def Euler_equation_solver(guesses, r, w, T_H, factor, j, params, chi_b, chi_n, tau_bq, rho, lambdas, weights, e):
J, S, T, beta, sigma, alpha, Z, delta, ltilde, nu, g_y, tau_payroll, retire, mean_income_data, a_tax_income, b_tax_income, c_tax_income, d_tax_income, h_wealth, p_wealth, m_wealth, b_ellipse, upsilon = params
b_guess = np.array(guesses[:S])
n_guess = np.array(guesses[S:])
b_s = np.array([0] + list(b_guess[:-1]))
b_splus1 = b_guess
b_splus2 = np.array(list(b_guess[1:]) + [0])
BQ = (1+r) * (b_guess * weights[:, j] * rho).sum()
theta = tax.replacement_rate_vals(n_guess, w, factor, e[:,j], J, weights[:, j])
error1 = house.euler_savings_func(w, r, e[:, j], n_guess, b_s, b_splus1, b_splus2, BQ, factor, T_H, chi_b[j], params, theta, tau_bq[j], rho, lambdas[j])
error2 = house.euler_labor_leisure_func(w, r, e[:, j], n_guess, b_s, b_splus1, BQ, factor, T_H, chi_n, params, theta, tau_bq[j], lambdas[j])
# Put in constraints for consumption and savings. According to the euler equations, they can be negative. When
# Chi_b is large, they will be. This prevents that from happening.
# I'm not sure if the constraints are needed for labor. But we might as well put them in for now.
mask1 = n_guess < 0
mask2 = n_guess > ltilde
mask3 = b_guess <= 0
error2[mask1] += 1e14
error2[mask2] += 1e14
error1[mask3] += 1e14
tax1 = tax.total_taxes(r, b_s, w, e[:, j], n_guess, BQ, lambdas[j], factor, T_H, None, 'SS', False, params, theta, tau_bq[j])
cons = house.get_cons(r, b_s, w, e[:, j], n_guess, BQ, lambdas[j], b_splus1, params, tax1)
mask4 = cons < 0
error1[mask4] += 1e14
# print np.append(error1.flatten(), error2.flatten()).max()
return list(error1.flatten()) + list(error2.flatten())
def SS_TPI_firstdoughnutring(guesses, winit, rinit, BQinit, T_H_init):
# This function does not work. The tax functions need to be changed.
b2 = float(guesses[0])
n1 = float(guesses[1])
b1 = float(initial_b[-2, j])
# Euler 1 equations
tax11 = tax.total_taxes_eul3_TPI(rinit, b1, winit, e[-1, j], n1, BQinit, lambdas[j], factor_ss, T_H_init, j)
cons11 = house.get_cons(rinit, b1, winit, e[-1, j], n1, BQinit, lambdas[j], b2, g_y, tax11)
bequest_ut = rho * np.exp(-sigma * g_y) * chi_b[-1, j] * b2 ** (-sigma)
error1 = house.marg_ut_cons(cons11) - bequest_ut
# Euler 2 equations
tax2 = tax.total_taxes_eul3_TPI(rinit, b1, winit, e[-1, j], n1, BQinit, lambdas[j], factor_ss, T_H_init, j)
cons2 = house.get_cons(rinit, b1, winit, e[-1, j], n1, BQinit, lambdas[j], b2, g_y, tax2)
income2 = (rinit * b1 + winit * e[-1, j] * n1) * factor_ss
deriv2 = 1 - tau_payroll - tax.tau_income(rinit, b1, winit, e[
-1, j], n1, factor_ss) - tax.tau_income_deriv(
rinit, b1, winit, e[-1, j], n1, factor_ss) * income2
error2 = house.marg_ut_cons(cons2) * winit * e[-1, j] * deriv2 - house.marg_ut_labor(n1, chi_n[-1])
if n1 <= 0:
error2 += 1e12
return [error1] + [error2]
def Euler_equation_solver(guesses, r, w, T_H, factor, j, params, chi_b, chi_n, tau_bq, rho, lambdas, weights, e):
'''
Finds the euler error for certain b and n, one ability type at a time.
Inputs:
guesses = guesses for b and n (2Sx1 list)
r = rental rate (scalar)
w = wage rate (scalar)
T_H = lump sum tax (scalar)
factor = scaling factor to dollars (scalar)
j = which ability group is being solved for (scalar)
params = list of parameters (list)
chi_b = chi^b_j (scalar)
chi_n = chi^n_s (Sx1 array)
tau_bq = bequest tax rate (scalar)
rho = mortality rates (Sx1 array)
lambdas = ability weights (scalar)
weights = population weights (Sx1 array)
e = ability levels (Sx1 array)
Outputs:
2Sx1 list of euler errors
'''
J, S, T, beta, sigma, alpha, Z, delta, ltilde, nu, g_y, g_n_ss, tau_payroll, retire, mean_income_data, a_tax_income, b_tax_income, c_tax_income, d_tax_income, h_wealth, p_wealth, m_wealth, b_ellipse, upsilon = params
b_guess = np.array(guesses[:S])
n_guess = np.array(guesses[S:])
b_s = np.array([0] + list(b_guess[:-1]))
b_splus1 = b_guess
b_splus2 = np.array(list(b_guess[1:]) + [0])
BQ = house.get_BQ(r, b_splus1, weights, lambdas[j], rho, g_n_ss)
theta = tax.replacement_rate_vals(n_guess, w, factor, e[:,j], J, weights, lambdas[j])
error1 = house.euler_savings_func(w, r, e[:, j], n_guess, b_s, b_splus1, b_splus2, BQ, factor, T_H, chi_b[j], params, theta, tau_bq[j], rho, lambdas[j])
error2 = house.euler_labor_leisure_func(w, r, e[:, j], n_guess, b_s, b_splus1, BQ, factor, T_H, chi_n, params, theta, tau_bq[j], lambdas[j])
