381 lines
14 KiB
Modula-2
381 lines
14 KiB
Modula-2
% Original model by: J. Linde, M. Trabandt (2019: Should We Use Linearized Models to Calculate Fiscal Multipliers?
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% Journal of Applied Econometrics, 2018, 33: 937-965. http://dx.doi.org/10.1002/jae.2641
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% This version has some additional dynamics for capital and investment
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% Created by Willi Mutschler (@wmutschl, willi@mutschler.eu)
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% =========================================================================
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% Copyright (C) 2019-2020 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
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% =========================================================================
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% =========================================================================
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% Declare endogenous variables
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% =========================================================================
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var
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% Staggered-price economy
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c ${c}$ (long_name='consumption')
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lam ${\lambda}$ (long_name='lagrange multiplier budget')
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n ${n}$ (long_name='labor')
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w ${w}$ (long_name='real wage')
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r ${r}$ (long_name='interest rate')
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pie ${\pi}$ (long_name='inflation')
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y ${y}$ (long_name='output')
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pstar ${p^\ast}$ (long_name='price dispersion')
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vartheta ${\vartheta}$ (long_name='aggregate price index')
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mc ${mc}$ (long_name='marginal costs')
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s ${s}$ (long_name='auxiliary variable for nonlinear pricing 1')
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f ${f}$ (long_name='auxiliary variable for nonlinear pricing 2')
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a ${a}$ (long_name='auxiliary variable for nonlinear pricing 3')
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ptilde ${\tilde{p}}$ (long_name='reoptimized price')
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delta1 ${\Delta_1}$ (long_name='price dispersion 1')
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delta2 ${\Delta_2}$ (long_name='price dispersion 2')
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delta3 ${\Delta_3}$ (long_name='price dispersion 3')
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b ${b}$ (long_name='bonds')
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tau ${\tau}$ (long_name='lump-sum tax')
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g
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nu %consumption preference shock
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% Flex-price economy
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cpot ${c^{pot}}$ (long_name='flex-price consumption')
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npot ${n^{pot}}$ (long_name='flex-price labor')
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wpot ${w^{pot}}$ (long_name='flex-price real wage')
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rrpot ${r^{pot}}$ (long_name='flex-price interest rate')
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ypot ${y^{pot}}$ (long_name='flex-price output')
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bpot ${b^{pot}}$ (long_name='flex-price bonds')
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taupot ${\tau^{pot}}$ (long_name='flex-price lump-sum tax')
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% Added variables for capital and investment
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k ${k}$ (long_name='capital')
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rk ${r^{K}}$ (long_name='rental rate on capital')
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iv ${i}$ (long_name='investment')
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q ${q}$ (long_name='Tobins Q')
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kpot ${k^{pot}}$ (long_name='flex-price capital')
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rkpot ${r^{K,pot}}$ (long_name='flex-price rental rate on capital')
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ivpot ${i^{pot}}$ (long_name='flex-price investment')
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;
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% =========================================================================
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% Declare observable variables
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% =========================================================================
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varobs c y;
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% =========================================================================
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% Declare exogenous variables
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% =========================================================================
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varexo
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epsg
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epsnu
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;
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% =========================================================================
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% Declare parameters
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% =========================================================================
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parameters
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ALPHA ${\alpha}$ (long_name='capital share')
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BETA ${\beta}$ (long_name='discount rate')
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PIEBAR ${\bar{\pi}}$ (long_name='target inflation rate')
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IOTA ${\iota}$ (long_name='indexation')
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CHI ${\chi}$ (long_name='inverse frisch elasticity')
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THETAP ${\theta_p}$ (long_name='net markup')
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XIP ${\xi_p}$ (long_name='Calvo probability')
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PSI ${\psi}$ (long_name='Kimbal curvature')
