Pension Functions

pensions.py modules

ogcore.pensions

ogcore.pensions.DB_1dim_loop(w, e, n, S_ret, S, g_y, L_inc_avg_s, L_inc_avg, DB, avg_earn_num_years, alpha_db, yr_contr)

Calculate public pension from a defined benefits system.

Parameters:

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• n (Numpy array) – labor supply

• S_ret (int) – retirement age

• S (int) – number of periods in the model

• g_y (array_like) – GDP growth rate

• L_inc_avg_s (Numpy array) – average labor income

• L_inc_avg (scalar) – average labor income

• DB (Numpy array) – pension amount for each household

• avg_earn_num_years (int) – number of years AIME is computed from

• alpha_db (scalar) – replacement rate

• yr_contr (scalar) – years of contribution

Returns:

pension amount for each household

Return type:

DB (Numpy array)

ogcore.pensions.DB_2dim_loop(w, e, n, S_ret, S, g_y, L_inc_avg_sj, L_inc_avg, DB, avg_earn_num_years, alpha_db, yr_contr)

Calculate public pension from a defined benefits system.

Parameters:

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• n (Numpy array) – labor supply

• S_ret (int) – retirement age

• S (int) – number of periods in the model

• g_y (array_like) – GDP growth rate

• L_inc_avg_sj (Numpy array) – average labor income

• L_inc_avg (scalar) – average labor income

• DB (Numpy array) – pension amount for each household

• avg_earn_num_years (int) – number of years AIME is computed from

• alpha_db (scalar) – replacement rate

• yr_contr (scalar) – years of contribution

Returns:

pension amount for each household

Return type:

DB (Numpy array)

ogcore.pensions.DB_3dim_loop(w_path, e_ss, n, n_preTP, S_ret, S, J, g_y, avg_earn_num_years, alpha_db, yr_contr)

Calculate public pension from a defined benefits system over the full time path.

Used when the TPI solution evaluates taxes for all periods at once to compute aggregates. Each retiree’s benefit is computed from their own cohort’s wage and labor history: the wage entering the average for a household aged u at time t, earned at age s, is the wage at time t - (u - s). Histories that predate the time path use the period-0 wage (trend growth is handled by the de-trending in the benefit formula) and the model’s initial labor supply, so this reproduces exactly what the per-cohort household solves compute and household behavior and aggregates are consistent. The inner work is vectorized rather than looped.

Parameters:

• w_path (Numpy array) – real wage rate path, length T

• e_ss (Numpy array) – effective labor units, size SxJ

• n (Numpy array) – labor supply, size TxSxJ

• n_preTP (Numpy array) – pre-time-path labor supply, size SxJ

• S_ret (int) – retirement age

• S (int) – number of periods in the model

• J (int) – number of lifetime income groups

• g_y (scalar) – growth rate of technology

• avg_earn_num_years (int) – number of years earnings are averaged over

• alpha_db (scalar) – replacement rate per year of contribution

• yr_contr (int) – years of contribution

Returns:

pension amount for each household, size TxSxJ

Return type:

DB (Numpy array)

ogcore.pensions.DB_amount(w, e, n, j, p)

Calculate public pension from a defined benefits system.

\[pension{j,s,t} = \biggl[\frac{\sum_{s=R-ny}^{R-1}w_{t}e_{j,s,t} n_{j,s,t}}{ny}\biggr]\times Cy \times \alpha_{DB} \quad \forall s > R\]

Parameters:

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• n (Numpy array) – labor supply; may be a partial lifetime path (length < S, for cohorts alive when the time path begins), a steady-state vector (S,) or matrix (S, J), or the full time path (T, S, J)

• j (int) – index of lifetime income group

• p (OG-Core Specifications object) – model parameters

Returns:

pension amount for each household

Return type:

DB (Numpy array)

ogcore.pensions.NDC_1dim_loop(w, e, n, S_ret, S, g_y, tau_p, g_ndc, delta_ret, NDC_s, NDC)

Calculate public pension from a notional defined contribution

Parameters:

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• n (Numpy array) – labor supply

