Dynare Reference Manual: 4.10 Steady state

4.10 Steady state

There are two ways of computing the steady state (i.e. the static equilibrium) of a model. The first way is to let Dynare compute the steady state using a nonlinear Newton-type solver; this should work for most models, and is relatively simple to use. The second way is to give more guidance to Dynare, using your knowledge of the model, by providing it with a “steady state file”.

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4.10.1 Finding the steady state with Dynare nonlinear solver

Command: steady ;

Command: steady (OPTIONS…);

Description

This command computes the steady state of a model using a nonlinear Newton-type solver and displays it. When a steady state file is used steady displays the steady state and checks that it is a solution of the static model.

More precisely, it computes the equilibrium value of the endogenous variables for the value of the exogenous variables specified in the previous initval or endval block.

steady uses an iterative procedure and takes as initial guess the value of the endogenous variables set in the previous initval or endval block.

For complicated models, finding good numerical initial values for the endogenous variables is the trickiest part of finding the equilibrium of that model. Often, it is better to start with a smaller model and add new variables one by one.

Options

maxit = INTEGER

Determines the maximum number of iterations used in the non-linear solver. The default value of maxit is 50.

tolf = DOUBLE

Convergence criterion for termination based on the function value. Iteration will cease when the residuals are smaller than tolf. Default: eps^(1/3)

solve_algo = INTEGER

Determines the non-linear solver to use. Possible values for the option are:

0: Use fsolve (under MATLAB, only available if you have the Optimization Toolbox; always available under Octave)
1: Use Dynare’s own nonlinear equation solver (a Newton-like algorithm with line-search)
2: Splits the model into recursive blocks and solves each block in turn using the same solver as value 1
3: Use Chris Sims’ solver
4: Splits the model into recursive blocks and solves each block in turn using a trust-region solver with autoscaling.
5: Newton algorithm with a sparse Gaussian elimination (SPE) (requires bytecode option, see section Model declaration)
6: Newton algorithm with a sparse LU solver at each iteration (requires bytecode and/or block option, see section Model declaration)
7: Newton algorithm with a Generalized Minimal Residual (GMRES) solver at each iteration (requires bytecode and/or block option, see section Model declaration; not available under Octave)
8: Newton algorithm with a Stabilized Bi-Conjugate Gradient (BICGSTAB) solver at each iteration (requires bytecode and/or block option, see section Model declaration)
9: Trust-region algorithm on the entire model.
10: Levenberg-Marquardt mixed compleproblem (LMMCP) solver (Kanzow and Petra 2004)
11: PATH mixed complementarity problem solver of Ferris and Munson (1999). The complementarity conditions are specified with an mcp equation tag, see lmmcp. Dynare only provides the interface for using the solver. Due to licence restrictions, you have to download the solver’s most current version yourself from http://pages.cs.wisc.edu/~ferris/path.html and place it in Matlab’s search path.

Default value is 4.

homotopy_mode = INTEGER

Use a homotopy (or divide-and-conquer) technique to solve for the steady state. If you use this option, you must specify a homotopy_setup block. This option can take three possible values:

1: In this mode, all the parameters are changed simultaneously, and the distance between the boundaries for each parameter is divided in as many intervals as there are steps (as defined by homotopy_steps option); the problem is solves as many times as there are steps.
2: Same as mode 1, except that only one parameter is changed at a time; the problem is solved as many times as steps times number of parameters.
3: Dynare tries first the most extreme values. If it fails to compute the steady state, the interval between initial and desired values is divided by two for all parameters. Every time that it is impossible to find a steady state, the previous interval is divided by two. When it succeeds to find a steady state, the previous interval is multiplied by two. In that last case homotopy_steps contains the maximum number of computations attempted before giving up.

homotopy_steps = INTEGER

Defines the number of steps when performing a homotopy. See homotopy_mode option for more details.

homotopy_force_continue = INTEGER

This option controls what happens when homotopy fails.

0: steady fails with an error message
1: steady keeps the values of the last homotopy step that was successful and continues. BE CAREFUL: parameters and/or exogenous variables are NOT at the value expected by the user

Default is 0.

nocheck

Don’t check the steady state values when they are provided explicitly either by a steady state file or a steady_state_model block. This is useful for models with unit roots as, in this case, the steady state is not unique or doesn’t exist.

markowitz = DOUBLE

Value of the Markowitz criterion, used to select the pivot. Only used when solve_algo = 5. Default: 0.5.

Example

See section Initial and terminal conditions.

After computation, the steady state is available in the following variable:

MATLAB/Octave variable: oo_.steady_state

Contains the computed steady state.

Endogenous variables are ordered in order of declaration used in var command (which is also the order used in M_.endo_names).

Block: homotopy_setup ;

Description

This block is used to declare initial and final values when using a homotopy method. It is used in conjunction with the option homotopy_mode of the steady command.

The idea of homotopy (also called divide-and-conquer by some authors) is to subdivide the problem of finding the steady state into smaller problems. It assumes that you know how to compute the steady state for a given set of parameters, and it helps you finding the steady state for another set of parameters, by incrementally moving from one to another set of parameters.

The purpose of the homotopy_setup block is to declare the final (and possibly also the initial) values for the parameters or exogenous that will be changed during the homotopy. It should contain lines of the form:

VARIABLE_NAME, EXPRESSION, EXPRESSION;

This syntax specifies the initial and final values of a given parameter/exogenous.

There is an alternative syntax:

VARIABLE_NAME, EXPRESSION;

Here only the final value is specified for a given parameter/exogenous; the initial value is taken from the preceeding initval block.

A necessary condition for a successful homotopy is that Dynare must be able to solve the steady state for the initial parameters/exogenous without additional help (using the guess values given in the initval block).