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GAMMAPIE ${\gamma_\pi}$ (long_name='Taylor rule parameter inflation')
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GAMMAX ${\gamma_x}$ (long_name='Taylor rule parameter output')
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DELTA ${\delta}$ (long_name='depreciation rate')
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QBAR ${\bar{Q}}$ (long_name='steady state Q')
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NUBAR ${\bar{\nu}}$ (long_name='steady state consumption preference shock')
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BBAR %government debt to annualized output ratio
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GBAR_o_YBAR % government consumption ratio
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TAUBAR
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TAUNBAR
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PHI
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RHONU
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RHOG
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SIGNU
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SIGG
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;
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% =========================================================================
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% Calibration
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% =========================================================================
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BETA = 0.995;
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PIEBAR = 1 + 0.005;
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ALPHA = 0.3;
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NUBAR = 0;%0.01;
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BBAR = 0;%0.6;
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GBAR_o_YBAR = 0;%0.2;
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TAUBAR = 0;
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PHI = 0.0101;
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IOTA = 0.5;
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CHI = 1/0.4;
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XIP = 2/3;
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PSI = -12.2;
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THETAP = 0.1;
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GAMMAPIE = 1.5;
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GAMMAX = 0.125;
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QBAR=1;
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DELTA = 0;%0.025;
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RHONU = 0.95;
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RHOG = 0.95;
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SIGG = 0.6/100;
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SIGNU = 0.6/100;
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TAUNBAR = 0;
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% ========================================================================
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% Model equations
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% =========================================================================
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model;
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% -------------------------------------------------------------------------
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% Auxiliary expressions
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% -------------------------------------------------------------------------
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#OMEGAP = (1+THETAP)*(1+PSI)/(1+PSI+THETAP*PSI); %gross markup
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#gammap = pie^IOTA*PIEBAR^(1-IOTA); %indexation rule
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%#gammap = PIEBAR;
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% -------------------------------------------------------------------------
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% Added equations for capital and investment
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% -------------------------------------------------------------------------
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[name='foc household wrt i']
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lam = lam*q;
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[name='foc household wrt k']
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lam*q = BETA*lam(+1)*(rk(+1)+ (1-DELTA)*q(+1) );
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[name='capital accumulation']
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k = (1-DELTA)*k(-1) + iv;
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[name='capital demand']
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rk=ALPHA*mc*y/k(-1);
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[name='flex price foc household wrt k']
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(cpot-steady_state(cpot)*nu)^(-1) = BETA*(cpot(+1)-steady_state(cpot)*nu(+1))^(-1)*(rkpot(+1) + (1-DELTA) );
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[name='flex price capital accumulation']
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kpot = (1-DELTA)*kpot(-1) + ivpot;
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[name='flex price capital demand']
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rkpot=ALPHA/(1+THETAP)*(npot/kpot(-1))^(1-ALPHA);
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% -------------------------------------------------------------------------
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% Actual model equations
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% -------------------------------------------------------------------------
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[name='foc household wrt c (marginal utility)']
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(c-steady_state(c)*nu)^(-1) = lam;
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[name='foc household wrt l (leisure vs. labor)']
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n^CHI = (1-TAUNBAR)*lam*w;
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[name='foc household wrt b (Euler equation)']
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lam = BETA*r/pie(+1)*lam(+1);
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[name='aggregate resource constraint']
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y = c + g + iv;
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[name='production function']
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y=pstar^(-1)*k(-1)^ALPHA*n^(1-ALPHA);
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[name='Nonlinear pricing 1']
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s = OMEGAP*lam*y*vartheta^(OMEGAP/(OMEGAP-1))*mc + BETA*XIP*(gammap/pie(+1))^(OMEGAP/(1-OMEGAP))*s(+1);
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[name='Nonlinear pricing 2']
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f = lam*y*vartheta^(OMEGAP/(OMEGAP-1)) + BETA*XIP*(gammap/pie(+1))^(1/(1-OMEGAP))*f(+1);