• S_ret (int) – retirement age

• S (int) – number of periods in the model

• g_y (array_like) – GDP growth rate

• tau_p (scalar) – tax rate

• g_ndc (scalar) – growth rate of NDC pension

• delta_ret (scalar) – conversion coefficient for the NDC pension amount

• NDC_s (Numpy array) – average labor income

• NDC (Numpy array) – pension amount for each household

Returns:

pension amount for each household

Return type:

NDC (Numpy array)

ogcore.pensions.NDC_2dim_loop(w, e, n, S_ret, S, g_y, tau_p, g_ndc, delta_ret, NDC_sj, NDC)

Calculate public pension from a notional defined contribution

Parameters:

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• n (Numpy array) – labor supply

• S_ret (int) – retirement age

• S (int) – number of periods in the model

• g_y (array_like) – GDP growth rate

• tau_p (scalar) – tax rate

• g_ndc (scalar) – growth rate of NDC pension

• delta_ret (scalar) – conversion coefficient for the NDC pension amount

• NDC_sj (Numpy array) – average labor income

• NDC (Numpy array) – pension amount for each household

Returns:

pension amount for each household

Return type:

NDC (Numpy array)

ogcore.pensions.NDC_amount(w, e, n, r, Y, j, p)

Calculate public pension from a notional defined contribution system.

\[pension{j,s,t} = \biggl[\sum_{s=E}^{R-1}\tau^{p}_{t}w_{t} e_{j,s,t}n_{j,s,t}(1 + g_{NDC,t})^{R-s-1}\biggr] \delta_{R, t} \quad \forall s > R\]

Parameters:

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• n (Numpy array) – labor supply

• r (array_like) – interest rate

• Y (array_like) – GDP

• j (int) – index of lifetime income group

• p (OG-Core Specifications object) – model parameters

Returns:

pension amount for each household

Return type:

NDC (Numpy array)

ogcore.pensions.PS_1dim_loop(w, e, n, S_ret, S, g_y, vpoint, factor, L_inc_avg_s, PS)

Calculate public pension from a points system.

Parameters:

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• n (Numpy array) – labor supply

• S_ret (int) – retirement age

• S (int) – number of periods in the model

• g_y (array_like) – GDP growth rate

• vpoint (scalar) – value of points

• factor (scalar) – scaling factor converting model units to local currency

• L_inc_avg_s (Numpy array) – average labor income

• PS (Numpy array) – pension amount for each household

Returns:

pension amount for each household

Return type:

PS (Numpy array)

ogcore.pensions.PS_2dim_loop(w, e, n, S_ret, S, J, g_y, vpoint, factor, L_inc_avg_sj, PS)

Calculate public pension from a points system.

Parameters:

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• n (Numpy array) – labor supply

• S_ret (int) – retirement age

• S (int) – number of periods in the model

• J (int) – number of lifetime income groups

• g_y (array_like) – GDP growth rate

• vpoint (scalar) – value of points

• factor (scalar) – scaling factor converting model units to local currency

• L_inc_avg_sj (Numpy array) – average labor income

• PS (Numpy array) – pension amount for each household

Returns:

pension amount for each household

Return type:

PS (Numpy array)

ogcore.pensions.PS_amount(w, e, n, j, factor, p)

Calculate public pension from a points system.

\[pension{j,s,t} = \sum_{s=E}^{R-1}w_{t}e_{j,s,t}n_{j,s,t} \times v_{t} \quad \forall s > R\]

Parameters:

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• n (Numpy array) – labor supply

• j (int) – index of lifetime income group

• factor (scalar) – scaling factor converting model units to dollars

• p (OG-Core Specifications object) – model parameters

Returns:

pension amount for each household

Return type:

PS (Numpy array)

ogcore.pensions.SS_amount(w, n, theta, t, j, shift, method, e, p)

Calculate public pension benefit amounts for each household under a US-style social security system.

