NashRamsey Toolbox  2.0

A MATLAB/Dynare toolbox that automates the derivation of welfare-maximizing Ramsey policies and open-loop Nash equilibria for any model you can write in Dynare.

Martin Bodenstein  ·  Luca Guerrieri

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Supports Dynare 6.5 and 7.0  |  MATLAB Symbolic Math Toolbox

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Published paper

Macroeconomic Policy Games

Martin Bodenstein, Luca Guerrieri & Joe LaBriola  —  Journal of Monetary Economics, Vol. 101 (2019), pp. 64–81  ·  doi:10.1016/j.jmoneco.2018.07.015

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What it does

Strategic interactions, solved automatically

When policymakers have distinct objectives, deviating from full cooperation can produce large welfare losses. This toolbox characterizes optimal policies by symbolic differentiation—no tedious manual Lagrangians required.

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Cooperative Ramsey

Computes the joint welfare-maximizing policy under full commitment, using the timeless-perspective approach of Benigno & Woodford (2012).

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Open-loop Nash

Solves the open-loop Nash game between two policymakers, each taking the other's complete contingent plan as given.

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Any Dynare Model

Reads a standard .mod file and outputs ready-to-run Dynare files for Ramsey, Nash, or single-agent Ramsey problems.

Prerequisites

What you need before you start

Package contents

What's inside the zip

The download is a single zip archive containing the toolbox root and several subfolders. Example subdirectories ship with all generated output files so you can inspect results without re-running the toolbox.

nash_ramsey_toolbox_051626/

Toolbox root. Contains setup_toolbox.m, setpathdynare6_5.m, setpathdynare7_0.m, run_tests.m, and call_cleanup.m.

toolbox_programs/

83 core .m files: convertmodfiles.m, get_nash.m, get_ramsey.m, cleanup_dynare_dir.m, templates, and more.

utilities/

Plotting helpers: makechart.m, makeirfsecondorder.m, and band-pass filter routines used by the replication scripts.

BBCDL_model_excl2ndorder/

Two-country open-economy model (Benigno & Benigno 2006, Corsetti, Dedola & Leduc 2010). Replicates Figures 1–3 of the paper.

GK_model/

Macroprudential regulation model with financial frictions (Gertler & Karadi 2011). Replicates Figures 4–5.

LQ_BBCDL_model/

Linear-quadratic implementation of the Corsetti, Dedola & Leduc (2010) model. Used to cross-check the toolbox output.

tests/

Test infrastructure. Run run_tests.m from the toolbox root to verify compatibility; results are saved as timestamped .mat files in tests/test_results/.

Appendix A.1.1

Preparing your model file

The toolbox reads a Dynare-formatted .mod file and generates all first-order conditions automatically. Follow these six steps to put your model file into the required format. Throughout, example.mod stands for your model's filename.

1

Declare utility variables in the var block

Define Util1 and Util2 as endogenous variables and add their equations to the model block. These are the period utility functions of the two policymakers. 

// Endogenous variables
var Util1, Util2, c1y, c1c, c2y, c2c, ...;

model;
  // Utility, agent 1 (e.g., country 1 central bank)
  Util1 = (c1c^(1-sigma) - 1) / (1 - sigma) - chi0*c1n^(1+chi);

  // Utility, agent 2 (e.g., country 2 central bank)
  Util2 = (c2c^(1-sigma) - 1) / (1 - sigma) - chi0*c2n^(1+chi);
  ...
end;
2

Split the var block into two blocks

Create one block for endogenous variables (including Util1, Util2) and a second block for exogenous shock processes. Precede each block with the comment markers the toolbox looks for. 

// Endogenous variables
var Util1, Util2, c1y, c1c, c2y, c2c, c1pid, c2pid, ...;

// Exogenous variables
var c1z, c2z, c1taupvar, c2taupvar, ...;
varexo c1zerr, c2zerr, c1taupvarerr, c2taupvarerr;
3

Move parameter values to a separate script

If parameter values are assigned directly inside the .mod file, remove them and save them as example_paramfile.m in the same folder. Dynare will call this script via the steadystate file. 