If the homotopy fails, a possible solution is to increase the number of steps (given in homotopy_steps option of steady).

Example

In the following example, Dynare will first compute the steady state for the initial values (gam=0.5 and x=1), and then subdivide the problem into 50 smaller problems to find the steady state for the final values (gam=2 and x=2).

var c k;
varexo x;

parameters alph gam delt bet aa;
alph=0.5;
delt=0.02;
aa=0.5;
bet=0.05;

model;
c + k - aa*x*k(-1)^alph - (1-delt)*k(-1);
c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam);
end;

initval;
x = 1;
k = ((delt+bet)/(aa*x*alph))^(1/(alph-1));
c = aa*x*k^alph-delt*k;
end;

homotopy_setup;
gam, 0.5, 2;
x, 2;
end;

steady(homotopy_mode = 1, homotopy_steps = 50);

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4.10.2 Using a steady state file

If you know how to compute the steady state for your model, you can provide a MATLAB/Octave function doing the computation instead of using steady. Again, there are two options for doing that:

The easiest way is to write a steady_state_model block, which is described below in more details. See also ‘fs2000.mod’ in the ‘examples’ directory for an example.
The steady state file generated by Dynare will be called ‘FILENAME_steadystate2.m’.
You can write the corresponding MATLAB function by hand. If your MOD-file is called ‘FILENAME.mod’, the steady state file must be called ‘FILENAME_steadystate.m’. See ‘NK_baseline_steadystate.m’ in the ‘examples’ directory for an example. This option gives a bit more flexibility, at the expense of a heavier programming burden and a lesser efficiency.

Note that both files allow to update parameters in each call of the function. This allows for example to calibrate a model to a labor supply of 0.2 in steady state by setting the labor disutility parameter to a corresponding value (see ‘NK_baseline_steadystate.m’ in the ‘examples’ directory). They can also be used in estimation where some parameter may be a function of an estimated parameter and needs to be updated for every parameter draw. For example, one might want to set the capital utilization cost parameter as a function of the discount rate to ensure that capacity utilization is 1 in steady state. Treating both parameters as independent or not updating one as a function of the other would lead to wrong results. But this also means that care is required. Do not accidentally overwrite your parameters with new values as it will lead to wrong results.

Block: steady_state_model ;

Description

When the analytical solution of the model is known, this command can be used to help Dynare find the steady state in a more efficient and reliable way, especially during estimation where the steady state has to be recomputed for every point in the parameter space.

Each line of this block consists of a variable (either an endogenous, a temporary variable or a parameter) which is assigned an expression (which can contain parameters, exogenous at the steady state, or any endogenous or temporary variable already declared above). Each line therefore looks like:

VARIABLE_NAME = EXPRESSION;

Note that it is also possible to assign several variables at the same time, if the main function in the right hand side is a MATLAB/Octave function returning several arguments:

[ VARIABLE_NAME, VARIABLE_NAME… ] = EXPRESSION;

Dynare will automatically generate a steady state file (of the form ‘FILENAME_steadystate2.m’) using the information provided in this block.

Steady state file for deterministic models

steady_state_model block works also with deterministic models. An initval block and, when necessary, an endval block, is used to set the value of the exogenous variables. Each initval or endval block must be followed by steady to execute the function created by steady_state_model and set the initial, respectively terminal, steady state.

Example

var m P c e W R k d n l gy_obs gp_obs y dA;
varexo e_a e_m;

parameters alp bet gam mst rho psi del;

…
// parameter calibration, (dynamic) model declaration, shock calibration…
…

steady_state_model;
  dA = exp(gam);
  gst = 1/dA; // A temporary variable
  m = mst;

  // Three other temporary variables
  khst = ( (1-gst*bet*(1-del)) / (alp*gst^alp*bet) )^(1/(alp-1));
  xist = ( ((khst*gst)^alp - (1-gst*(1-del))*khst)/mst )^(-1);
  nust = psi*mst^2/( (1-alp)*(1-psi)*bet*gst^alp*khst^alp );

  n  = xist/(nust+xist);
  P  = xist + nust;
  k  = khst*n;

  l  = psi*mst*n/( (1-psi)*(1-n) );
  c  = mst/P;
  d  = l - mst + 1;
  y  = k^alp*n^(1-alp)*gst^alp;
  R  = mst/bet;

  // You can use MATLAB functions which return several arguments
  [W, e] = my_function(l, n);

  gp_obs = m/dA;
  gy_obs = dA;
end;

steady;

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4.10.3 Replace some equations during steady state computations

When there is no steady state file, Dynare computes the steady state by solving the static model, i.e. the model from the ‘.mod’ file from which leads and lags have been removed.

In some specific cases, one may want to have more control over the way this static model is created. Dynare therefore offers the possibility to explicitly give the form of equations that should be in the static model.

More precisely, if an equation is prepended by a [static] tag, then it will appear in the static model used for steady state computation, but that equation will not be used for other computations. For every equation tagged in this way, you must tag another equation with [dynamic]: that equation will not be used for steady state computation, but will be used for other computations.

This functionality can be useful on models with a unit root, where there is an infinity of steady states. An equation (tagged [dynamic]) would give the law of motion of the nonstationary variable (like a random walk). To pin down one specific steady state, an equation tagged [static] would affect a constant value to the nonstationary variable.

Example

This is a trivial example with two endogenous variables. The second equation takes a different form in the static model.

var c k;
varexo x;

…

model;
c + k - aa*x*k(-1)^alph - (1-delt)*k(-1);
[dynamic] c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam);
[static] k = ((delt+bet)/(x*aa*alph))^(1/(alph-1));
end;

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