# Put in constraints for consumption and savings. According to the euler equations, they can be negative. When
# Chi_b is large, they will be. This prevents that from happening.
# I'm not sure if the constraints are needed for labor. But we might as well put them in for now.
mask1 = n_guess < 0
mask2 = n_guess > ltilde
mask3 = b_guess <= 0
error2[mask1] += 1e14
error2[mask2] += 1e14
error1[mask3] += 1e14
tax1 = tax.total_taxes(r, b_s, w, e[:, j], n_guess, BQ, lambdas[j], factor, T_H, None, 'SS', False, params, theta, tau_bq[j])
cons = house.get_cons(r, b_s, w, e[:, j], n_guess, BQ, lambdas[j], b_splus1, params, tax1)
mask4 = cons < 0
error1[mask4] += 1e14
return list(error1.flatten()) + list(error2.flatten())
def Steady_State_SS(guesses, chi_params, params, weights_SS, rho_vec, lambdas, theta, tau_bq, e):
'''
Parameters: Steady state distribution of capital guess as array
size 2*S*J
Returns: Array of 2*S*J Euler equation errors
'''
J, S, T, beta, sigma, alpha, Z, delta, ltilde, nu, g_y, tau_payroll, retire, mean_income_data, a_tax_income, b_tax_income, c_tax_income, d_tax_income, h_wealth, p_wealth, m_wealth, b_ellipse, upsilon = params
chi_b = np.tile(np.array(chi_params[:J]).reshape(1, J), (S, 1))
chi_n = np.array(chi_params[J:])
b_guess = guesses[0: S * J].reshape((S, J))
K = house.get_K(b_guess, weights_SS)
n_guess = guesses[S * J:-1].reshape((S, J))
L = firm.get_L(e, n_guess, weights_SS)
Y = firm.get_Y(K, L, params)
w = firm.get_w(Y, L, params)
r = firm.get_r(Y, K, params)
BQ = (1 + r) * (b_guess * weights_SS * rho_vec.reshape(S, 1)).sum(0)
b_s = np.array(list(np.zeros(J).reshape(1, J)) + list(b_guess[:-1, :]))
b_splus1 = b_guess
b_splus2 = np.array(list(b_guess[1:]) + list(np.zeros(J).reshape(1, J)))
factor = guesses[-1]
T_H = tax.get_lump_sum(r, b_s, w, e, n_guess, BQ, lambdas, factor, weights_SS, 'SS', params, theta, tau_bq)
error1 = house.euler_savings_func(w, r, e, n_guess, b_s, b_splus1, b_splus2, BQ.reshape(1, J), factor, T_H, chi_b, params, theta, tau_bq, rho_vec, lambdas)
error2 = house.euler_labor_leisure_func(w, r, e, n_guess, b_s, b_splus1, BQ.reshape(1, J), factor, T_H, chi_n, params, theta, tau_bq, lambdas)
average_income_model = ((r * b_s + w * e * n_guess) * weights_SS).sum()
error3 = [mean_income_data - factor * average_income_model]
# Check and punish constraint violations
mask1 = n_guess < 0
error2[mask1] += 1e9
mask2 = n_guess > ltilde
error2[mask2] += 1e9
if b_guess.sum() <= 0:
error1 += 1e9
tax1 = tax.total_taxes(r, b_s, w, e, n_guess, BQ, lambdas, factor, T_H, None, 'SS', False, params, theta, tau_bq)
cons = house.get_cons(r, b_s, w, e, n_guess, BQ.reshape(1, J), lambdas, b_splus1, params, tax1)
mask3 = cons < 0
error2[mask3] += 1e9
mask4 = b_guess[:-1] <= 0
error1[mask4] += 1e9
# print np.abs(np.array(list(error1.flatten()) + list(
# error2.flatten()) + error3)).max()
return list(error1.flatten()) + list(
error2.flatten()) + error3
def SS_TPI_firstdoughnutring(guesses, winit, rinit, BQinit, T_H_init, j):
'''
Solves the first entries of the upper triangle of the twist doughnut. This is
separate from the main TPI function because the the values of b and n are scalars,
so it is easier to just have a separate function for these cases.
