%These are notes on the use of code for computing optimal monetary %policy that was prepared by Andy Levin (Federal Reserve Board). This code %may be used freely. However, if you use it, please include an acknowledgement, %'the code for this is taken from Andrew Levin, Lopez-Salido, J.D., 2004. %"Optimal Monetary Policy with Endogenous Capital Accumulation", manuscript, %Federal Reserve Board, and Andrew Levin, Onatski, A., Williams, J., %Williams, N., 2005. "Monetary Policy under Uncertainty in Microfounded %Macroeconometric Models." In: NBER Macroeconomics Annual 2005, %Gertler, M., Rogoff, K., eds. Cambridge, MA: MIT Press. %The Levin code may be downloaded from %http://faculty.wcas.northwestern.edu/~lchrist/d16/d1606/Get_Ramsey.zip %Dynare (another software package that can be freely used, with %attribution) may be downloaded from %http://faculty.wcas.northwestern.edu/~lchrist/d16/d1606/dyn_mat_v3_064.zip %Instructions for running the optimal policy code % %Make sure that dynare and Andy Levin's code are in your MATLAB path. For %example, in my computer I enter this at the MATLAB prompt: % % addpath('c:\projects\solve\dynare') % addpath('c:\projects\solve\levin') % %In case dynare and Levin's code are in the same path where your programs %reside, then there is no need to bother with the addpath commands. % %The basic monetary model with the exogenous policy rule is in model.mod %This *.mod file is ready to run if what you want to study is equilibrium %in the version of the model with exogenous policy. Simply type % % dynare model % %at the MATLAB command prompt. % %To get model.mod ready for optimal policy, you have to do several things. %You must have two var statements, one with the %variables to be optimized in the Ramsey problem, and one with the rest. %(One exception is that Util and Welf must be included in the endogenous variables list and Andy %Levin's code, sensibly, does not optimize with respect to these variables.) % %The var statement with the variables to be optimized in the %Ramsey problem must be preceded by the comment: % % // Endogenous variables % %The var statement with the other 'endogenous' variables must be preceded %by % // Exogenous variables % %In practice, you would include the name of your economic shocks in the %latter var statement. % %You must have, among the equations of the model, an equation that %specifies an exogenous policy, preceded by the statement: % % // Monetary Policy Rule % %You must apply Levin's translation code to model.mod. %This requires making sure the model.mod file is set up correctly. You will %need to have two additional equations. One is exactly the following % % Welf = Util + nbeta*Welf(+1); % %These exact names MUST be used. The variables, Welf and Util must be included %in the list of endogenous variables. The parameter, nbeta, must be set and %included in the list of parameters. nbeta is the Ramsey problem discount %rate which you might just want to equate to the private agents' %discount rate. %The second equation you must include is one that defines Util % % Util = ..... ; % %where the definition of your period utility should be placed where the dots %are. Now you are ready to apply Levin's translation code. %At the Matlab prompt, type % % infilename = 'model'; % outfilename = 'model_OUT'; % get_ramsey % %this writes a new dynare file, model_OUT.mod, in which the monetary policy %rule is replaced by the first order conditions associated with the %Lagrangian representation of the Ramsey problem. % %Once the translation is complete, things need to be done to the new %program, model_OUT.mod, so that it can be run by Dynare. % %1. three var statements must be merged - the ones for endogenous and %exogenous variables, and the one for the multipliers. % %2. you need to do a little (not much!) work to compute the steady state of %the Ramsey problem. The call to the steady state program in model.mod %computes the steady state in the version of the model with the exogenous %monetary policy rule. In the optimal policy problem, the policy rule has %been dropped and replaced by the Ramsey first order conditions. With two %exceptions, get_ramsey computes the first order conditions with respect to %all the variables you listed in the var statement after the %// Endogenous variables command in model_OUT.mod. The two exceptions are %Util and Welf. Getramsey does not differentiate with respect %to these variables, because they do not represent binding restrictions %on the level of utility that can be achieved in the Ramsey problem (i.e., %we know that their multipliers would be exactly zero). %Suppose the number of variables with which you differentiate is N (N=4 in %modelans_OUT.mod, which is what is output by getramsey when getramsey is %applied to modelans.mod, my answer to homework #9). %So, application of getramsey to your model_OUT.mod program replaces the %monetary policy rule by N first order conditions. In addition, there are %N-1 multipliers. The steady state value of these multipliers must be %determined, in addition to the steady state value of the %N endogenous variables. So, the steady state problem is different in the %Ramsey problem than it is in the model with exogenous policy. %The good news is that getramsey takes care of the most difficult part of %this steady state problem. % %The strategy for finding the steady state of the Ramsey problem is as %follows. You specify a conjectured value for the %steady state inflation rate, say x. Then, you compute the values of the N-1 %endogenous variables that solve the steady state %equilibrium conditions. For this step, you can simply use the steady state %program that you use for the case when monetary policy was exogenous). %Then, the steady state multipliers are found which %solve the N first order conditions of the Ramsey problem. In general, you %will not be able to solve this system, because you only have N-1 multipliers. %The way the problem is solved, is you adjust the value of x until the N-1 %multipliers you compute solve all N Ramsey first order conditions. %Then you are done. In our example, life is simple. The optimal inflation %rate in the Ramsey problem is unity. (Many problems have this form, where you %can easily conjecture the Ramsey steady state inflation rate. To see %a series of these problems, have a look at the examples in % http://faculty.wcas.northwestern.edu/~lchrist/d16/d1606/ramsey.htm %A research paper which uses getramsey, and where it is easy to conjecture %the steady state can be found in %http://www.faculty.econ.northwestern.edu/faculty/christiano/research/ECB/m %irage.htm). % %The only painful part of the Ramsey steady state computation is the %computation of the N-1 multipliers conditional on the steady state values %of the endogenous variables. Fortunately, getramsey.m writes a program that %does this for you! To see how this works, have a look at the program, ssnew. % %The program which computes the steady state of the N-1 equilibrium %conditions is called sstate.m. The program which computes the steady state %of the multipliers is Levin's endogenously generated code, model_out_lmss.m % %make sure ssnew appears after the parameters statement with the multipliers % % %3.%model_out_lmss.m must be adjusted in two ways: % %a. comment out the global statements at the start of the code. % %b. it is a good idea to remove the following commands: % % if errcheck > 1e-08, % disp('Warning: steady states of lagrange multipliers cannot be accurately determined'); % disp([' errcheck = ',num2str(errcheck,12)]); % end; % %this check need not be put anywhere else because it appears at the end of ssnew. %leaving the check in model_out_lmss.m is a nuisance because %this program is evaluated repeatedly if you need to try out different values of x %in the process of computing a steady %state. Until the steady state is found, the check would generate a warning.