@@ -11,16 +11,17 @@ this key feature of the data is because the exogenous productivity (and
1111government spending shocks) are assumed to be highly persistent. Such large and
1212persistent shocks to productivity/technology are hard to identify in the data.
1313
14- Model suffers from other issues common to RBC models, for examples the model has
14+ Model suffers from other issues common to RBC models, for example the model has
1515no room for involuntary unemployment. In this model, workers choose to be
1616unemployed (or under employed!): workers optimally adjust their labor supply
1717across time in response to changes in the real wage (which in turn are driven by
1818exogenous productivity/technology shocks).
1919
2020Note that there are two sources of growth in Romer's RBC: technology growth and
2121population growth. Thus we will need to detrend variables accordingly in order
22- to have a stationary model. Below I work with per effective worker variables
23- unless otherwise noted.
22+ to have a stationary model. All variables, with the exception of l(t), are
23+ expressed in per effective worker terms. The labor supply, l(t), is in per
24+ capita terms.
2425
2526%Model Information++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
2627Name = Romer's RBC model;
@@ -31,8 +32,8 @@ Name = Romer's RBC model;
3132# discount rate
3233rho = 0.01;
3334
34- # weight (relative to consumption) that worker places on utility from leisure
35- b = 2.5 ;
35+ # Calibrated steady state labour, in the SS solver the parameter b now gets determined
36+ l_bar = 2.0 / 3.0 ;
3637
3738# Household size (normalized to unity)
3839H = 1.0;
@@ -47,11 +48,12 @@ n = 0.0025;
4748delta = 0.025;
4849
4950# capital's share of output
50- alpha = 1.0/ 3.0;
51+ alpha = 1.0 / 3.0;
5152
5253# government spending per effective worker is chosen so that in SS government
5354# spending roughly is 20% of output (which matches U.S. data)
54- gov = 0.5774;
55+ # g_bar now gets determined residually in the SS solver
56+ gov_share = 0.20;
5557
5658# persistence of technology shocks
5759rho_A = 0.95;
@@ -65,66 +67,91 @@ sigma_A = 0.011;
6567# standard deviation of government spending shocks
6668sigma_G = 0.011;
6769
70+ # Various easily computed steady-state values
6871z_A_bar = 1.0;
6972z_G_bar = 1.0;
70- A_bar = 1.0;
71- G_bar = 1.0;
72- g_bar = gov;
7373
7474%Variable Vectors+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
75- # Per capita consumption, c, and labor_supply, l, are the control variables ; the
75+ # Only endogenous state variable is capital per effective worker, k ; the
7676# two exogenous variables are technology, A, and government spending, G. Both
7777# are driven by AR(1) processes z_A and z_G, respectively.
7878
79- [1] y(t):output{con}[log,hp]
80- [2] k(t):physical_capital{endo}[log,hp]
81- [3] L(t):leisure{con}[log,hp]
82- [4] N(t):labur{con}[log,hp]
83- [4] c(t):consumption{con}[log,hp]
84- [5] i(t):investment{con}[log,hp]
85- [6] w(t):real_wage{con}[log,hp]
86- [7] R(t):gross_interest_rate{con}[log,hp]
87- [8] A(t):technology{con}[log,hp]
88- [9] g(t):gov_spending{con}[log,hp]
89- [10] z_A(t):eps_A(t):tech_shocks{exo}[log,hp]
90- [11] z_G(t):eps_G(t):gov_shocks{exo}[log,hp]
79+ [0] y(t):output{con}[log,cf]
80+ [1] k(t):physical_capital{endo}[log,cf]
81+ [2] l(t):labor{con}[log,cf]
82+ [3] c(t):consumption{con}[log,cf]
83+ [4] i(t):investment{con}[log,cf]
84+ [5] w(t):real_wage{con}[log]
85+ [6] R(t):gross_interest_rate{con}
86+ [7] g(t):gov_spending{con}[log,cf]
87+ [8] z_A(t):eps_A(t):tech_shocks{exo}[log,cf]
88+ [9] z_G(t):eps_G(t):gov_shocks{exo}[log,cf]
9189
9290%Boundary Conditions++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
9391None
9492
9593%Variable Substitution Non-Linear System++++++++++++++++++++++++++++++++++++++++
96- # Following Romer, I work with per effective worker variables.
