def euler_labor_leisure_func(w, r, e, n_guess, b_s, b_splus1, BQ, factor, T_H, chi_n, params, theta, tau_bq, lambdas):
"""
This function is usually looped through over J, so it does one ability group at a time.
Inputs:
w = wage rate (scalar)
r = rental rate (scalar)
e = ability levels (Sx1 array)
n_guess = labor distribution (Sx1 array)
b_s = wealth holdings at the start of a period (Sx1 array)
b_splus1 = wealth holdings for the next period (Sx1 array)
BQ = aggregate bequests for a certain ability (scalar)
factor = scaling factor to convert to dollars (scalar)
T_H = lump sum tax (scalar)
chi_n = chi^n_s (Sx1 array)
params = parameter list (list)
theta = replacement rate for a certain ability (scalar)
tau_bq = bequest tax rate (scalar)
lambdas = ability weight (scalar)
Output:
euler = Value of labor leisure euler error (Sx1 array)
"""
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
)
tax1 = tax.total_taxes(r, b_s, w, e, n_guess, BQ, lambdas, factor, T_H, None, "SS", False, params, theta, tau_bq)
cons = get_cons(r, b_s, w, e, n_guess, BQ, lambdas, b_splus1, params, tax1)
income = (r * b_s + w * e * n_guess) * factor
deriv = (
1
- tau_payroll
- tax.tau_income(r, b_s, w, e, n_guess, factor, params)
- tax.tau_income_deriv(r, b_s, w, e, n_guess, factor, params) * income
)
euler = marg_ut_cons(cons, params) * w * deriv * e - marg_ut_labor(n_guess, chi_n, params)
return euler
def euler_labor_leisure_func(w, r, e, n_guess, b_s, b_splus1, BQ, factor, T_H,
chi_n, params, theta, tau_bq, lambdas):
'''
This function is usually looped through over J, so it does one ability group at a time.
Inputs:
w = wage rate (scalar)
r = rental rate (scalar)
e = ability levels (Sx1 array)
n_guess = labor distribution (Sx1 array)
b_s = wealth holdings at the start of a period (Sx1 array)
b_splus1 = wealth holdings for the next period (Sx1 array)
BQ = aggregate bequests for a certain ability (scalar)
factor = scaling factor to convert to dollars (scalar)
T_H = lump sum tax (scalar)
chi_n = chi^n_s (Sx1 array)
params = parameter list (list)
theta = replacement rate for a certain ability (scalar)
tau_bq = bequest tax rate (scalar)
lambdas = ability weight (scalar)
Output:
euler = Value of labor leisure euler error (Sx1 array)
'''
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
tax1 = tax.total_taxes(r, b_s, w, e, n_guess, BQ, lambdas, factor, T_H,
None, 'SS', False, params, theta, tau_bq)
cons = get_cons(r, b_s, w, e, n_guess, BQ, lambdas, b_splus1, params, tax1)
income = (r * b_s + w * e * n_guess) * factor
deriv = 1 - tau_payroll - tax.tau_income(
r, b_s, w, e, n_guess, factor, params) - tax.tau_income_deriv(
r, b_s, w, e, n_guess, factor, params) * income
euler = marg_ut_cons(cons, params) * w * deriv * e - \
marg_ut_labor(n_guess, chi_n, params)
return euler
def euler_savings_func(w, r, e, n_guess, b_s, b_splus1, b_splus2, BQ, factor,
T_H, chi_b, params, theta, tau_bq, rho, lambdas):
'''
This function is usually looped through over J, so it does one ability group at a time.
