Solve for b, UI benefits share of unemployment income loss recovered #18

Given P(U|Age, Edu) in 2009, given unemploymnet duration parameter 
xi=0.532, solve for the b parameter, the share of lost income recovred 
by unemployment UI benefits.

We know total wage income spent on UI 2.1 percent in 2009. So:

1. we find total wage income given xi and unemployment probabilities in 
2009
2. We multiply (1) by 0.021 to get total spending on UI
3. We divide (2) by lost income due to unemployment given xi and 
unemployment probability by age and education to find b.

PrjOptiSNW/calibrate/snw_calibrate_2009_b.m

@@ -0,0 +1,275 @@
+%% SNW_calibrate_2009 UI Support for Lost Wages Calibration
+%    The ratio of UI benefits to wages and salary is 2.1 percent in 2009.
+%    xi in [0,1] governs the duration of unemployment shock for those
+%    unemployed. This equals to 0.532 in 2009 (xi=0 no wages earned).
+%
+%    We solve for total wage earnings from unemployed and employed in 2009,
+%    for employed, same as under steady-state. For unemployed, they lose
+%    (1-xi) share of the wage they would otherwise have earned. Our
+%    unemployment probability in 2009 is conditional on age and edu groups
+%    (SNW_UNEMP_2008.m) computed based on rectiilnear restriction.
+%
+%    We know total UI amount (multiply its share of total "Wages and
+%    salary" by total "wages and salary". We know how much wage was lost
+%    due to xi. The ratio of these two levels is b, which is the
+%    parameter that is the share of lost-wage recovered. Note that this is
+%    based on exogenous wage earnings, so we do not have to worry about
+%    endogenous changes to savings. We will solve for the steady-state
+%    distribution, which generates mass of people by age, education,
+%    marital status, kids count, etc.
+%
+%    [FL_B_CALIBRATED_BY_UI_SHARE, MP_STATS_WAGE_UI_SPENDING,
+%    MN_EARN_TOT_WGTED, MN_EARN_UNEMP_WGTED, MN_EARN_UNEMP_TOT_WGTED,
+%    MN_EARN_UNEMP_WEIGHTED_WGTED] = SNW_CALIBRATE_2009_B(MP_PARAMS,
+%    MP_CONTROLS, FL_RATIO_UI_BENEFITS_TO_WAGE) where
+%    FL_RATIO_UI_BENEFITS_TO_WAGE is the ratio of UI benefits to teh total
+%    wages and salary bill, including wage for the household head and
+%    pouse, considering unemployment probability by states and loss of
+%    income due to unemployment duration.
+%
+%    See also SNWX_V0808_JAEEMK, SNW_V0808_JAEEMK, SNWX_UNEMP_2008,
+%    SNW_UNEMP_2008, SNW_UNEMP_2008
+%
+
+%%
+function [varargout]=snw_calibrate_2009_b(varargin)
+
+%% Default and Parse
+if (~isempty(varargin))
+
+    fl_ratio_ui_benefits_to_wage = 0.021;
+    if (length(varargin)==2)
+        [mp_params, mp_controls] = varargin{:};
+    elseif (length(varargin)==3)
+        [mp_params, mp_controls, fl_ratio_ui_benefits_to_wage] = varargin{:};
+    else
+        error('Need to provide 2/3 parameter inputs');
+    end
+
+else
+    clc;
+    close all;
+
+    % Solve the VFI Problem and get Value Function
+    mp_more_inputs = containers.Map('KeyType','char', 'ValueType','any');
+    mp_more_inputs('fl_ss_non_college') = 0.225;
+    mp_more_inputs('fl_ss_college') = 0.271;
+    mp_more_inputs('fl_scaleconvertor') = 54831;
+    st_param_group = 'default_small';
+%     st_param_group = 'default_dense';
+    mp_params = snw_mp_param(st_param_group, false, 'tauchen', false, 8, 8, mp_more_inputs);
+    mp_controls = snw_mp_control('default_test');
+
+    % no b, solving for b, b set to 0 when solving for wages
+    % set Unemployment Related Variables
+    xi=0.532; % Proportional reduction in income due to unemployment (xi=0 refers to 0 labor income; xi=1 refers to no drop in labor income)
+    % TR parameter does not matter below
+    TR=100/54831; % Value of a welfare check (can receive multiple checks). TO DO: Update with alternative values
+
+    mp_params('xi') = xi;
+    mp_params('TR') = TR;
+
+    % Solve for Unemployment Values
+    mp_controls('bl_print_calibrate_2009') = true;
+    mp_controls('bl_print_calibrate_2009_verbose') = true;
+
+    fl_ratio_ui_benefits_to_wage = 0.021;
+
+end
+
+%% Set b = 0
+% solving for b, b=0 to get wages without b
+fl_b_zero = 0;
+mp_params('b') = 0;
+
+%% Parse Model Parameters