97- None
94+ [0] @U(t) = LOG(c(t)) + b*LOG(1-l(t));
95+ [1] @Uc(t) = DIFF{@U(t),c(t)};
96+ [2] @Uc(t+1) = FF_1{@Uc(t)};
97+ [10] @F(t) = k(t-1)**alpha;
98+ [11] @Fk(t) = alpha * k(t-1)**(alpha - 1);
99+ [12] @Fl(t) = z_A(t) * (1 - alpha) * k(t-1)**alpha;
98100
99101%Non-Linear First-Order Conditions++++++++++++++++++++++++++++++++++++++++++++++
100- [1] EXP(n + g) * k(t) - ((1 - delta) * k(t-1) + i(t)) = 0;
101- [2] y(t) - (c(t) + i(t) + g(t)) = 0;
102- [3] y(t) - k(t-1)**alpha = 0;
103- [4] 1 - L(t) - N(t) = 0;
104- [4] w(t) - (1 - alpha) * k(t-1)**alpha = 0;
105- [5] R(t) - (alpha * k(t-1)**(alpha - 1) + (1 - delta)) = 0;
106- [6] (1 / c(t)) - EXP(-(rho + g)) * (1 / E(t)|c(t+1)) * E(t)|R(t+1) = 0;
107- [7] b * c(t) * L(t) - (1 - L(t))*w(t) = 0;
108- [8] E(t)|A(t+1) - EXP(g + E(t)|z_A(t+1)) * A(t) = 0;
109- [9] E(t)|g(t+1) - EXP(n + g + E(t)|z_G(t+1)) * g(t) = 0;
110- [10] E(t)|z_A(t+1) - rho_A * z_A(t) = 0;
111- [11] E(t)|z_G(t+1) - rho_G * z_G(t) = 0;
102+ # Evolution of physical capital
103+ [0] (E(t)|z_A(t+1) * E(t)|l(t+1))*k(t) * EXP(n + g) - (z_A(t) * l(t))*((1 - delta) * k(t-1) + i(t)) = 0;
104+
105+ # Aggregate resource constraint
106+ [1] (z_A(t) * l(t))*(y(t) - c(t) - i(t) - g(t)) = 0;
107+
108+ # production process
109+ [2] y(t)*(z_A(t) * l(t)) - w(t)*l(t) - (R(t)-1.0)*k(t-1)*(z_A(t) * l(t)) = 0;
110+
111+ # labor is paid its marginal product
112+ [3] w(t)/(z_A(t) * l(t)) - @Fl(t) = 0;
113+
114+ # capital is paid its marginal product
115+ [4] R(t) - (1 + @Fk(t) - delta) = 0;
116+
117+ # consumption Euler equation
118+ [5] 1 - EXP(-(rho + g)) * ((z_A(t)*l(t)*c(t)) / (E(t)|z_A(t+1)*E(t)|l(t+1)*E(t)|c(t+1))) * E(t)|R(t+1) = 0;
119+
120+ # intra-temporal consumption/labor supply trade-off
121+ [6] b * c(t) * (z_A(t) * l(t)) - w(t)*(1 - l(t)) = 0;
122+
123+ # evolution of government spending (per effective worker!)
124+ [7] LOG((E(t)|g(t+1)* (E(t)|z_A(t+1) * E(t)|l(t+1)))/(g(t)* (z_A(t) * l(t)))) - (n+g) - (E(t)|z_G(t+1)/z_G(t)) = 0;
125+
126+ # technology shocks
127+ [8] LOG(E(t)|z_A(t+1)) - rho_A * LOG(z_A(t)) = 0;
128+
129+ # government spending shocks
130+ [9] LOG(E(t)|z_G(t+1)) - rho_G * LOG(z_G(t)) = 0;
112131
113132%Steady States [Closed form]++++++++++++++++++++++++++++++++++++++++++++++++++++
114133None
115134
116-
135+
117136%Steady State Non-Linear System [Manual]++++++++++++++++++++++++++++++++++++++++
118- USE_FOCS=[0,1,2,3,4,5,6,7];
119-
120- [1] k_bar = 35.0;
121- [2] y_bar = k_bar**alpha;
122- [3] w_bar = (1-alpha)*k_bar**alpha;
123- [4] R_bar = (alpha*k_bar**(alpha-1)+(1- delta));
124- [5] L_bar = 0.7;
125- [6] N_bar = 0.3;
126- [7] c_bar = 2.0;
127- [8] i_bar = y_bar-c_bar-g_bar;
137+ [0] (EXP(n + g) - 1 + delta) * k_bar - i_bar = 0;
138+ [1] y_bar - c_bar - i_bar - g_bar = 0;
139+ [2] y_bar - k_bar**alpha = 0;
140+ [3] w_bar - (1 - alpha) * k_bar**alpha = 0;
141+ [4] R_bar - (1 + alpha * k_bar**(alpha - 1) - delta) = 0;
142+ [5] 1 - EXP(-(rho + g)) * R_bar = 0;
143+ [6] b * c_bar * l_bar - (1 - l_bar) * w_bar = 0;
144+ [7] g_bar - y_bar * gov_share = 0;
145+
146+ # initial values for numerical steady-state computation
147+ [0] k_bar = 20.0;
148+ [1] y_bar = k_bar**alpha;
149+ [2] w_bar = (1 - alpha) * k_bar**alpha;
150+ [3] R_bar = 1 + alpha * k_bar**(alpha - 1) - delta;
151+ [4] b = 1.0;
152+ [5] c_bar = 2.0;
153+ [6] g_bar = y_bar * gov_share;
154+ [7] i_bar = y_bar - c_bar - g_bar;
128155
129156%Log-Linearized Model Equations+++++++++++++++++++++++++++++++++++++++++++++++++
130157None
@@ -134,3 +161,4 @@ Sigma = [sigma_A**2 0;
134161 0 sigma_G**2];
135162
136163%End Of Model File++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
164+
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