Inputs:
guesses = guess for b and n (2x1 list)
winit = initial wage rate (scalar)
rinit = initial rental rate (scalar)
BQinit = initial aggregate bequest (scalar)
T_H_init = initial lump sum tax (scalar)
j = which ability type is being solved for (scalar)
Output:
euler errors (2x1 list)
'''
b2 = float(guesses[0])
n1 = float(guesses[1])
b1 = float(initial_b[-2, j])
# Euler 1 equations
tax1 = tax.total_taxes(rinit, b1, winit, e[-1, j], n1, BQinit, lambdas[j],
factor_ss, T_H_init, j, 'TPI_scalar', False,
parameters, theta, tau_bq)
cons1 = house.get_cons(rinit, b1, winit, e[-1, j], n1, BQinit, lambdas[j],
b2, parameters, tax1)
bequest_ut = rho[-1] * np.exp(-sigma * g_y) * chi_b[-1, j] * b2**(-sigma)
error1 = house.marg_ut_cons(cons1, parameters) - bequest_ut
# Euler 2 equations
income2 = (rinit * b1 + winit * e[-1, j] * n1) * factor_ss
deriv2 = 1 - tau_payroll - tax.tau_income(
rinit, b1, winit, e[-1, j],
n1, factor_ss, parameters) - tax.tau_income_deriv(
rinit, b1, winit, e[-1, j], n1, factor_ss, parameters) * income2
error2 = house.marg_ut_cons(
cons1, parameters) * winit * e[-1, j] * deriv2 - house.marg_ut_labor(
n1, chi_n[-1], parameters)
if n1 <= 0 or n1 >= 1:
error2 += 1e12
if b2 <= 0:
error1 += 1e12
if cons1 <= 0:
error1 += 1e12
return [error1] + [error2]
def wrguess(X, bssmat, nssmat, params, chi_b, chi_n, tau_bq, rho, lambdas, weights, e):
J, S, T, beta, sigma, alpha, Z, delta, ltilde, nu, g_y, tau_payroll, retire, mean_income_data, a_tax_income, b_tax_income, c_tax_income, d_tax_income, h_wealth, p_wealth, m_wealth, b_ellipse, upsilon = params
w, r, T_H, factor = X
for j in xrange(J):
# Solve the euler equations
guesses = np.append(bssmat[:, j], nssmat[:, j])
solutions = opt.fsolve(Euler_equation_solver, guesses * .9, args=(r, w, T_H, factor, j, params, chi_b, chi_n, tau_bq, rho, lambdas, weights, e), xtol=1e-13)
bssmat[:,j] = solutions[:S]
nssmat[:,j] = solutions[S:]
theta = tax.replacement_rate_vals(nssmat[:, j], w, factor, e[:, j], J, weights[:, j])
# print np.array(Euler_equation_solver(np.append(bssmat[:, j], nssmat[:, j]), r, w, T_H, factor, j, params, chi_b, chi_n, tau_bq, rho, lambdas, weights, e)).max()
#Calculate demand for C, I, and Y
BQ = (1+r) * (weights * rho.reshape(S,1)*bssmat).sum(0)
theta = tax.replacement_rate_vals(nssmat, w, factor, e, J, weights)
net_tax = tax.total_taxes(r, bssmat, w, e, nssmat, BQ, lambdas, factor, T_H, 0, 'SS', False, params, theta, tau_bq)
b_s = np.array(list(np.zeros(J).reshape(1,J)) + list(bssmat[:-1]))
cons = house.get_cons(r, b_s, w, e, nssmat, BQ, lambdas, bssmat, params, net_tax)
C = (weights*cons).sum()
B = house.get_K(bssmat, weights)
Y = C / (1-(delta*alpha/(r+delta)))
#Get demand for K and L
K_demand = alpha*Y / (r+delta)
L_demand = (1-alpha)*Y / w
#Get the 2 errors from difference between supply and demand of K and L
error1 = K_demand - B
error2 = L_demand - firm.get_L(e, nssmat, weights)
#Get other 2 errors from the factor equation and government constraint
error3 = T_H - tax.get_lump_sum(r, b_s, w, e, nssmat, BQ, lambdas, factor, weights, 'SS', params, theta, tau_bq)
# error3 = T_H - (weights*revenue).sum()
mean_income_model = ((r * b_s + w * e * nssmat) * weights).sum()
error4 = factor - mean_income_data / mean_income_model
error = [error1, error2, error3, error4]
# print error
print max([abs(error1), abs(error2), abs(error3), abs(error4)])
return error
def SS_TPI_firstdoughnutring(guesses, winit, rinit, BQinit, T_H_init):
b2 = float(guesses[0])
n1 = float(guesses[1])
b1 = float(initial_b[-2, j])
# Euler 1 equations
tax1 = tax.total_taxes(rinit, b1, winit, e[-1, j], n1, BQinit, lambdas[j], factor_ss, T_H_init, j, 'TPI_scalar', False, parameters, theta, tau_bq)
cons1 = house.get_cons(rinit, b1, winit, e[-1, j], n1, BQinit, lambdas[j], b2, parameters, tax1)
bequest_ut = rho[-1] * np.exp(-sigma * g_y) * chi_b[-1, j] * b2 ** (-sigma)
error1 = house.marg_ut_cons(cons1, parameters) - bequest_ut
# Euler 2 equations
income2 = (rinit * b1 + winit * e[-1, j] * n1) * factor_ss
deriv2 = 1 - tau_payroll - tax.tau_income(rinit, b1, winit, e[
-1, j], n1, factor_ss, parameters) - tax.tau_income_deriv(
rinit, b1, winit, e[-1, j], n1, factor_ss, parameters) * income2
error2 = house.marg_ut_cons(cons1, parameters) * winit * e[-1, j] * deriv2 - house.marg_ut_labor(n1, chi_n[-1], parameters)
if n1 <= 0 or n1 >= 1:
error2 += 1e12
if b2 <=0:
error1 += 1e12
if cons1 <= 0:
error1 += 1e12
return [error1] + [error2]
def Euler_equation_solver(guesses, r, w, T_H, factor, j, params, chi_b, chi_n, theta, tau_bq, rho, lambdas, weights):
b_guess = np.array(guesses[:S])
n_guess = np.array(guesses[S:])
b_s = np.array([0] + list(b_guess[:-1]))
b_splus1 = b_guess
b_splus2 = np.array(list(b_guess[1:]) + [0])
BQ = (1+r) * (b_guess * weights[:, j] * rho).sum()
error1 = house.euler_savings_func(w, r, e[:, j], n_guess, b_s, b_splus1, b_splus2, BQ, factor, T_H, chi_b[j], params, theta[j], tau_bq[j], rho, lambdas[j])
error2 = house.euler_labor_leisure_func(w, r, e[:, j], n_guess, b_s, b_splus1, BQ, factor, T_H, chi_n, params, theta[j], tau_bq[j], lambdas[j])
# Put in constraints
mask1 = n_guess < 0
mask2 = n_guess > ltilde
mask3 = b_guess <= 0
error2[mask1] += 1e9
error2[mask2] += 1e9
error1[mask3] += 1e9
tax1 = tax.total_taxes(r, b_s, w, e[:, j], n_guess, BQ, lambdas[j], factor, T_H, None, 'SS', False, params, theta[j], tau_bq[j])
cons = house.get_cons(r, b_s, w, e[:, j], n_guess, BQ, lambdas[j], b_splus1, params, tax1)
mask4 = cons < 0
error1[mask4] += 1e9
return list(error1.flatten()) + list(error2.flatten())
def SS_TPI_firstdoughnutring(guesses, winit, rinit, BQinit, T_H_init, j):
'''
Solves the first entries of the upper triangle of the twist doughnut. This is
separate from the main TPI function because the the values of b and n are scalars,
so it is easier to just have a separate function for these cases.