Parameters:

• w (array_like) – real wage rate

• n (Numpy array) – labor supply

• theta (Numpy array) – social security replacement rate value for lifetime income group j

• t (int) – time period

• j (int) – index of lifetime income group

• shift (bool) – whether computing for periods 0–s or 1–(s+1), =True for 1–(s+1)

• method (str) – adjusts calculation dimensions based on ‘SS’ or ‘TPI’

• e (Numpy array) – effective labor units

• p (OG-Core Specifications object) – model parameters

Returns:

pension amount for each household

Return type:

pension (Numpy array)

ogcore.pensions.delta_point(r, Y, g_n, g_y, p)

Compute growth rate used for contributions to points system pension

Parameters:

• r (array_like) – interest rate

• Y (array_like) – GDP

• g_n (array_like) – population growth rate

• g_y (array_like) – GDP growth rate

• p (OG-Core Specifications object) – model parameters

Returns:

growth rate used for contributions to

points

Return type:

delta_point (Numpy array)

ogcore.pensions.delta_ret(r, Y, p)

Compute conversion coefficient for the NDC pension amount

\[\delta_{R} = (dir_{R} + ind_{R} - k)^{-1}\]

Parameters:

• r (array_like) – interest rate

• Y (array_like) – GDP

• p (OG-Core Specifications object) – model parameters

Returns:

conversion coefficient for the NDC

pension amount

Return type:

delta_ret (Numpy array)

ogcore.pensions.delta_ret_loop(S, S_ret, surv_rates, g_dir_value, dir_delta_s)

Compute conversion coefficient for the NDC pension amount

Parameters:

• S (int) – number of periods in the model

• S_ret (int) – retirement age

• surv_rates (Numpy array) – survival rates

• g_dir_value (scalar) – growth rate of NDC pension

• dir_delta_s (Numpy array) – conversion coefficient for the NDC pension amount

Returns:

conversion coefficient for the NDC pension

amount

Return type:

dir_delta (scalar)

ogcore.pensions.deriv_DB(w, e, per_rmn, p)

Change in DB pension benefits for another unit of labor supply

\[\begin{split}\frac{\partial \theta_{j,u,t+u-s}}{\partial n_{j,s,t}} = \begin{cases} 0 , & \text{if}\ s < R - Cy \\ w_{t}e_{j,s}\alpha_{DB}\times \frac{Cy}{ny}, & \text{if}\ R - Cy <= s < R \\ 0, & \text{if}\ s \geq R \\ \end{cases}\end{split}\]

Parameters:

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• per_rmn (int) – number of periods remaining in the model

• p (OG-Core Specifications object) – model parameters

Returns:

change in DB pension benefits for another unit of labor

supply

Return type:

d_theta

ogcore.pensions.deriv_DB_loop(w, e, S, S_ret, per_rmn, avg_earn_num_years, alpha_db, yr_contr)

Change in DB pension benefits for another unit of labor supply

Parameters:

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• S (int) – number of periods in the model

• S_ret (int) – retirement age

• per_rmn (int) – number of periods remaining in the model

• avg_earn_num_years (int) – number of years AIME is computed from

• alpha_db (scalar) – replacement rate

• yr_contr (scalar) – years of contribution

Returns:

change in DB pension benefits for

another unit of labor supply

Return type:

d_theta (Numpy array)

ogcore.pensions.deriv_NDC(r, w, e, Y, per_rmn, p)

Change in NDC pension benefits for another unit of labor supply

\[\begin{split}\frac{\partial \theta_{j,u,t+u-s}}{\partial n_{j,s,t}} = \begin{cases} \tau^{p}_{t}w_{t}e_{j,s}(1+g_{NDC,t})^{u - s}\delta_{R,t}, & \text{if}\ s<R-1 \\ 0, & \text{if}\ s \geq R \\ \end{cases}\end{split}\]

Parameters:

• r (array_like) – interest rate

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• Y (array_like) – GDP

• per_rmn (int) – number of periods remaining in the model

• p (OG-Core Specifications object) – model parameters

Returns:

change in NDC pension benefits for

another unit of labor supply

Return type:

d_theta (Numpy array)

ogcore.pensions.deriv_NDC_loop(w, e, per_rmn, S, S_ret, tau_p, g_ndc_value, delta_ret_value, d_theta)

Change in NDC pension benefits for another unit of labor supply

Parameters:

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• per_rmn (int) – number of periods remaining in the model

• S (int) – number of periods in the model

• S_ret (int) – retirement age

• tau_p (scalar) – tax rate

• g_ndc_value (scalar) – growth rate of NDC pension

• delta_ret_value (scalar) – conversion coefficient for the NDC pension amount

• d_theta (Numpy array) – change in NDC pension benefits for another unit of labor supply