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[name='Nonlinear pricing 3']
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a = PSI*(OMEGAP-1)*lam*y + BETA*XIP*(gammap/pie(+1))*a(+1);
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[name='Nonlinear pricing 4']
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s = f*ptilde -a*ptilde^(1+OMEGAP/(OMEGAP-1));
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[name='zero profit condition']
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vartheta = 1 + PSI - PSI*delta2;
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[name='aggregate price index']
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vartheta = delta3;
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[name='overall price dispersion']
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pstar = 1/(1+PSI)*vartheta^(OMEGAP/(OMEGAP-1))*delta1^(OMEGAP/(1-OMEGAP)) + PSI/(1+PSI);
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[name='price dispersion 1']
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delta1^(OMEGAP/(1-OMEGAP)) = (1-XIP)*ptilde^(OMEGAP/(1-OMEGAP)) + XIP*(gammap/pie*delta1(-1))^(OMEGAP/(1-OMEGAP));
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[name='price dispersion 2']
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delta2 = (1-XIP)*ptilde+XIP*gammap/pie*delta2(-1);
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[name='price dispersion 3']
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delta3^(1/(1-OMEGAP)) = (1-XIP)*ptilde^(1/(1-OMEGAP)) + XIP*(gammap/pie*delta3(-1))^(1/(1-OMEGAP));
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[name='marginal costs']
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(1-ALPHA)*mc = w*k(-1)^(-ALPHA)*n^ALPHA;
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[name='Taylor Rule']
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r = steady_state(r)*(pie/PIEBAR)^GAMMAPIE*( (y/steady_state(y))/(ypot/steady_state(ypot)) )^GAMMAX;
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[name='government budget']
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b = r(-1)/pie*b(-1) + g/steady_state(y) -TAUNBAR*w*n/steady_state(y) - tau;
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[name='fiscal rule']
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tau = TAUBAR + PHI*(b(-1)-BBAR);
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[name='flex price Euler equation']
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(cpot-steady_state(cpot)*nu)^(-1) = BETA*rrpot*(cpot(+1)-steady_state(cpot)*nu(+1))^(-1);
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[name='flex price foc household wrt l (leisure vs. labor)']
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npot^CHI = (1-TAUNBAR)*(cpot-steady_state(cpot)*nu)^(-1)*wpot;
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[name='flex price wage']
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(1-ALPHA)/(1+THETAP)*kpot(-1)^ALPHA = wpot*npot^ALPHA;
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[name='flex price aggregate resource constraint']
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ypot = cpot + g + ivpot;
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[name='flex price production function']
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ypot=kpot(-1)^ALPHA*npot^(1-ALPHA);
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[name='flex price government budget']
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bpot = rrpot(-1)*bpot(-1) + g/steady_state(y) -TAUNBAR*wpot*npot/steady_state(y) - taupot;
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[name='fiscal rule ']
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taupot = TAUBAR + PHI*(bpot(-1)-BBAR);
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g/steady_state(y) - GBAR_o_YBAR = RHOG*(g(-1)/steady_state(y) - GBAR_o_YBAR) + epsg;
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nu - NUBAR = RHONU*(nu(-1) - NUBAR) + epsnu;
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end;
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% =========================================================================
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% Steady state using a steady_state_model block
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% =========================================================================
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steady_state_model;
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OMEGP = (1+THETAP)*(1+PSI)/(1+PSI+THETAP*PSI);
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q = QBAR;
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pie = PIEBAR;
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GAMMAP = PIEBAR^IOTA*PIEBAR^(1-IOTA);
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b = BBAR;
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tau = TAUBAR;
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taun = TAUNBAR;
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nu = NUBAR;
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r = pie/BETA;
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RK = (1/BETA+DELTA-1)*q; % foc k
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aux1 = ( (1-XIP)/(1-XIP*(GAMMAP/pie)^(1/(1-OMEGP)) ) )^(1-OMEGP);
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aux2 = (1-XIP)/(1-XIP*(GAMMAP/pie));
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ptilde = (1+PSI)/ ( aux1 + PSI*aux2 );
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delta2 = aux2*ptilde;
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delta3 = aux1*ptilde;
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vartheta = delta3;
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delta1 = ( (1-XIP)/(1-XIP*(GAMMAP/pie)^(OMEGP/(1-OMEGP) )) )^((1-OMEGP)/OMEGP)*ptilde;
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pstar = 1/(1+PSI)*vartheta^(OMEGP/(OMEGP-1))*delta1^(OMEGP/(1-OMEGP)) + PSI/(1+PSI);
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f_o_a = ( vartheta^(OMEGP/(OMEGP-1)) / (1-BETA*XIP*(GAMMAP/pie)^(1/(1-OMEGP))) ) / ( PSI*(OMEGP-1) /(1-BETA*XIP*GAMMAP/pie) );
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s_o_a = ( OMEGP*vartheta^(OMEGP/(OMEGP-1)) / (1-BETA*XIP*(GAMMAP/pie)^(OMEGP/(1-OMEGP))) ) / ( PSI*(OMEGP-1) /(1-BETA*XIP*GAMMAP/pie) );
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mc = (f_o_a*ptilde-ptilde^(1+OMEGP/(OMEGP-1)))*s_o_a^(-1);
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k_o_n = (RK/(mc*ALPHA))^(1/(ALPHA-1));
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w = (1-ALPHA)*mc*k_o_n^ALPHA; % labor demand
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y_o_n = pstar^(-1)*k_o_n^ALPHA; % production function
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iv_o_n = DELTA*k_o_n;
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iv_o_y = iv_o_n / y_o_n;
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g_o_y = GBAR_o_YBAR;
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c_o_n = y_o_n - iv_o_n - g_o_y; % market clearing
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% The labor level, closed-form solution for labor