Inputs:
guesses = guess for b and n (2x1 list)
winit = initial wage rate (scalar)
rinit = initial rental rate (scalar)
BQinit = initial aggregate bequest (scalar)
T_H_init = initial lump sum tax (scalar)
j = which ability type is being solved for (scalar)
Output:
euler errors (2x1 list)
'''
b2 = float(guesses[0])
n1 = float(guesses[1])
b1 = float(initial_b[-2, j])
# Euler 1 equations
tax1 = tax.total_taxes(rinit, b1, winit, e[-1, j], n1, BQinit, lambdas[j], factor_ss, T_H_init, j, 'TPI_scalar', False, parameters, theta, tau_bq)
cons1 = house.get_cons(rinit, b1, winit, e[-1, j], n1, BQinit, lambdas[j], b2, parameters, tax1)
bequest_ut = rho[-1] * np.exp(-sigma * g_y) * chi_b[-1, j] * b2 ** (-sigma)
error1 = house.marg_ut_cons(cons1, parameters) - bequest_ut
# Euler 2 equations
income2 = (rinit * b1 + winit * e[-1, j] * n1) * factor_ss
deriv2 = 1 - tau_payroll - tax.tau_income(rinit, b1, winit, e[
-1, j], n1, factor_ss, parameters) - tax.tau_income_deriv(
rinit, b1, winit, e[-1, j], n1, factor_ss, parameters) * income2
error2 = house.marg_ut_cons(cons1, parameters) * winit * e[-1, j] * deriv2 - house.marg_ut_labor(n1, chi_n[-1], parameters)
if n1 <= 0 or n1 >= 1:
error2 += 1e12
if b2 <=0:
error1 += 1e12
if cons1 <= 0:
error1 += 1e12
return [error1] + [error2]
K0 = house.get_K(initial_b, omega_stationary[0].reshape(S, 1), lambdas,
g_n_vector[0])
b_sinit = np.array(list(np.zeros(J).reshape(1, J)) + list(initial_b[:-1]))
b_splus1init = initial_b
L0 = firm.get_L(e, initial_n, omega_stationary[0].reshape(S, 1), lambdas)
Y0 = firm.get_Y(K0, L0, parameters)
w0 = firm.get_w(Y0, L0, parameters)
r0 = firm.get_r(Y0, K0, parameters)
BQ0 = house.get_BQ(r0, initial_b, omega_stationary[0].reshape(S, 1), lambdas,
rho.reshape(S, 1), g_n_vector[0])
T_H_0 = tax.get_lump_sum(r0, b_sinit, w0, e, initial_n, BQ0, lambdas,
factor_ss, omega_stationary[0].reshape(S, 1), 'SS',
parameters, theta, tau_bq)
tax0 = tax.total_taxes(r0, b_sinit, w0, e, initial_n, BQ0, lambdas, factor_ss,
T_H_0, None, 'SS', False, parameters, theta, tau_bq)
c0 = house.get_cons(r0, b_sinit, w0, e, initial_n, BQ0.reshape(1, J),
lambdas.reshape(1, J), b_splus1init, parameters, tax0)
'''
------------------------------------------------------------------------
Solve for equilibrium transition path by TPI
------------------------------------------------------------------------
'''
def SS_TPI_firstdoughnutring(guesses, winit, rinit, BQinit, T_H_init, j):
'''
Solves the first entries of the upper triangle of the twist doughnut. This is
separate from the main TPI function because the the values of b and n are scalars,
so it is easier to just have a separate function for these cases.
Inputs:
guesses = guess for b and n (2x1 list)
winit = initial wage rate (scalar)
bq = solutions[(S-1)*J:S*J]
bssmat_s = np.array(list(np.zeros(J).reshape(1, J)) + list(bssmat))
bssmat_splus1 = np.array(list(bssmat) + list(bq.reshape(1, J)))
nssmat = solutions[S * J:2*S*J].reshape(S, J)
wss, rss, factor_ss, T_Hss = solutions[2*S*J:]
Kss = house.get_K(bssmat_splus1, omega_SS)
Lss = firm.get_L(e, nssmat, omega_SS)
Yss = firm.get_Y(Kss, Lss, parameters)
theta = tax.replacement_rate_vals(nssmat, wss, factor_ss, e, J, omega_SS)
BQss = (1+rss)*(np.array(list(bssmat) + list(bq.reshape(1, J))).reshape(
S, J) * omega_SS * rho.reshape(S, 1)).sum(0)
b_s = np.array(list(np.zeros(J).reshape((1, J))) + list(bssmat))
taxss = tax.total_taxes(rss, b_s, wss, e, nssmat, BQss, lambdas, factor_ss, T_Hss, None, 'SS', False, parameters, theta, tau_bq)
cssmat = house.get_cons(rss, b_s, wss, e, nssmat, BQss.reshape(1, J), lambdas.reshape(1, J), bssmat_splus1, parameters, taxss)
house.constraint_checker_SS(bssmat, nssmat, cssmat, parameters)
'''
------------------------------------------------------------------------
Generate variables for graphs
------------------------------------------------------------------------
b_s = SxJ array of bssmat in period t
b_splus1 = SxJ array of bssmat in period t+1
b_splus2 = SxJ array of bssmat in period t+2
euler_savings = euler errors from savings euler equation
euler_labor_leisure = euler errors from labor leisure euler equation
------------------------------------------------------------------------
'''
b_s = np.array(list(np.zeros(J).reshape((1, J))) + list(bssmat))