Returns:

change in NDC pension benefits for

another unit of labor supply

Return type:

d_theta (Numpy array)

ogcore.pensions.deriv_PS(w, e, per_rmn, factor, p)

Change in points system pension benefits for another unit of labor supply

\[\begin{split}\frac{\partial \theta_{j,u,t+u-s}}{\partial n_{j,s,t}} = \begin{cases} 0 , & \text{if}\ s < R \\ w_{t}e_{j,s}v_{t}, & \text{if}\ s \geq R \\ \end{cases}\end{split}\]

Parameters:

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• per_rmn (int) – number of periods remaining in the model

• factor (scalar) – scaling factor converting model units to

• p (OG-Core Specifications object) – model parameters

Returns:

change in points system pension benefits

for another unit of labor supply

Return type:

d_theta (Numpy array)

ogcore.pensions.deriv_PS_loop(w, e, S, S_ret, per_rmn, d_theta, vpoint, factor)

Change in points system pension benefits for another unit of labor supply

Parameters:

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• S (int) – number of periods in the model

• S_ret (int) – retirement age

• per_rmn (int) – number of periods remaining in the model

• d_theta (Numpy array) – change in points system pension benefits for another unit of labor supply

• vpoint (scalar) – value of points

• factor (scalar) – scaling factor converting model units to local currency

Returns:

change in points system pension benefits

for another unit of labor supply

Return type:

d_theta (Numpy array)

ogcore.pensions.deriv_theta(r, w, e, Y, per_rmn, factor, p)

Change in pension benefits for another unit of labor supply for pension system selected

Parameters:

• r (array_like) – interest rate

• w (array_like) – real wage rate

• e (Numpy array) – effective labor units

• Y (array_like) – GDP

• per_rmn (int) – number of periods remaining in the model

• factor (scalar) – scaling factor converting model units to

Returns:

change in pension benefits for another

unit of labor supply

Return type:

d_theta (Numpy array)

ogcore.pensions.g_dir(r, Y, g_y, g_n, dir_growth_rate)

Compute growth rate used for contributions to NDC pension

Parameters:

• r (array_like) – interest rate

• Y (array_like) – GDP

• g_y (array_like) – GDP growth rate

• g_n (array_like) – population growth rate

• dir_growth_rate (str) – growth rate used for contributions to NDC

Returns:

growth rate used for contributions to NDC

Return type:

g_dir (Numpy array)

ogcore.pensions.g_ndc(r, Y, p)

Compute growth rate used for contributions to NDC pension

Parameters:

• r (array_like) – interest rate

• Y (array_like) – GDP

• p (OG-Core Specifications object) – model parameters

Returns:

growth rate used for contributions to NDC

Return type:

g_ndc (Numpy array)

ogcore.pensions.pension_amount(r, w, n, Y, theta, t, j, shift, method, e, factor, p)

Calculate public pension benefit amounts for each household.

Parameters:

• w (array_like) – real wage rate

• n (Numpy array) – labor supply

• theta (Numpy array) – social security replacement rate value for lifetime income group j

• t (int) – time period

• j (int) – index of lifetime income group

• shift (bool) – whether computing for periods 0–s or 1–(s+1), =True for 1–(s+1)

• method (str) – adjusts calculation dimensions based on ‘SS’ or ‘TPI’

• e (Numpy array) – effective labor units

• p (OG-Core Specifications object) – model parameters

Returns:

pension amount for each household

Return type:

pension (Numpy array)

ogcore.pensions.replacement_rate_vals(nssmat, wss, factor_ss, j, p)

Calculates replacement rate values for the social security system.

\[\theta_{j,R,t+R} = \frac{PIA_{j,R,t+R} \times 12}{factor \times w_{t+R}}\]

Parameters:

• nssmat (Numpy array) – initial guess at labor supply, size = SxJ

• new_w (scalar) – steady state real wage rate

• factor_ss (scalar) – scaling factor converting model units to dollars

• j (int) – index of lifetime income group

• p (OG-Core Specifications object) – model parameters

Returns:

social security replacement rate value for

lifetime income group j

Return type:

theta (Numpy array)