Example #1
0
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
Example #2
0
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
Example #3
0
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())
Example #4
0
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())
Example #5
0
def FOC_savings(r, w, b, b_splus1, b_splus2, n, BQ, factor, T_H,
                params):
    '''
    Computes Euler errors for the FOC for savings in the steady state.
    This function is usually looped through over J, so it does one
    lifetime income group at a time.

    Inputs:
        r           = scalar, interest rate
        w           = scalar, wage rate
        b           = [S,J] array, distribution of wealth/capital
        b_splus1    = [S,J] array, distribution of wealth/capital,
                        one period ahead
        b_splus2    = [S,J] array, distribution of wealth/capital, two
                        periods ahead
        n           = [S,J] array, distribution of labor supply
        BQ          = [J,] vector, aggregate bequests by lifetime income
                        group
        factor      = scalar, scaling factor to convert model income to
                        dollars
        T_H         = scalar, lump sum transfer
        params      = length 18 tuple (e, sigma, beta, g_y, chi_b,
                                       theta, tau_bq, rho, lambdas, J,
                                       S, etr_params, mtry_params,
                                       h_wealth, p_wealth, m_wealth,
                                       tau_payroll, tau_bq)
        e           = [S,J] array, effective labor units
        sigma       = scalar, coefficient of relative risk aversion
        beta        = scalar, discount factor
        g_y         = scalar, exogenous labor augmenting technological
                        growth
        chi_b       = [J,] vector, utility weight on bequests for each
                        lifetime income group
        theta       = [J,] vector, replacement rate for each lifetime
                        income group
        tau_bq      = scalar, bequest tax rate (scalar)
        rho         = [S,] vector, mortality rates
        lambdas     = [J,] vector, ability weights
        J           = integer, number of lifetime income groups
        S           = integer, number of economically active periods in
                        lifetime
        etr_params  = [S,12] array, parameters of effective income tax
                        rate function
        mtry_params = [S,12] array, parameters of marginal tax rate on
                        capital income function
        h_wealth    = scalar, parameter in wealth tax function
        p_wealth    = scalar, parameter in wealth tax function
        m_wealth    = scalar, parameter in wealth tax function
        tau_payroll = scalar, payroll tax rate
        tau_bq      = scalar, bequest tax rate

    Functions called:
        get_cons
        marg_ut_cons
        tax.total_taxes
        tax.MTR_capital

    Objects in function:
        tax1 = [S,J] array, net taxes in the current period
        tax2 = [S,J] array, net taxes one period ahead
        cons1 = [S,J] array, consumption in the current period
        cons2 = [S,J] array, consumption one period ahead
        deriv = [S,J] array, after-tax return on capital
        savings_ut = [S,J] array, marginal utility from savings
        euler = [S,J] array, Euler error from FOC for savings

    Returns: euler
    '''
    (e, sigma, beta, g_y, chi_b, theta, tau_bq, rho, lambdas, j, J, S,
     analytical_mtrs, etr_params, mtry_params, h_wealth, p_wealth,
     m_wealth, tau_payroll, retire, method) = 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) + [0])
    etr_params_extended = np.append(etr_params,
                                    np.reshape(etr_params[-1, :],
                                               (1, etr_params.shape[1])),
                                    axis=0)[1:, :]
    mtry_params_extended = np.append(mtry_params,
                                     np.reshape(mtry_params[-1, :],
                                                (1, mtry_params.shape[1])),
                                     axis=0)[1:, :]
    if method == 'TPI':
        r_extended = np.append(r, r[-1])
        w_extended = np.append(w, w[-1])
        BQ_extended = np.append(BQ, BQ[-1])
        T_H_extended = np.append(T_H, T_H[-1])
    elif method == 'SS':
        r_extended = np.array([r, r])
        w_extended = np.array([w, w])
        BQ_extended = np.array([BQ, BQ])
        T_H_extended = np.array([T_H, T_H])

    tax1_params = (e, lambdas, method, retire, etr_params, h_wealth,
                   p_wealth, m_wealth, tau_payroll, theta, tau_bq, J, S)
    tax1 = tax.total_taxes(r, w, b, n, BQ, factor, T_H, j, False,
                           tax1_params)
    tax2_params = (e_extended[1:], lambdas, method, retire,
                   etr_params_extended, h_wealth, p_wealth, m_wealth,
                   tau_payroll, theta, tau_bq, J, S)
    tax2 = tax.total_taxes(r_extended[1:], w_extended[1:], b_splus1,
                           n_extended[1:], BQ_extended[1:], factor,
                           T_H_extended[1:], j, True, tax2_params)
    cons1_params = (e, lambdas, g_y)
    cons1 = get_cons(r, w, b, b_splus1, n, BQ, tax1, cons1_params)
    cons2_params = (e_extended[1:], lambdas, g_y)
    cons2 = get_cons(r_extended[1:], w_extended[1:], b_splus1, b_splus2,
                     n_extended[1:], BQ_extended[1:], tax2,
                     cons2_params)
    cons2[-1] = 0.01  # set to small positive number to avoid exception
    # errors when negative b/c this period value doesn't matter -
    # it's consumption after the last period of life
    mtr_cap_params = (e_extended[1:], etr_params_extended,
                      mtry_params_extended, analytical_mtrs)
    deriv = ((1 + r_extended[1:]) - r_extended[1:] *
             (tax.MTR_capital(r_extended[1:], w_extended[1:], b_splus1,
                              n_extended[1:], factor, mtr_cap_params)) -
             (tax.tau_w_prime(b_splus1, (h_wealth, p_wealth, m_wealth)) *
             b_splus1) - tax.tau_wealth(b_splus1, (h_wealth, p_wealth,
                                                   m_wealth)))

    savings_ut = (rho * np.exp(-sigma * g_y) * chi_b * b_splus1 **
                  (-sigma))

    euler_error = (marg_ut_cons(cons1, sigma) - beta * (1 - rho) *
                   deriv * marg_ut_cons(cons2, sigma) *
                   np.exp(-sigma * g_y) - savings_ut)

    return euler_error