def Steady_state_TPI_solver(guesses, winit, rinit, BQinit, T_H_init, factor, j,
s, t, params, theta, tau_bq, rho, lambdas, e,
initial_b, chi_b, chi_n):
'''
Parameters:
guesses = distribution of capital and labor (various length list)
winit = wage rate ((T+S)x1 array)
rinit = rental rate ((T+S)x1 array)
BQinit = aggregate bequests ((T+S)x1 array)
T_H_init = lump sum tax over time ((T+S)x1 array)
factor = scaling factor (scalar)
j = which ability type is being solved for (scalar)
s = which upper triangle loop is being solved for (scalar)
t = which diagonal is being solved for (scalar)
params = list of parameters (list)
theta = replacement rates (Jx1 array)
tau_bq = bequest tax rate (Jx1 array)
rho = mortalit rate (Sx1 array)
lambdas = ability weights (Jx1 array)
e = ability type (SxJ array)
initial_b = capital stock distribution in period 0 (SxJ array)
chi_b = chi^b_j (Jx1 array)
chi_n = chi^n_s (Sx1 array)
Output:
Value of Euler error (various length list)
'''
J, S, T, beta, sigma, alpha, Z, delta, ltilde, nu, g_y, g_n_ss, tau_payroll, retire, mean_income_data, a_tax_income, b_tax_income, c_tax_income, d_tax_income, h_wealth, p_wealth, m_wealth, b_ellipse, upsilon = params
length = len(guesses) / 2
b_guess = np.array(guesses[:length])
n_guess = np.array(guesses[length:])
if length == S:
b_s = np.array([0] + list(b_guess[:-1]))
else:
b_s = np.array([(initial_b[-(s + 3), j])] + list(b_guess[:-1]))
b_splus1 = b_guess
b_splus2 = np.array(list(b_guess[1:]) + [0])
w_s = winit[t:t + length]
w_splus1 = winit[t + 1:t + length + 1]
r_s = rinit[t:t + length]
r_splus1 = rinit[t + 1:t + length + 1]
n_s = n_guess
n_extended = np.array(list(n_guess[1:]) + [0])
e_s = e[-length:, j]
e_extended = np.array(list(e[-length + 1:, j]) + [0])
BQ_s = BQinit[t:t + length]
BQ_splus1 = BQinit[t + 1:t + length + 1]
T_H_s = T_H_init[t:t + length]
T_H_splus1 = T_H_init[t + 1:t + length + 1]
# Savings euler equations
tax_s = tax.total_taxes(r_s, b_s, w_s, e_s, n_s, BQ_s, lambdas[j], factor,
T_H_s, j, 'TPI', False, params, theta, tau_bq)
tax_splus1 = tax.total_taxes(r_splus1, b_splus1, w_splus1, e_extended,
n_extended, BQ_splus1, lambdas[j], factor,
T_H_splus1, j, 'TPI', True, params, theta,
tau_bq)
cons_s = house.get_cons(r_s, b_s, w_s, e_s, n_s, BQ_s, lambdas[j],
b_splus1, params, tax_s)
cons_splus1 = house.get_cons(r_splus1, b_splus1, w_splus1, e_extended,
n_extended, BQ_splus1, lambdas[j], b_splus2,
params, tax_splus1)
income_splus1 = (r_splus1 * b_splus1 +
w_splus1 * e_extended * n_extended) * factor
savings_ut = rho[-(length):] * np.exp(
-sigma * g_y) * chi_b[-(length):, j] * b_splus1**(-sigma)
deriv_savings = 1 + r_splus1 * (
1 - tax.tau_income(r_splus1, b_splus1, w_splus1, e_extended,
n_extended, factor, params) -
tax.tau_income_deriv(r_splus1, b_splus1, w_splus1, e_extended,
n_extended, factor, params) * income_splus1
) - tax.tau_w_prime(b_splus1, params) * b_splus1 - tax.tau_wealth(
b_splus1, params)
error1 = house.marg_ut_cons(
cons_s, params) - beta * (1 - rho[-(length):]) * np.exp(
-sigma * g_y) * deriv_savings * house.marg_ut_cons(
cons_splus1, params) - savings_ut
# Labor leisure euler equations
income_s = (r_s * b_s + w_s * e_s * n_s) * factor
deriv_laborleisure = 1 - tau_payroll - tax.tau_income(
r_s, b_s, w_s, e_s, n_s, factor, params) - tax.tau_income_deriv(
r_s, b_s, w_s, e_s, n_s, factor, params) * income_s
error2 = house.marg_ut_cons(cons_s, params) * w_s * e[
-(length):, j] * deriv_laborleisure - house.marg_ut_labor(
n_s, chi_n[-length:], params)
# Check and punish constraint violations
mask1 = n_guess < 0
error2[mask1] += 1e12
mask2 = n_guess > ltilde
error2[mask2] += 1e12
mask3 = cons_s < 0
error2[mask3] += 1e12
mask4 = b_guess <= 0
error2[mask4] += 1e12
mask5 = cons_splus1 < 0
error2[mask5] += 1e12
return list(error1.flatten()) + list(error2.flatten())
def SS_solver(b_guess_init, n_guess_init, wguess, rguess, T_Hguess, factorguess, chi_n, chi_b, params, iterative_params, tau_bq, rho, lambdas, weights, e):
J, S, T, beta, sigma, alpha, Z, delta, ltilde, nu, g_y, tau_payroll, retire, mean_income_data, a_tax_income, b_tax_income, c_tax_income, d_tax_income, h_wealth, p_wealth, m_wealth, b_ellipse, upsilon = params
maxiter, mindist_SS = iterative_params
w = wguess
r = rguess
T_H = T_Hguess
factor = factorguess
bssmat = b_guess_init
nssmat = n_guess_init
