optimal policy problem and declared with option instruments. Note that choosing the instruments is partly a matter of interpretation and you can choose instruments that are handy from a mathematical point of view but different from the instruments you would refer to in the analysis of the paper. A typical example is choosing inflation or nominal interest rate as an instrument.

Command: ramsey_policy [VARIABLE_NAME…];

Command: ramsey_policy (OPTIONS…) [VARIABLE_NAME…];

Description

This command computes the first order approximation of the policy that maximizes the policy maker objective function submitted to the constraints provided by the equilibrium path of the economy.

The planner objective must be declared with the planner_objective command.

See section Auxiliary variables, for an explanation of how this operator is handled internally and how this affects the output.

Options

This command accepts all options of stoch_simul, plus:

planner_discount = EXPRESSION

Declares the discount factor of the central planner. Default: 1.0

instruments = (VARIABLE_NAME,…)

Declares instrument variables for the computation of the steady state under optimal policy. Requires a steady_state_model block or a …_steadystate.m file. See below.

Note that only first order approximation is available (i.e. order=1 must be specified).

Output

This command generates all the output variables of stoch_simul.

In addition, it stores the value of planner objective function under Ramsey policy in oo_.planner_objective_value.

Steady state

Dynare takes advantage of the fact that the Lagrange multipliers appear linearly in the equations of the steady state of the model under optimal policy. Nevertheless, it is in general very difficult to compute the steady state with simply a numerical guess in initval for the endogenous variables.

It greatly facilitates the computation, if the user provides an analytical solution for the steady state (in steady_state_model block or in a …_steadystate.m file). In this case, it is necessary to provide a steady state solution CONDITIONAL on the value of the instruments in the optimal policy problem and declared with option instruments. Note that choosing the instruments is partly a matter of interpretation and you can choose instruments that are handy from a mathematical point of view but different from the instruments you would refer to in the analysis of the paper. A typical example is choosing inflation or nominal interest rate as an instrument.

Command: discretionary_policy [VARIABLE_NAME…];

Command: discretionary_policy (OPTIONS…) [VARIABLE_NAME…];

Description

This command computes an approximation of the optimal policy under discretion. The algorithm implemented is essentially an LQ solver, and is described by Dennis (2007).

You should ensure that your model is linear and your objective is quadratic. Also, you should set the linear option of the model block.

Options

This command accepts the same options than ramsey_policy, plus:

discretionary_tol = NON-NEGATIVE DOUBLE

Sets the tolerance level used to assess convergence of the solution algorithm. Default: 1e-7.

maxit = INTEGER

Maximum number of iterations. Default: 3000.

Command: planner_objective MODEL_EXPRESSION;

This command declares the policy maker objective, for use with ramsey_policy or discretionary_policy.

You need to give the one-period objective, not the discounted lifetime objective. The discount factor is given by the planner_discount option of ramsey_policy and discretionary_policy. The objective function can only contain current endogenous variables and no exogenous ones. This limitation is easily circumvented by defining an appropriate auxiliary variable in the model.

With ramsey_policy, you are not limited to quadratic objectives: you can give any arbitrary nonlinear expression.

With discretionary_policy, the objective function must be quadratic.

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