% example_paramfile.m
beta   = 0.99;
sigma  = 2;
chi    = 1;
chi0   = 1;
xi_p   = 0.75;   % Calvo parameter
nu_p   = 0.15;   % price markup
omega_c = 0.85;  % home bias in consumption
4

Label each policy equation

In the model block, insert the comment // Policy Rule, agent 1 immediately before the equation that will be replaced by the optimal policy rule for player 1, and // Policy Rule, agent 2 for player 2. These can be placeholder Taylor rules or simple definitions. 

model;
  ...
  // Policy Rule, agent 1
  i1_t = c1pid;    // instrument: country-1 producer price inflation

  // Policy Rule, agent 2
  i2_t = c2pid;    // instrument: country-2 producer price inflation
  ...
end;
5

Replace initval with a steady-state script

Delete the initval block and save steady-state values for all endogenous variables as example_ss_defs.m. Then create example_steadystate.m using the template below — Dynare calls this function to initialize the solver. 

% example_steadystate.m
function [ys, check] = example_steadystate(junk, ys)
global M_
check = 0;

%% assign parameter values
example_paramfile

%% assign steady-state values
example_ss_defs

%% send parameters and steady states to Dynare
nparams = size(M_.param_names, 1);
for icount = 1:nparams
  eval(['M_.params(icount) = ', M_.param_names{icount}, ';'])
end

nvars = M_.endo_nbr;
ys    = zeros(nvars, 1);
for i_indx = 1:nvars
  eval(['ys(i_indx) = ', M_.endo_names{i_indx}, ';'])
end
end
6

Move exogenous shock equations to the end of the model block

Collect all equations that describe the paths of exogenous variables (AR(1) shock processes, etc.) and place them at the very end of the model block, after all structural equations. 

model;
  // ... all structural equations first ...

  // Shock processes — place these last
  c1z       = rho_z * c1z(-1)     + sigma_z * c1zerr;
  c1taupvar = rho_p * c1taupvar(-1) + sigma_p * c1taupvarerr;
  c2z       = rho_z * c2z(-1)     + sigma_z * c2zerr;
  c2taupvar = rho_p * c2taupvar(-1) + sigma_p * c2taupvarerr;
end;

Appendix A.1.2

Key toolbox programs

Once your model file is in the correct format, a single call to convertmodfiles generates all the Dynare files you need.

convertmodfiles(infilename, policy_problem, instrument1, instrument2, discount_factor_name [, welfare_correction_flag])

Argument Type Description
infilename string Name of your Dynare model file (with or without the .mod extension). Example: 'BBCDLmodelcomp'
policy_problem 'ramsey' | 'nash' | 'one_agent_ramsey' ramsey — joint welfare-maximizing cooperative policy.
nash — open-loop Nash game: each player best-responds to the other's full contingent plan.
one_agent_ramsey — one player optimizes while the other follows the arbitrary rule in the original file.
instrument1 string Name of the policy instrument variable for player 1. Example: 'c1pid' (country-1 producer price inflation).
instrument2 string Name of the policy instrument variable for player 2, or '1'/'2' if policy_problem = 'one_agent_ramsey'.
discount_factor_name string Name of the discount-factor parameter as it appears in infilename_paramfile.m. Example: 'c1beta'. The toolbox automatically assigns this value to nbeta in the generated policy paramfile — no manual editing required.
welfare_correction_flag 0 | 1  (optional, default 0) When set to 1, appends a welfare-correction term that accounts for violations of commitment under the timeless perspective.

Output files generated

For a call convertmodfiles('example','nash','instr1','instr2','beta') the toolbox writes the following files:

FilePurpose
example_nash.mod Complete Dynare model with the optimal first-order conditions replacing the original policy rules.
example_nash_lmss.m Subset of first-order conditions that are linear in the Lagrange multipliers; used to initialize the steady-state solver.
example_nash_steadystate.m Steadystate function called by Dynare. Calls the paramfile and ss_defs, then solves for the optimal steady state numerically.
guess_example_nash_steadystate.m Uses the original model's steady state as a starting guess for the numerical steady-state solver.
example_nash_ss_defs.m Steady-state values for the structural (non-multiplier) variables.
example_nash_paramfile.m Parameter file; sets omega_welf1 = 0.5, omega_welf2 = 0.5, and nbeta = beta (the value of discount_factor_name passed to convertmodfiles) automatically.

Additional utility scripts

add_welfare_vars.m

Augments any processed Dynare model with cumulative welfare variables Welf1, Welf2, and Welf.

edit_shocks.m

Turns shocks on or off across all .mod files in the current folder. Useful when comparing responses to different shock groups.

Appendix A.2.1  ·  Worked example

Reproducing Figures 1 and 2

Figures 1 and 2 of the paper compare impulse responses under cooperative Ramsey policy and open-loop Nash play in a two-country sticky-price model following Benigno & Benigno (2006) and Corsetti, Dedola & Leduc (2010). Figure 1 shows responses to a technology shock; Figure 2 shows responses to a cost-push shock. All code lives in BBCDL_model_excl2ndorder/.