dist = 10
iteration = 0
dist_vec = np.zeros(maxiter)
w_step = .1
r_step = .01
w_down = True
r_down = True
while (dist > mindist_SS) and (iteration < maxiter):
for j in xrange(J):
# Solve the euler equations
guesses = np.append(bssmat[:, j], nssmat[:, j])
solutions = opt.fsolve(Euler_equation_solver, guesses * .9, args=(r, w, T_H, factor, j, params, chi_b, chi_n, tau_bq, rho, lambdas, weights, e), xtol=1e-13)
bssmat[:,j] = solutions[:S]
nssmat[:,j] = solutions[S:]
# print np.array(Euler_equation_solver(np.append(bssmat[:, j], nssmat[:, j]), r, w, T_H, factor, j, params, chi_b, chi_n, theta, tau_bq, rho, lambdas, e)).max()
# Update factor, T_H
b_s = np.array(list(np.zeros(J).reshape(1, J)) + list(bssmat[:-1, :]))
average_income_model = ((r * b_s + w * e * nssmat) * weights).sum()
new_factor = mean_income_data / average_income_model
BQ = (1+r)*(bssmat * weights * rho.reshape(S, 1)).sum(0)
theta = tax.replacement_rate_vals(nssmat, w, factor, e, J, weights)
new_T_H = tax.get_lump_sum(r, b_s, w, e, nssmat, BQ, lambdas, new_factor, weights, 'SS', params, theta, tau_bq)
# Update w, r
B_supply = house.get_K(bssmat, weights)
L_supply = firm.get_L(e, nssmat, weights)
total_tax = tax.total_taxes(r, b_s, w, e, nssmat, BQ, lambdas, new_factor, new_T_H, None, 'SS', False, params, theta, tau_bq)
c_mat = house.get_cons(r, b_s, w, e, nssmat, BQ, lambdas, bssmat, params, total_tax)
C = (c_mat*weights).sum()
Y = C / (1-(delta*alpha/(r+delta)))
B_demand = alpha * Y / (r + delta)
L_demand = (1-alpha) * Y / w
if B_demand - B_supply > mindist_SS:
if r_down:
r_step /= 2.0
r_down = False
r += r_step
else:
if not(r_down):
r_step /= 2.0
r_down = True
r -= r_step
if L_demand - L_supply > mindist_SS:
if w_down:
w_step /=2.0
w_down = False
w += w_step
else:
if not(w_down):
w_step /= 2.0
w_down = True
w -= w_step
factor = misc_funcs.convex_combo(new_factor, factor, params)
T_H = misc_funcs.convex_combo(new_T_H, T_H, params)
dist = np.array([misc_funcs.perc_dif_func(new_T_H, T_H)] + [misc_funcs.perc_dif_func(new_factor, factor)] + [misc_funcs.perc_dif_func(B_demand, B_supply)] + [misc_funcs.perc_dif_func(L_demand, L_supply)]).max()
dist_vec[iteration] = dist
if iteration > 10:
if dist_vec[iteration] - dist_vec[iteration-1] > 0:
nu /= 2.0
print 'New value of nu:', nu
iteration += 1
print "Iteration: %02d" % iteration, " Distance: ", dist
eul_errors = np.ones(J)
b_mat = np.zeros((S, J))
n_mat = np.zeros((S, J))
for j in xrange(J):
solutions1 = opt.fsolve(Euler_equation_solver, np.append(bssmat[:, j], nssmat[:, j])* .9, args=(r, w, T_H, factor, j, params, chi_b, chi_n, tau_bq, rho, lambdas, weights, e), xtol=1e-13)
eul_errors[j] = np.array(Euler_equation_solver(solutions1, r, w, T_H, factor, j, params, chi_b, chi_n, tau_bq, rho, lambdas, weights, e)).max()
b_mat[:, j] = solutions1[:S]
n_mat[:, j] = solutions1[S:]
print 'SS fsolve euler error:', eul_errors.max()
solutions = np.append(b_mat.flatten(), n_mat.flatten())
other_vars = np.array([w, r, factor, T_H])
solutions = np.append(solutions, other_vars)
return solutions
def Steady_state_TPI_solver(guesses, winit, rinit, BQinit, T_H_init, factor, j, s, t, params, theta, tau_bq, rho, lambdas, e, initial_b, chi_b, chi_n):
'''
Parameters:
guesses = distribution of capital and labor in period t
((S-1)*S*J x 1 list)
winit = wage rate (scalar)
rinit = rental rate (scalar)
t = time period
Returns:
Value of Euler error. (as an 2*S*J x 1 list)
'''
J, S, T, beta, sigma, alpha, Z, delta, ltilde, nu, g_y, tau_payroll, retire, mean_income_data, a_tax_income, b_tax_income, c_tax_income, d_tax_income, h_wealth, p_wealth, m_wealth, b_ellipse, upsilon = params
length = len(guesses)/2
b_guess = np.array(guesses[:length])
n_guess = np.array(guesses[length:])
if length == S:
b_s = np.array([0] + list(b_guess[:-1]))
else:
b_s = np.array([(initial_b[-(s+3), j])] + list(b_guess[:-1]))
b_splus1 = b_guess
b_splus2 = np.array(list(b_guess[1:]) + [0])
w_s = winit[t:t+length]
w_splus1 = winit[t+1:t+length+1]
r_s = rinit[t:t+length]
r_splus1 = rinit[t+1:t+length+1]
n_s = n_guess
n_extended = np.array(list(n_guess[1:]) + [0])
e_s = e[-length:, j]
e_extended = np.array(list(e[-length+1:, j]) + [0])
BQ_s = BQinit[t:t+length]
BQ_splus1 = BQinit[t+1:t+length+1]
T_H_s = T_H_init[t:t+length]
T_H_splus1 = T_H_init[t+1:t+length+1]