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n = (c_o_n^(-1)*w)^(1/(CHI+1)) ;
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% Value of remaining variables
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c = c_o_n*n;
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rk = RK;
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iv = iv_o_n *n;
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k = k_o_n*n;
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y = y_o_n*n;
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lam = n^CHI/w;
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g = g_o_y*y;
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a = PSI*(OMEGP-1)*y*lam/(1-BETA*XIP*GAMMAP/pie);
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f = f_o_a*a;
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s = f*ptilde - a*ptilde^(1+OMEGP/(OMEGP-1));
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%flex price economy
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bpot = BBAR;
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taupot = TAUBAR;
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taun = TAUNBAR;
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rrpot = 1/BETA;
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RKpot = (1/BETA+DELTA-1)*QBAR; % foc k
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rkpot = RKpot;
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kpot_o_npot = (RKpot/(ALPHA/(1+THETAP)))^(1/(ALPHA-1));
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wpot = (1-ALPHA)/(1+THETAP)*kpot_o_npot^ALPHA; % labor demand
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ypot_o_npot = kpot_o_npot^ALPHA; % production function
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ivpot_o_npot = DELTA*kpot_o_npot;
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ivpot_o_ypot = ivpot_o_npot / ypot_o_npot;
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cpot_o_npot = ypot_o_npot - ivpot_o_npot - g_o_y; % market clearing
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% The labor level, closed-form solution for labor
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npot = (cpot_o_npot^(-1)*wpot)^(1/(CHI+1)) ;
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% Value of remaining variables
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cpot = cpot_o_npot*npot;
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ivpot = ivpot_o_npot *npot;
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kpot = kpot_o_npot*npot;
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ypot = ypot_o_npot*npot;
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end;
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% =========================================================================
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% Declare settings for shocks
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% =========================================================================
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shocks;
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var epsg = 1;
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var epsnu = 1;
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end;
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estimated_params;
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ALPHA, 0.3, normal_pdf, 0.3 , 0.1;
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BETA, 0.995, normal_pdf, 0.995, 0.1;
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PIEBAR, 1.005, normal_pdf, 1.005, 0.1;
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IOTA, 0.5, normal_pdf, 0.5, 0.1;
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CHI, 1/0.4, normal_pdf, 1/0.4, 0.1;
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THETAP, 0.1, normal_pdf, 0.1, 0.1;
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XIP, 2/3, normal_pdf, 2/3, 0.1;
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PSI, -12.2, normal_pdf, -12.2, 0.1;
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GAMMAPIE, 1.5, normal_pdf, 1.5, 0.1;
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GAMMAX, 0.125, normal_pdf, 0.125, 0.1;
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DELTA, 0, normal_pdf, 0, 0.1;
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QBAR, 1, normal_pdf, 1, 0.1;
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NUBAR, 0, normal_pdf, 0, 0.1;
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BBAR, 0, normal_pdf, 0, 0.1;
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GBAR_o_YBAR, 0, normal_pdf, 0, 0.1; %commenting this solves the analytic_derivation_mode=-2|-1 problem
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TAUBAR, 0, normal_pdf, 0, 0.1;
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TAUNBAR, 0, normal_pdf, 0, 0.1;
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PHI, 0.0101, normal_pdf, 0.0101, 0.1;
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RHONU, 0.95, normal_pdf, 0.95, 0.1;
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RHOG, 0.95, normal_pdf, 0.95, 0.1;
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SIGNU, 0.6/100, normal_pdf, 0.6/100, 0.1;
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SIGG, 0.6/100, normal_pdf, 0.6/100, 0.1;
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end;
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% =========================================================================
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% Computations
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% =========================================================================
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model_diagnostics;
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steady;
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resid;
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check;
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identification(order=1,no_identification_strength,analytic_derivation_mode= 0,ar=5); %works
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%identification(no_identification_strength,analytic_derivation_mode= 1,ar=5); %works, this takes the longest due to Kronecker products
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options_.dynatol.x = 1e-9;
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identification(order=1,no_identification_strength,analytic_derivation_mode=-1,ar=5); %works, but tolerance is way too small
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identification(order=1,no_identification_strength,analytic_derivation_mode=-2,ar=5); %works, but tolerance is way to small
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options_.dynatol.x = 1e-5; %this is the default value
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identification(order=1,no_identification_strength,analytic_derivation_mode=-1,ar=5); %works only if GBAR_o_YBAR is uncommented
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identification(order=1,no_identification_strength,analytic_derivation_mode=-2,ar=5); %works only if GBAR_o_YBAR is uncommented
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