Generated on 2023-07-10 01:36:03 by gEcon ver. 1.2.2 (2023-07-10) http://gecon.r-forge.r-project.org/ Model name: tc_rs Variables selected for reduction: pi[], pi_ast[], PI_ast[], PI[], T_ast[], K_d[], H_d[], T[], K_d_ast[], H_d_ast[] Block: CONSUMER Definitions: u[] = (1 - eta)^-1 * (C[]^mu * (1 - H[])^(1 - mu))^(1 - eta) Controls: K[], C[], H[], I[] Objective: U[] = beta * E[][U[1]] + (1 - eta)^-1 * (C[]^mu * (1 - H[])^(1 - mu))^(1 - eta) Constraints: pi[] - C[] - I[] - T[] + TR[] + K[-1] * r[] + H[] * W[] - psi * K[-1] * (-delta + K[-1]^-1 * I[])^2 = 0 (lambda_c[]) I[] - K[] + K[-1] * (1 - delta) = 0 (lambda__CONSUMER_2[]) First order conditions: -lambda__CONSUMER_2[] + beta * ((1 - delta) * E[][lambda__CONSUMER_2[1]] + E[][lambda_c[1] * (r[1] - psi * (-delta + K[]^-1 * I[1])^2 + 2 * psi * K[]^-1 * I[1] * (-delta + K[]^-1 * I[1]))]) = 0 (K[]) -lambda_c[] + mu * C[]^(-1 + mu) * (1 - H[])^(1 - mu) * (C[]^mu * (1 - H[])^(1 - mu))^(-eta) = 0 (C[]) lambda_c[] * W[] + (-1 + mu) * C[]^mu * (1 - H[])^(-mu) * (C[]^mu * (1 - H[])^(1 - mu))^(-eta) = 0 (H[]) lambda__CONSUMER_2[] + lambda_c[] * (-1 - 2 * psi * (-delta + K[-1]^-1 * I[])) = 0 (I[]) Block: FIRM Controls: K_d[], H_d[], Y[], pi[] Objective: PI[] = pi[] Constraints: -Y[] + Z[] * H_d[]^(1 - alpha) * K_d[]^alpha = 0 (lambda__FIRM_1[]) -pi[] + Y[] - r[] * K_d[] - H_d[] * W[] = 0 (lambda__FIRM_2[]) First order conditions: -lambda__FIRM_2[] * r[] + alpha * lambda__FIRM_1[] * Z[] * H_d[]^(1 - alpha) * K_d[]^(-1 + alpha) = 0 (K_d[]) -lambda__FIRM_2[] * W[] + lambda__FIRM_1[] * Z[] * (1 - alpha) * H_d[]^(-alpha) * K_d[]^alpha = 0 (H_d[]) -lambda__FIRM_1[] + lambda__FIRM_2[] = 0 (Y[]) 1 - lambda__FIRM_2[] = 0 (pi[]) First order conditions after reduction: -r[] + alpha * Z[] * H_d[]^(1 - alpha) * K_d[]^(-1 + alpha) = 0 (K_d[]) -W[] + Z[] * (1 - alpha) * H_d[]^(-alpha) * K_d[]^alpha = 0 (H_d[]) Block: CONSUMER_ast Definitions: u_ast[] = (1 - eta)^-1 * (C_ast[]^mu * (1 - H_ast[])^(1 - mu))^(1 - eta) Controls: K_ast[], C_ast[], H_ast[], I_ast[] Objective: U_ast[] = beta * E[][U_ast[1]] + (1 - eta)^-1 * (C_ast[]^mu * (1 - H_ast[])^(1 - mu))^(1 - eta) Constraints: pi_ast[] - C_ast[] - I_ast[] - T_ast[] - TR[] + K_ast[-1] * r_ast[] + H_ast[] * W_ast[] - psi * K_ast[-1] * (-delta + K_ast[-1]^-1 * I_ast[])^2 = 0 (lambda_c_ast[]) I_ast[] - K_ast[] + K_ast[-1] * (1 - delta) = 0 (lambda__CONSUMER_ast_2[]) First order conditions: -lambda__CONSUMER_ast_2[] + beta * ((1 - delta) * E[][lambda__CONSUMER_ast_2[1]] + E[][lambda_c_ast[1] * (r_ast[1] - psi * (-delta + K_ast[]^-1 * I_ast[1])^2 + 2 * psi * K_ast[]^-1 * I_ast[1] * (-delta + K_ast[]^-1 * I_ast[1]))]) = 0 (K_ast[]) -lambda_c_ast[] + mu * C_ast[]^(-1 + mu) * (1 - H_ast[])^(1 - mu) * (C_ast[]^mu * (1 - H_ast[])^(1 - mu))^(-eta) = 0 (C_ast[]) lambda_c_ast[] * W_ast[] + (-1 + mu) * C_ast[]^mu * (1 - H_ast[])^(-mu) * (C_ast[]^mu * (1 - H_ast[])^(1 - mu))^(-eta) = 0 (H_ast[]) lambda__CONSUMER_ast_2[] + lambda_c_ast[] * (-1 - 2 * psi * (-delta + K_ast[-1]^-1 * I_ast[])) = 0 (I_ast[]) Block: FIRM_ast Controls: K_d_ast[], H_d_ast[], Y_ast[], pi_ast[] Objective: PI_ast[] = pi_ast[] Constraints: -Y_ast[] + Z_ast[] * H_d_ast[]^(1 - alpha) * K_d_ast[]^alpha = 0 (lambda__FIRM_ast_1[]) -pi_ast[] + Y_ast[] - r_ast[] * K_d_ast[] - H_d_ast[] * W_ast[] = 0 (lambda__FIRM_ast_2[]) First order conditions: -lambda__FIRM_ast_2[] * r_ast[] + alpha * lambda__FIRM_ast_1[] * Z_ast[] * H_d_ast[]^(1 - alpha) * K_d_ast[]^(-1 + alpha) = 0 (K_d_ast[]) -lambda__FIRM_ast_2[] * W_ast[] + lambda__FIRM_ast_1[] * Z_ast[] * (1 - alpha) * H_d_ast[]^(-alpha) * K_d_ast[]^alpha = 0 (H_d_ast[]) -lambda__FIRM_ast_1[] + lambda__FIRM_ast_2[] = 0 (Y_ast[]) 1 - lambda__FIRM_ast_2[] = 0 (pi_ast[]) First order conditions after reduction: -r_ast[] + alpha * Z_ast[] * H_d_ast[]^(1 - alpha) * K_d_ast[]^(-1 + alpha) = 0 (K_d_ast[]) -W_ast[] + Z_ast[] * (1 - alpha) * H_d_ast[]^(-alpha) * K_d_ast[]^alpha = 0 (H_d_ast[]) Block: EQUILIBRIUM Identities: K[-1] - K_d[] = 0 H[] - H_d[] = 0 G_d[] - T[] = 0 K_ast[-1] - K_d_ast[] = 0 -H_d_ast[] + H_ast[] = 0 G_d_ast[] - T_ast[] = 0 -lambda_c[] + lambda_c_ast[] = 0 Block: EXOG Identities: epsilon_G[] - G_d[] + phi_G * G_d[-1] = 0 -Z[] + exp(epsilon_Z[] + phi_Z * log(Z[-1])) = 0 epsilon_G_ast[] - G_d_ast[] + phi_G * G_d_ast[-1] = 0 -Z_ast[] + exp(epsilon_Z_ast[] + phi_Z * log(Z_ast[-1])) = 0 Variables (23): lambda_c[], lambda_c_ast[], r[], r_ast[], C[], C_ast[], G_d[], G_d_ast[], H[], H_ast[], I[], I_ast[], K[], K_ast[], TR[], U[], U_ast[], W[], W_ast[], Y[], Y_ast[], Z[], Z_ast[] Shocks (4): epsilon_Z[], epsilon_G[], epsilon_G_ast[], epsilon_Z_ast[] Parameters (8): alpha, beta, delta, eta, mu, phi_G, phi_Z, psi Free parameters (8): alpha, beta, delta, eta, mu, phi_G, phi_Z, psi Equations (23): (1) -lambda_c[] + lambda_c_ast[] = 0 (2) -lambda_c[] + mu * C[]^(-1 + mu) * (1 - H[])^(1 - mu) * (C[]^mu * (1 - H[])^(1 - mu))^(-eta) = 0 (3) -lambda_c_ast[] + mu * C_ast[]^(-1 + mu) * (1 - H_ast[])^(1 - mu) * (C_ast[]^mu * (1 - H_ast[])^(1 - mu))^(-eta) = 0 (4) -r[] + alpha * Z[] * K[-1]^(-1 + alpha) * H[]^(1 - alpha) = 0 (5) -r_ast[] + alpha * Z_ast[] * K_ast[-1]^(-1 + alpha) * H_ast[]^(1 - alpha) = 0 (6) -W[] + Z[] * (1 - alpha) * K[-1]^alpha * H[]^(-alpha) = 0 (7) -W_ast[] + Z_ast[] * (1 - alpha) * K_ast[-1]^alpha * H_ast[]^(-alpha) = 0 (8) -Y[] + Z[] * K[-1]^alpha * H[]^(1 - alpha) = 0 (9) -Y_ast[] + Z_ast[] * K_ast[-1]^alpha * H_ast[]^(1 - alpha) = 0 (10) -Z[] + exp(epsilon_Z[] + phi_Z * log(Z[-1])) = 0 (11) -Z_ast[] + exp(epsilon_Z_ast[] + phi_Z * log(Z_ast[-1])) = 0 (12) beta * (-(1 - delta) * E[][lambda_c[1] * (-1 - 2 * psi * (-delta + K[]^-1 * I[1]))] + E[][lambda_c[1] * (r[1] - psi * (-delta + K[]^-1 * I[1])^2 + 2 * psi * K[]^-1 * I[1] * (-delta + K[]^-1 * I[1]))]) + lambda_c[] * (-1 - 2 * psi * (-delta + K[-1]^-1 * I[])) = 0 (13) beta * (-(1 - delta) * E[][lambda_c_ast[1] * (-1 - 2 * psi * (-delta + K_ast[]^-1 * I_ast[1]))] + E[][lambda_c_ast[1] * (r_ast[1] - psi * (-delta + K_ast[]^-1 * I_ast[1])^2 + 2 * psi * K_ast[]^-1 * I_ast[1] * (-delta + K_ast[]^-1 * I_ast[1]))]) + lambda_c_ast[] * (-1 - 2 * psi * (-delta + K_ast[-1]^-1 * I_ast[])) = 0 (14) lambda_c[] * W[] + (-1 + mu) * C[]^mu * (1 - H[])^(-mu) * (C[]^mu * (1 - H[])^(1 - mu))^(-eta) = 0 (15) lambda_c_ast[] * W_ast[] + (-1 + mu) * C_ast[]^mu * (1 - H_ast[])^(-mu) * (C_ast[]^mu * (1 - H_ast[])^(1 - mu))^(-eta) = 0 (16) epsilon_G[] - G_d[] + phi_G * G_d[-1] = 0 (17) epsilon_G_ast[] - G_d_ast[] + phi_G * G_d_ast[-1] = 0 (18) I[] - K[] + K[-1] * (1 - delta) = 0 (19) I_ast[] - K_ast[] + K_ast[-1] * (1 - delta) = 0 (20) U[] - beta * E[][U[1]] - (1 - eta)^-1 * (C[]^mu * (1 - H[])^(1 - mu))^(1 - eta) = 0 (21) U_ast[] - beta * E[][U_ast[1]] - (1 - eta)^-1 * (C_ast[]^mu * (1 - H_ast[])^(1 - mu))^(1 - eta) = 0 (22) -C[] - G_d[] - I[] + TR[] + Y[] - psi * K[-1] * (-delta + K[-1]^-1 * I[])^2 = 0 (23) -C_ast[] - G_d_ast[] - I_ast[] - TR[] + Y_ast[] - psi * K_ast[-1] * (-delta + K_ast[-1]^-1 * I_ast[])^2 = 0 Steady state equations (23): (1) -lambda_c[ss] + lambda_c_ast[ss] = 0 (2) -lambda_c[ss] + mu * C[ss]^(-1 + mu) * (1 - H[ss])^(1 - mu) * (C[ss]^mu * (1 - H[ss])^(1 - mu))^(-eta) = 0 (3) -lambda_c_ast[ss] + mu * C_ast[ss]^(-1 + mu) * (1 - H_ast[ss])^(1 - mu) * (C_ast[ss]^mu * (1 - H_ast[ss])^(1 - mu))^(-eta) = 0 (4) -r[ss] + alpha * Z[ss] * H[ss]^(1 - alpha) * K[ss]^(-1 + alpha) = 0 (5) -r_ast[ss] + alpha * Z_ast[ss] * H_ast[ss]^(1 - alpha) * K_ast[ss]^(-1 + alpha) = 0 (6) -W[ss] + Z[ss] * (1 - alpha) * H[ss]^(-alpha) * K[ss]^alpha = 0 (7) -W_ast[ss] + Z_ast[ss] * (1 - alpha) * H_ast[ss]^(-alpha) * K_ast[ss]^alpha = 0 (8) -Y[ss] + Z[ss] * H[ss]^(1 - alpha) * K[ss]^alpha = 0 (9) -Y_ast[ss] + Z_ast[ss] * H_ast[ss]^(1 - alpha) * K_ast[ss]^alpha = 0 (10) -Z[ss] + exp(phi_Z * log(Z[ss])) = 0 (11) -Z_ast[ss] + exp(phi_Z * log(Z_ast[ss])) = 0 (12) beta * (lambda_c[ss] * (r[ss] - psi * (-delta + I[ss] * K[ss]^-1)^2 + 2 * psi * I[ss] * K[ss]^-1 * (-delta + I[ss] * K[ss]^-1)) - lambda_c[ss] * (-1 - 2 * psi * (-delta + I[ss] * K[ss]^-1)) * (1 - delta)) + lambda_c[ss] * (-1 - 2 * psi * (-delta + I[ss] * K[ss]^-1)) = 0 (13) beta * (lambda_c_ast[ss] * (r_ast[ss] - psi * (-delta + I_ast[ss] * K_ast[ss]^-1)^2 + 2 * psi * I_ast[ss] * K_ast[ss]^-1 * (-delta + I_ast[ss] * K_ast[ss]^-1)) - lambda_c_ast[ss] * (-1 - 2 * psi * (-delta + I_ast[ss] * K_ast[ss]^-1)) * (1 - delta)) + lambda_c_ast[ss] * (-1 - 2 * psi * (-delta + I_ast[ss] * K_ast[ss]^-1)) = 0 (14) lambda_c[ss] * W[ss] + (-1 + mu) * C[ss]^mu * (1 - H[ss])^(-mu) * (C[ss]^mu * (1 - H[ss])^(1 - mu))^(-eta) = 0 (15) lambda_c_ast[ss] * W_ast[ss] + (-1 + mu) * C_ast[ss]^mu * (1 - H_ast[ss])^(-mu) * (C_ast[ss]^mu * (1 - H_ast[ss])^(1 - mu))^(-eta) = 0 (16) -G_d[ss] + phi_G * G_d[ss] = 0 (17) -G_d_ast[ss] + phi_G * G_d_ast[ss] = 0 (18) I[ss] - K[ss] + K[ss] * (1 - delta) = 0 (19) I_ast[ss] - K_ast[ss] + K_ast[ss] * (1 - delta) = 0 (20) U[ss] - beta * U[ss] - (1 - eta)^-1 * (C[ss]^mu * (1 - H[ss])^(1 - mu))^(1 - eta) = 0 (21) U_ast[ss] - beta * U_ast[ss] - (1 - eta)^-1 * (C_ast[ss]^mu * (1 - H_ast[ss])^(1 - mu))^(1 - eta) = 0 (22) -C[ss] - G_d[ss] - I[ss] + TR[ss] + Y[ss] - psi * K[ss] * (-delta + I[ss] * K[ss]^-1)^2 = 0 (23) -C_ast[ss] - G_d_ast[ss] - I_ast[ss] - TR[ss] + Y_ast[ss] - psi * K_ast[ss] * (-delta + I_ast[ss] * K_ast[ss]^-1)^2 = 0 Parameter settings (8): (1) alpha = 0.4 (2) beta = 0.99 (3) delta = 0.025 (4) eta = 2 (5) mu = 0.3 (6) phi_G = 0.95 (7) phi_Z = 0.95 (8) psi = 0.8