# Savings euler equations
tax_s = tax.total_taxes(r_s, b_s, w_s, e_s, n_s, BQ_s, lambdas[j], factor, T_H_s, j, 'TPI', False, params, theta, tau_bq)
tax_splus1 = tax.total_taxes(r_splus1, b_splus1, w_splus1, e_extended, n_extended, BQ_splus1, lambdas[j], factor, T_H_splus1, j, 'TPI', True, params, theta, tau_bq)
cons_s = house.get_cons(r_s, b_s, w_s, e_s, n_s, BQ_s, lambdas[j], b_splus1, params, tax_s)
cons_splus1 = house.get_cons(r_splus1, b_splus1, w_splus1, e_extended, n_extended, BQ_splus1, lambdas[j], b_splus2, params, tax_splus1)
income_splus1 = (r_splus1 * b_splus1 + w_splus1 * e_extended * n_extended) * factor
savings_ut = rho[-(length):] * np.exp(-sigma * g_y) * chi_b[-(length):, j] * b_splus1 ** (-sigma)
deriv_savings = 1 + r_splus1 * (1 - tax.tau_income(
r_splus1, b_splus1, w_splus1, e_extended, n_extended, factor, params) - tax.tau_income_deriv(
r_splus1, b_splus1, w_splus1, e_extended, n_extended, factor, params) * income_splus1) - tax.tau_w_prime(
b_splus1, params)*b_splus1 - tax.tau_wealth(b_splus1, params)
error1 = house.marg_ut_cons(cons_s, params) - beta * (1-rho[-(length):]) * np.exp(-sigma * g_y) * deriv_savings * house.marg_ut_cons(
cons_splus1, params) - savings_ut
# Labor leisure euler equations
income_s = (r_s * b_s + w_s * e_s * n_s) * factor
deriv_laborleisure = 1 - tau_payroll - tax.tau_income(r_s, b_s, w_s, e_s, n_s, factor, params) - tax.tau_income_deriv(
r_s, b_s, w_s, e_s, n_s, factor, params) * income_s
error2 = house.marg_ut_cons(cons_s, params) * w_s * e[-(length):, j] * deriv_laborleisure - house.marg_ut_labor(n_s, chi_n[-length:], params)
# Check and punish constraint violations
mask1 = n_guess < 0
error2[mask1] += 1e12
mask2 = n_guess > ltilde
error2[mask2] += 1e12
mask3 = cons_s < 0
error2[mask3] += 1e12
mask4 = b_guess <= 0
error2[mask4] += 1e12
mask5 = cons_splus1 < 0
error2[mask5] += 1e12
return list(error1.flatten()) + list(error2.flatten())
bssmat_s = np.array(list(np.zeros(J).reshape(1, J)) + list(bssmat)) bssmat_splus1 = np.array(list(bssmat) + list(bq.reshape(1, J))) nssmat = solutions[S * J:2*S*J].reshape(S, J) wss, rss, factor_ss, T_Hss = solutions[2*S*J:] Kss = house.get_K(bssmat_splus1, omega_SS.reshape(S, 1), lambdas, g_n_ss) Lss = firm.get_L(e, nssmat, omega_SS.reshape(S, 1), lambdas) Yss = firm.get_Y(Kss, Lss, parameters) Iss = firm.get_I(Kss, Kss, delta, g_y, g_n_ss) theta = tax.replacement_rate_vals(nssmat, wss, factor_ss, e, J, omega_SS.reshape(S, 1), lambdas) BQss = house.get_BQ(rss, bssmat_splus1, omega_SS.reshape(S, 1), lambdas, rho.reshape(S, 1), g_n_ss) b_s = np.array(list(np.zeros(J).reshape((1, J))) + list(bssmat)) taxss = tax.total_taxes(rss, b_s, wss, e, nssmat, BQss, lambdas, factor_ss, T_Hss, None, 'SS', False, parameters, theta, tau_bq) cssmat = house.get_cons(rss, b_s, wss, e, nssmat, BQss.reshape(1, J), lambdas.reshape(1, J), bssmat_splus1, parameters, taxss) Css = house.get_C(cssmat, omega_SS.reshape(S, 1), lambdas) resource_constraint = Yss - (Css + Iss) print 'Resource Constraint Difference:', resource_constraint house.constraint_checker_SS(bssmat, nssmat, cssmat, parameters) b_s = np.array(list(np.zeros(J).reshape((1, J))) + list(bssmat)) b_splus1 = bssmat_splus1 b_splus2 = np.array(list(bssmat_splus1[1:]) + list(np.zeros(J).reshape((1, J)))) chi_b = np.tile(chi_params[:J].reshape(1, J), (S, 1)) chi_n = np.array(chi_params[J:])
initial_b = bssmat_splus1
initial_n = nssmat
else:
initial_b = bssmat_init
initial_n = nssmat_init
K0 = house.get_K(initial_b, omega_stationary[0])
b_sinit = np.array(list(np.zeros(J).reshape(1, J)) + list(initial_b[:-1]))
b_splus1init = initial_b
L0 = firm.get_L(e, initial_n, omega_stationary[0])
Y0 = firm.get_Y(K0, L0, parameters)
w0 = firm.get_w(Y0, L0, parameters)
r0 = firm.get_r(Y0, K0, parameters)
BQ0 = (1+r0)*(initial_b * omega_stationary[0] * rho.reshape(S, 1)).sum(0)
T_H_0 = tax.get_lump_sum(r0, b_sinit, w0, e, initial_n, BQ0, lambdas, factor_ss, omega_stationary[0], 'SS', parameters, theta, tau_bq)
tax0 = tax.total_taxes(r0, b_sinit, w0, e, initial_n, BQ0, lambdas, factor_ss, T_H_0, None, 'SS', False, parameters, theta, tau_bq)
c0 = house.get_cons(r0, b_sinit, w0, e, initial_n, BQ0.reshape(1, J), lambdas.reshape(1, J), b_splus1init, parameters, tax0)
'''
------------------------------------------------------------------------