Figure 1 — Cooperative and Open-loop Nash Policies: Technology Shock
Figure 1: Cooperative and Open-loop Nash Policies
Responses to a Technology Shock

Output (home & foreign), output price inflation, relative consumption, real exchange rate — 10 quarters.

Figure 1 — Responses to a one-std-dev technology shock. Blue solid: Ramsey (cooperative). Red dashed: Open-loop Nash.

Figure 2 — Cooperative and Open-loop Nash Policies: Cost-Push Shock
Figure 2: Cooperative and Open-loop Nash Policies
Responses to a Cost-Push Shock

Output (home & foreign), output price inflation, relative consumption, real exchange rate — 10 quarters.

Figure 2 — Responses to a one-std-dev cost-push (markup) shock. Blue solid: Ramsey (cooperative). Red dashed: Open-loop Nash.

1

Set up the Dynare path

After unzipping, open MATLAB, navigate to the toolbox root (nash_ramsey_toolbox_051626/), and run setup_toolbox.m once. This auto-detects Dynare 6.5 and 7.0 and embeds the correct absolute paths into both setpathdynare6_5.m and setpathdynare7_0.m. Then navigate to BBCDL_model_excl2ndorder/ and run the appropriate path script for your Dynare version: 

% Step 1 – run once from the toolbox root:
cd 'path/to/nash_ramsey_toolbox_051626'
setup_toolbox       % configures both setpathdynare6_5.m and setpathdynare7_0.m

% Step 2 – before each session, from the model folder:
cd 'path/to/nash_ramsey_toolbox_051626/JME_paper_replication_and_examples/BBCDL_model_excl2ndorder'
setpathdynare6_5    % use this for Dynare 6.5
% — or —
setpathdynare7_0    % use this for Dynare 7.0
2

Generate Ramsey and Nash model files

Run CREATE_RAMSEY_AND_NASH.m. This calls convertmodfiles once for the cooperative (Ramsey) problem and once for the Nash problem, both using producer price inflation (c1pid, c2pid) as instruments. Pass 'c1beta' as the fifth argument so the toolbox automatically sets the planner's discount factor to match the households' — no manual editing of any generated file is needed. 

% CREATE_RAMSEY_AND_NASH.m (run this script)
setpathdynare6_5

% Cooperative Ramsey — 5th arg sets nbeta = c1beta automatically
convertmodfiles('BBCDLmodelcomp', 'ramsey', 'c1pid', 'c2pid', 'c1beta')

% Open-loop Nash — producer price inflation as instruments
convertmodfiles('BBCDLmodelcomp', 'nash', 'c1pid', 'c2pid', 'c1beta')
What gets generated: BBCDLmodelcomp_ramsey_c1pid_c2pid.mod and BBCDLmodelcomp_nash_c1pid_c2pid.mod, together with their companion steadystate, paramfile, ss_defs, and lmss files. The generated paramfiles already contain nbeta = c1beta and the shock blocks are preserved — the files are ready to run in Dynare immediately with no further edits.
Welfare weights: The generated paramfiles default to omega_welf1 = 0.5 and omega_welf2 = 0.5 (equal country weights). Leave these unchanged for Figures 1–2.
3

Run BBCDL_figure1and2.m to generate the figures

This script solves both models with Dynare, extracts impulse responses using makeirfsecondorder (first-order approximation), and plots Figures 1 and 2 via makechart. 

BBCDL_figure1and2   % run from BBCDL_model_excl2ndorder/

The script loops over two shocks — technology (c1zerr) and cost-push (c1taupvarerr) — and plots six panels for each:

  • Output, home country
  • Output, foreign country
  • Output price inflation, home
  • Output price inflation, foreign
  • Relative consumption
  • Real exchange rate
Expected output: Two MATLAB figure windows appear, matching Figures 1 and 2 of the paper. Under cooperation (blue solid), output price inflation is perfectly stabilized in response to the technology shock. Under the Nash game (red dashed), small but non-zero inflation deviations emerge due to strategic instrument competition.

Download the toolbox

All source code, replication scripts, and generated model files are packaged in a single zip archive.

⇓  nash_ramsey_toolbox_051626.zip

Contains: nash_ramsey_toolbox_051626/  ·  Dynare 6.5 & 7.0 compatible