Inputs:
w = wage rate (scalar)
r = rental rate (scalar)
e = ability levels (Sx1 array)
n_guess = labor distribution (Sx1 array)
b_s = wealth holdings at the start of a period (Sx1 array)
b_splus1 = wealth holdings for the next period (Sx1 array)
b_splus2 = wealth holdings for 2 periods ahead (Sx1 array)
BQ = aggregate bequests for a certain ability (scalar)
factor = scaling factor to convert to dollars (scalar)
T_H = lump sum tax (scalar)
chi_b = chi^b_j for a certain ability (scalar)
params = parameter list (list)
theta = replacement rate for a certain ability (scalar)
tau_bq = bequest tax rate (scalar)
rho = mortality rate (Sx1 array)
lambdas = ability weight (scalar)
Output:
euler = Value of savings euler error (Sx1 array)
'''
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
# In order to not have 2 savings euler equations (one that solves the first S-1 equations, and one that solves the last one),
# we combine them. In order to do this, we have to compute a consumption term in period t+1, which requires us to have a shifted
# e and n matrix. We append a zero on the end of both of these so they will be the right size. We could append any value to them,
# since in the euler equation, the coefficient on the marginal utility of
# consumption for this term will be zero (since rho is one).
e_extended = np.array(list(e) + [0])
n_extended = np.array(list(n_guess) + [0])
tax1 = tax.total_taxes(r, b_s, w, e, n_guess, BQ, lambdas, factor, T_H,
None, 'SS', False, params, theta, tau_bq)
tax2 = tax.total_taxes(r, b_splus1, w, e_extended[1:], n_extended[1:], BQ,
lambdas, factor, T_H, None, 'SS', True, params,
theta, tau_bq)
cons1 = get_cons(r, b_s, w, e, n_guess, BQ, lambdas, b_splus1, params,
tax1)
cons2 = get_cons(r, b_splus1, w, e_extended[1:], n_extended[1:], BQ,
lambdas, b_splus2, params, tax2)
income = (r * b_splus1 + w * e_extended[1:] * n_extended[1:]) * factor
deriv = (1 + r *
(1 - tax.tau_income(r, b_splus1, w, e_extended[1:],
n_extended[1:], factor, params) -
tax.tau_income_deriv(r, b_splus1, w, e_extended[1:],
n_extended[1:], factor, params) * income) -
tax.tau_w_prime(b_splus1, params) * b_splus1 -
tax.tau_wealth(b_splus1, params))
savings_ut = rho * np.exp(-sigma * g_y) * chi_b * b_splus1**(-sigma)
# Again, not who in this equation, the (1-rho) term will zero out in the last period, so the last entry of cons2 can be complete
# gibberish (which it is). It just has to exist so cons2 is the right
# size to match all other arrays in the equation.
euler = marg_ut_cons(cons1,
params) - beta * (1 - rho) * deriv * marg_ut_cons(
cons2, params) * np.exp(-sigma * g_y) - savings_ut
return euler
def SS_TPI_firstdoughnutring(guesses, winit, rinit, BQinit, T_H_init,
initial_b, factor_ss, j, parameters, theta,
tau_bq):
'''
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)
initial_b = initial distribution of capital (SxJ array)
factor_ss = steady state scaling factor (scalar)
j = which ability type is being solved for (scalar)
parameters = list of parameters (list)
theta = replacement rates (Jx1 array)
tau_bq = bequest tax rates (Jx1 array)
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 = household.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 = household.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 = household.marg_ut_cons(cons1, parameters) * winit * \
e[-1, j] * deriv2 - household.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 SS_TPI_firstdoughnutring(guesses, winit, rinit, BQinit, T_H_init, initial_b, factor_ss, j, parameters, theta, tau_bq):
'''
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)
initial_b = initial distribution of capital (SxJ array)
factor_ss = steady state scaling factor (scalar)
j = which ability type is being solved for (scalar)
parameters = list of parameters (list)
theta = replacement rates (Jx1 array)
tau_bq = bequest tax rates (Jx1 array)
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 = household.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 = household.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 = household.marg_ut_cons(cons1, parameters) * winit * \
e[-1, j] * deriv2 - household.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 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 = household.get_cons(r_s, b_s, w_s, e_s, n_s, BQ_s, lambdas[j],
b_splus1, params, tax_s)
cons_splus1 = household.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 = household.marg_ut_cons(
cons_s, params) - beta * (1 - rho[-(length):]) * np.exp(
-sigma * g_y) * deriv_savings * household.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 = household.marg_ut_cons(cons_s, params) * w_s * e[
-(length):, j] * deriv_laborleisure - household.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 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 = household.get_cons(r_s, b_s, w_s, e_s, n_s, BQ_s, lambdas[
j], b_splus1, params, tax_s)
cons_splus1 = household.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 = household.marg_ut_cons(cons_s, params) - beta * (1 - rho[-(length):]) * np.exp(-sigma * g_y) * deriv_savings * household.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 = household.marg_ut_cons(cons_s, params) * w_s * e[-(
length):, j] * deriv_laborleisure - household.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 euler_savings_func(
w, r, e, n_guess, b_s, b_splus1, b_splus2, BQ, factor, T_H, chi_b, params, theta, tau_bq, rho, lambdas
):
"""
This function is usually looped through over J, so it does one ability group at a time.