Solve for equilibrium transition path by TPI
------------------------------------------------------------------------
'''
def SS_TPI_firstdoughnutring(guesses, winit, rinit, BQinit, T_H_init):
b2 = float(guesses[0])
n1 = float(guesses[1])
b1 = float(initial_b[-2, j])
# Euler 1 equations
tax1 = tax.total_taxes(rinit, b1, winit, e[-1, j], n1, BQinit, lambdas[j], factor_ss, T_H_init, j, 'TPI_scalar', False, parameters, theta, tau_bq)
cons1 = house.get_cons(rinit, b1, winit, e[-1, j], n1, BQinit, lambdas[j], b2, parameters, tax1)
def Steady_state_TPI_solver(guesses, winit, rinit, BQinit, T_H_init, factor, j, s, t, params, theta, tau_bq, rho, lambdas, e, initial_b, chi_b, chi_n):
'''
Parameters:
guesses = distribution of capital and labor (various length list)
winit = wage rate ((T+S)x1 array)
rinit = rental rate ((T+S)x1 array)
BQinit = aggregate bequests ((T+S)x1 array)
T_H_init = lump sum tax over time ((T+S)x1 array)
factor = scaling factor (scalar)
j = which ability type is being solved for (scalar)
s = which upper triangle loop is being solved for (scalar)
t = which diagonal is being solved for (scalar)
params = list of parameters (list)
theta = replacement rates (Jx1 array)
tau_bq = bequest tax rate (Jx1 array)
rho = mortalit rate (Sx1 array)
lambdas = ability weights (Jx1 array)
e = ability type (SxJ array)
initial_b = capital stock distribution in period 0 (SxJ array)
chi_b = chi^b_j (Jx1 array)
chi_n = chi^n_s (Sx1 array)
Output:
Value of Euler error (various length list)
'''
J, S, T, beta, sigma, alpha, Z, delta, ltilde, nu, g_y, g_n_ss, tau_payroll, retire, mean_income_data, a_tax_income, b_tax_income, c_tax_income, d_tax_income, h_wealth, p_wealth, m_wealth, b_ellipse, upsilon = params
length = len(guesses)/2
b_guess = np.array(guesses[:length])
n_guess = np.array(guesses[length:])
if length == S:
b_s = np.array([0] + list(b_guess[:-1]))
else:
b_s = np.array([(initial_b[-(s+3), j])] + list(b_guess[:-1]))
b_splus1 = b_guess
b_splus2 = np.array(list(b_guess[1:]) + [0])
w_s = winit[t:t+length]
w_splus1 = winit[t+1:t+length+1]
r_s = rinit[t:t+length]
r_splus1 = rinit[t+1:t+length+1]
n_s = n_guess
n_extended = np.array(list(n_guess[1:]) + [0])
e_s = e[-length:, j]
e_extended = np.array(list(e[-length+1:, j]) + [0])
BQ_s = BQinit[t:t+length]
BQ_splus1 = BQinit[t+1:t+length+1]
T_H_s = T_H_init[t:t+length]
T_H_splus1 = T_H_init[t+1:t+length+1]
# Savings euler equations
tax_s = tax.total_taxes(r_s, b_s, w_s, e_s, n_s, BQ_s, lambdas[j], factor, T_H_s, j, 'TPI', False, params, theta, tau_bq)
tax_splus1 = tax.total_taxes(r_splus1, b_splus1, w_splus1, e_extended, n_extended, BQ_splus1, lambdas[j], factor, T_H_splus1, j, 'TPI', True, params, theta, tau_bq)
cons_s = house.get_cons(r_s, b_s, w_s, e_s, n_s, BQ_s, lambdas[j], b_splus1, params, tax_s)
cons_splus1 = house.get_cons(r_splus1, b_splus1, w_splus1, e_extended, n_extended, BQ_splus1, lambdas[j], b_splus2, params, tax_splus1)
income_splus1 = (r_splus1 * b_splus1 + w_splus1 * e_extended * n_extended) * factor
savings_ut = rho[-(length):] * np.exp(-sigma * g_y) * chi_b[-(length):, j] * b_splus1 ** (-sigma)
deriv_savings = 1 + r_splus1 * (1 - tax.tau_income(
r_splus1, b_splus1, w_splus1, e_extended, n_extended, factor, params) - tax.tau_income_deriv(
r_splus1, b_splus1, w_splus1, e_extended, n_extended, factor, params) * income_splus1) - tax.tau_w_prime(
b_splus1, params)*b_splus1 - tax.tau_wealth(b_splus1, params)
error1 = house.marg_ut_cons(cons_s, params) - beta * (1-rho[-(length):]) * np.exp(-sigma * g_y) * deriv_savings * house.marg_ut_cons(
cons_splus1, params) - savings_ut
# Labor leisure euler equations
income_s = (r_s * b_s + w_s * e_s * n_s) * factor
deriv_laborleisure = 1 - tau_payroll - tax.tau_income(r_s, b_s, w_s, e_s, n_s, factor, params) - tax.tau_income_deriv(
r_s, b_s, w_s, e_s, n_s, factor, params) * income_s
error2 = house.marg_ut_cons(cons_s, params) * w_s * e[-(length):, j] * deriv_laborleisure - house.marg_ut_labor(n_s, chi_n[-length:], params)
# Check and punish constraint violations
mask1 = n_guess < 0
error2[mask1] += 1e12
mask2 = n_guess > ltilde
error2[mask2] += 1e12
mask3 = cons_s < 0
error2[mask3] += 1e12
mask4 = b_guess <= 0
error2[mask4] += 1e12
mask5 = cons_splus1 < 0
error2[mask5] += 1e12
return list(error1.flatten()) + list(error2.flatten())
