How to compute the steady-state of the RBC model both analytically and numerically, with preprocessing tips.
This video is part of a series of videos on the baseline Real Business Cycle model and its implementation in Dynare. We focus on computing the steady-state both analytically and numerically. First, we derive the steady-state using pen and paper and then implement this using either an initval or steady_state_model block in Dynare. We also cover “helper functions” that introduce numerical optimization in an otherwise analytical steady_state_model block, in order to compute the steady-state for variables for which we cannot derive closed-form expressions by hand.
Theory
Dynare Implementation
Dynare Preprocessor
Outro & References
function l = rbc_steady_state_helper(L0, w,C_L,ETAC,ETAL,PSI,GAMMA)
options = optimset('Display','off','TolX',1e-10,'TolFun',1e-10);
l = fsolve(@(l) w*C_L^(-ETAC) - PSI/GAMMA*(1-l)^(-ETAL)*l^ETAC , L0,options);
end
@#define LOGUTILITY = 1
var
y ${Y}$ (long_name='output')
c ${C}$ (long_name='consumption')
k ${K}$ (long_name='capital')
l ${L}$ (long_name='labor')
a ${A}$ (long_name='productivity')
r ${R}$ (long_name='interest Rate')
w ${W}$ (long_name='wage')
iv ${I}$ (long_name='investment')
mc ${MC}$ (long_name='marginal Costs')
;
model_local_variable
uc ${U_t^C}$
ucp ${E_t U_{t+1}^C}$
ul ${U_t^L}$
fk ${f_t^K}$
fl ${f_t^L}$
;
varexo
epsa ${\varepsilon^A}$ (long_name='Productivity Shock')
;
parameters
BETA ${\beta}$ (long_name='Discount Factor')
DELTA ${\delta}$ (long_name='Depreciation Rate')
GAMMA ${\gamma}$ (long_name='Consumption Utility Weight')
PSI ${\psi}$ (long_name='Labor Disutility Weight')
@#if LOGUTILITY != 1
ETAC ${\eta^C}$ (long_name='Risk Aversion')
ETAL ${\eta^L}$ (long_name='Inverse Frisch Elasticity')
@#endif
ALPHA ${\alpha}$ (long_name='Output Elasticity of Capital')
RHOA ${\rho^A}$ (long_name='Discount Factor')
;
% Parameter calibration
ALPHA = 0.35;
BETA = 0.99;
DELTA = 0.025;
GAMMA = 1;
PSI = 1.6;
RHOA = 0.9;
@#if LOGUTILITY == 0
ETAC = 2;
ETAL = 1;
@#endif
model;
%marginal utility of consumption and labor
@#if LOGUTILITY == 1
#uc = GAMMA*c^(-1);
#ucp = GAMMA*c(+1)^(-1);
#ul = -PSI*(1-l)^(-1);
@#else
#uc = GAMMA*c^(-ETAC);
#ucp = GAMMA*c(+1)^(-ETAC);
#ul = -PSI*(1-l)^(-ETAL);
@#endif
%marginal products of production
#fk = ALPHA*y/k(-1);
#fl = (1-ALPHA)*y/l;
[name='intertemporal optimality (Euler)']
uc = BETA*ucp*(1-DELTA+r(+1));
[name='labor supply']
w = -ul/uc;
[name='capital accumulation']
k = (1-DELTA)*k(-1) + iv;
[name='market clearing']
y = c + iv;
[name='production function']
y = a*k(-1)^ALPHA*l^(1-ALPHA);
[name='marginal costs']
mc = 1;
[name='labor demand']
w = mc*fl;
[name='capital demand']
r = mc*fk;
[name='total factor productivity']
log(a) = RHOA*log(a(-1)) + epsa;
end;
% ------------------------ %
% Steady State Computation %
% ------------------------ %
@#define AnalyticalSteadyState = 1
@#if AnalyticalSteadyState == 1
steady_state_model;
a = 1;
mc = 1;
r = 1/BETA + DELTA -1;
K_L = (mc*ALPHA*a/r)^(1/(1-ALPHA));
w = mc*(1-ALPHA)*a*K_L^ALPHA;
IV_L = DELTA*K_L;
Y_L = a*(K_L)^ALPHA;
C_L = Y_L - IV_L;
@#if LOGUTILITY==1
l = GAMMA/PSI*C_L^(-1)*w/(1+GAMMA/PSI*C_L^(-1)*w);
@#else
L0 = 1/3;
l = rbc_steady_state_helper(L0, w,C_L,ETAC,ETAL,PSI,GAMMA);
@#endif
c = C_L*l;
y = Y_L*l;
iv = IV_L*l;
k = K_L*l;
end;
@#else
initval;
a = 1;
mc = 1;
r = 0.03;
l = 1/3;
y = 1.2;
c = 0.9;
iv = 0.35;
k = 12;
w = 2.25;
end;
@#endif
steady;
write_latex_definitions;
write_latex_parameter_table;
write_latex_original_model;
%write_latex_dynamic_model;
write_latex_static_model;
write_latex_steady_state_model;
collect_latex_files;
if system(['/Library/TeX/texbin/pdflatex -halt-on-error -interaction=batchmode ' M_.fname '_TeX_binder.tex'])
warning('TeX-File did not compile; you need to compile it manually')
end