When the framework is deterministic, Dynare can be used for models with the assumption of perfect foresight. The system is supposed to be in a given state before a period 1 (often a steady state) when the news of a contemporaneous or of a future shock is learned by the agents in the model. The purpose of the simulation is to describe the reaction in anticipation of, then in reaction to the shock, until the system returns to equilibrium. This return to equilibrium is only an asymptotic phenomenon, which one must approximate by an horizon of simulation far enough in the future. Another exercise for which Dynare is well suited is to study the transition path to a new equilibrium following a permanent shock. For deterministic simulations, the numerical problem consists of solving a nonlinear system of simultaneous equations in n endogenous variables in T periods. Dynare uses a Newton-type method to solve the simultaneous equation system. Because the resulting Jacobian is in the order of n by T and hence will be very large for long simulations with many variables, Dynare makes use of the sparse matrix code .

Dynare commands

perfect_foresight_setup

Command: perfect\_foresight\_setup;

Command: perfect\_foresight\_setup (OPTIONS...);

Prepares a perfect foresight simulation, by extracting the information in the initval, endval and shocks blocks and converting them into simulation paths for exogenous and endogenous variables.

This command must always be called before running the simulation with perfect\_foresight\_solver.

Options

• periods = INTEGER

Number of periods of the simulation.

• datafile = FILENAME

Used to specify path for all endogenous and exogenous variables. Strictly equivalent to initval_file.

Output

The paths for the exogenous variables are stored into context.results.model_resultst[1].simulations.

The initial and terminal conditions for the endogenous variables and the initial guess for the path of endogenous variables are stored into context.results.model_results[1].simulations.

perfect_foresight_solver

Command: perfect\_foresight\_solver ;

Command: perfect\_foresight\_solver (OPTIONS...);

Computes the perfect foresight (or deterministic) simulation of the model.

Note that perfect\_foresight\_setup must be called before this command, in order to setup the environment for the simulation.

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 it proves impossible to improve the function value by more than tolf. Default: 1e-5

• tolx = DOUBLE

Convergence criterion for termination based on the change in the function argument. Iteration will cease when the solver attempts to take a step that is smaller than tolx. Default: 1e-5

• noprint

Don't print anything. Useful for loops.

• print

Print results (opposite of noprint).

• lmmcp

Solves mixed complementarity problems (the term refers to the LMMCP solver (Kanzow and Petra, 2004), that is used by DynareMatlab. DynareJulia uses the PATHSovler package)

• endogenous_terminal_period

The number of periods is not constant across Newton iterations when solving the perfect foresight model. The size of the nonlinear system of equations is reduced by removing the portion of the paths (and associated equations) for which the solution has already been identified (up to the tolerance parameter). This strategy can be interpreted as a mix of the shooting and relaxation approaches. Note that round off errors are more important with this mixed strategy (user should check the reported value of the maximum absolute error). Only available with option stack_solve_algo==0.

Remark

Be careful when employing auxiliary variables in the context of perfect
foresight computations. The same model may work for stochastic
simulations, but fail for perfect foresight simulations. The issue
arises when an equation suddenly only contains variables dated `t+1` (or
`t-1` for that matter). In this case, the derivative in the last (first)
period with respect to all variables will be 0, rendering the stacked
Jacobian singular.

Example

Consider the following specification of an Euler equation with log utility:

Lambda = beta*C(-1)/C;
Lambda(+1)*R(+1)= 1;

Clearly, the derivative of the second equation with respect to all endogenous variables at time t is zero, causing perfect_foresight_solver to generally fail. This is due to the use of the Lagrange multiplier Lambda as an auxiliary variable. Instead, employing the identical

beta*C/C(+1)*R(+1)= 1;

will work.

Julia function

perfect_foresight!(; periods, context = context, display = true,
                   linear_solve_algo=ilu, maxit = 50, mcp = false,
                   tolf = 1e-5, tolx = 1e-5)

Output

The simulated endogenous variables are available in context.results.model_results[1].simulations. This is a vector of AxisArrayTable, one for each simulations stored in context. Each AxisArrayTable contains the trajectories for endogenous and exogenous variables

Solving mixed complementarity problems

requires a particular model setup as the goal is to get rid of any min/max operators and complementary slackness conditions that might introduce a singularity into the Jacobian. This is done by attaching an equation tag (see model-decl) with the mcp keyword to affected equations. The format of the mcp tag is

[mcp = 'VARIABBLENAME OP CONSTANT']

where VARIABLENAME is an endogenous variable and OP is either > or <. For complicated occasionally binding constraints, it may be necessary to declare a new endogenous variable.

This tag states that the equation to which the tag is attached has to hold unless the expression within the tag is binding. For instance, a ZLB on the nominal interest rate would be specified as follows in the model block:

    model;
       ...
       [mcp = 'r > -1.94478']
       r = rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e;
       ...
    end;

where r is the nominal interest rate in deviation from the steady state. This construct implies that the Taylor rule is operative, unless the implied interest rate r<=-1.94478, in which case the r is fixed at -1.94478. This is equavalant to

\[(r_t > -1.94478)\;\; \bot\;\; r_t = \rho r_{t-1} + (1-\rho) (g_\pi Infl_t+g_y YGap_t) + e_t\]

By restricting the value of r coming out of this equation, the mcp tag also avoids using max(r,-1.94478) for other occurrences of r in the rest of the model. It is important to keep in mind that, because the mcp tag effectively replaces a complementary slackness condition, it cannot be simply attached to any equation.

Note that in the current implementation, the content of the mcp equation tag is not parsed by the preprocessor. The inequalities must therefore be as simple as possible: an endogenous variable, followed by a relational operator, followed by a number (not a variable, parameter or expression).