Inputs:
w = wage rate (scalar)
r = rental rate (scalar)
e = ability levels (Sx1 array)
n_guess = labor distribution (Sx1 array)
b_s = wealth holdings at the start of a period (Sx1 array)
b_splus1 = wealth holdings for the next period (Sx1 array)
b_splus2 = wealth holdings for 2 periods ahead (Sx1 array)
BQ = aggregate bequests for a certain ability (scalar)
factor = scaling factor to convert to dollars (scalar)
T_H = lump sum tax (scalar)
chi_b = chi^b_j for a certain ability (scalar)
params = parameter list (list)
theta = replacement rate for a certain ability (scalar)
tau_bq = bequest tax rate (scalar)
rho = mortality rate (Sx1 array)
lambdas = ability weight (scalar)
Output:
euler = Value of savings euler error (Sx1 array)
"""
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
)
# In order to not have 2 savings euler equations (one that solves the first S-1 equations, and one that solves the last one),
# we combine them. In order to do this, we have to compute a consumption term in period t+1, which requires us to have a shifted
# e and n matrix. We append a zero on the end of both of these so they will be the right size. We could append any value to them,
# since in the euler equation, the coefficient on the marginal utility of
# consumption for this term will be zero (since rho is one).
e_extended = np.array(list(e) + [0])
n_extended = np.array(list(n_guess) + [0])
tax1 = tax.total_taxes(r, b_s, w, e, n_guess, BQ, lambdas, factor, T_H, None, "SS", False, params, theta, tau_bq)
tax2 = tax.total_taxes(
r,
b_splus1,
w,
e_extended[1:],
n_extended[1:],
BQ,
lambdas,
factor,
T_H,
None,
"SS",
True,
params,
theta,
tau_bq,
)
cons1 = get_cons(r, b_s, w, e, n_guess, BQ, lambdas, b_splus1, params, tax1)
cons2 = get_cons(r, b_splus1, w, e_extended[1:], n_extended[1:], BQ, lambdas, b_splus2, params, tax2)
income = (r * b_splus1 + w * e_extended[1:] * n_extended[1:]) * factor
deriv = (
1
+ r
* (
1
- tax.tau_income(r, b_splus1, w, e_extended[1:], n_extended[1:], factor, params)
- tax.tau_income_deriv(r, b_splus1, w, e_extended[1:], n_extended[1:], factor, params) * income
)
- tax.tau_w_prime(b_splus1, params) * b_splus1
- tax.tau_wealth(b_splus1, params)
)
savings_ut = rho * np.exp(-sigma * g_y) * chi_b * b_splus1 ** (-sigma)
# Again, not who in this equation, the (1-rho) term will zero out in the last period, so the last entry of cons2 can be complete
# gibberish (which it is). It just has to exist so cons2 is the right
# size to match all other arrays in the equation.
euler = (
marg_ut_cons(cons1, params)
- beta * (1 - rho) * deriv * marg_ut_cons(cons2, params) * np.exp(-sigma * g_y)
- savings_ut
)
return euler