+params_group = values(mp_params, {'theta', 'r' , 'jret'});
+[theta,  r, jret] = params_group{:};
+
+params_group = values(mp_params, {'Bequests', 'bequests_option', 'throw_in_ocean'});
+[Bequests, bequests_option, throw_in_ocean] = params_group{:};
+
+params_group = values(mp_params, {'agrid', 'eta_H_grid', 'eta_S_grid'});
+[agrid, eta_H_grid, eta_S_grid] = params_group{:};
+
+params_group = values(mp_params, {'epsilon', 'SS'});
+[epsilon, SS] = params_group{:};
+
+params_group = values(mp_params, ...
+    {'n_jgrid', 'n_agrid', 'n_etagrid', 'n_educgrid', 'n_marriedgrid', 'n_kidsgrid'});
+[n_jgrid, n_agrid, n_etagrid, n_educgrid, n_marriedgrid, n_kidsgrid] = params_group{:};
+
+params_group = values(mp_params, {'pi_unemp_2009_edu_age'});
+[pi_unemp_2009_edu_age] = params_group{:};
+
+params_group = values(mp_params, {'xi','b'});
+[xi, b] = params_group{:};
+
+%% Parse Model Controls
+% Profiling Controls
+params_group = values(mp_controls, {'bl_timer'});
+[bl_timer] = params_group{:};
+
+% Display Controls
+params_group = values(mp_controls, {'bl_print_calibrate_2009', 'bl_print_calibrate_2009_verbose'});
+[bl_print_calibrate_2009, bl_print_calibrate_2009_verbose] = params_group{:};
+
+%% Timing and Profiling Start
+if (bl_timer)
+    tm_start = tic;
+end
+
+%% Solve for distribution
+% the savings distirbution will not impact later calculations, since
+% calcualtions based on wages only, not savings. But will compute the
+% distributions to get the joint distribution of age, children count,
+% marital status, etc.
+
+[V_ss, ap_ss, cons_ss] = snw_vfi_main_bisec_vec(mp_params, mp_controls);
+[Phi_true, Phi_adj_ss] = snw_ds_main_vec(mp_params, mp_controls, ap_ss, cons_ss);
+
+
+%% Solve 08 Policy and Value given 09 Unemployment shock
+% 09 has two states, employed or not (extensive), and if unemployed (intensive duration)
+
+% mn_earn_tot = wage earning household head + household-head-spouse (no interest no SS)
+mn_earn_tot = NaN(n_jgrid,n_agrid,n_etagrid,n_educgrid,n_marriedgrid,n_kidsgrid);
+% mn_earn_unemp = household head only wage earning under unemployment
+mn_earn_unemp = NaN(n_jgrid,n_agrid,n_etagrid,n_educgrid,n_marriedgrid,n_kidsgrid);
+% mn_earn_unemp_tot = wage earning household head + household-head-spouse (no interest no SS, no UI benefits) under unemployment
+mn_earn_unemp_tot = NaN(n_jgrid,n_agrid,n_etagrid,n_educgrid,n_marriedgrid,n_kidsgrid);
+% mn_earn_unemp_weighted = weighted sum of mn_earn_tot and mn_earn_unemp_tot
+mn_earn_unemp_weighted = NaN(n_jgrid,n_agrid,n_etagrid,n_educgrid,n_marriedgrid,n_kidsgrid);
+
+% Loop Over States and Pre-Store
+for j=1:n_jgrid
+    for a=1:n_agrid
+        for eta=1:n_etagrid
+            for educ=1:n_educgrid
+                for married=1:n_marriedgrid
+                    for kids=1:n_kidsgrid
+
+                        % 1. Get unemployment probability conditional on age and edu
+                        % columns are edu: 1st column high school; 2nd col college
+                        % rows are age groups: 18--24; 25-54; 55 to 65
+                        if (j<=7)
+                            it_age_grp = 1;
+                        elseif (j<=37)
+                            it_age_grp = 2;
+                        else
+                            it_age_grp = 3;
+                        end
+                        fl_unemp_edu_age = pi_unemp_2009_edu_age(it_age_grp, educ);
+
+                        % 2. Income and Wages employed
+                        % inc = SS + wages + interest earnings + bequest
+                        % earn = wages
+                        [inc, earn]=snw_hh_individual_income(j,a,eta,educ,...
+                            theta, r, agrid, epsilon, eta_H_grid, SS, Bequests, bequests_option);
+                        spouse_inc=snw_hh_spousal_income(j,educ,kids,earn,SS(j,educ), jret);
+                        % total household income and earnings
+                        earn_tot = earn+(married-1)*spouse_inc*exp(eta_S_grid(eta));
+
+                        % 3. Income unemployed
+                        % inc_umemp = SS + wages*(xi+b*(1-xi)) + interest earnings + bequest
+                        % earn_unemp = wages*(xi+b*(1-xi)), NOT multiplied by (xi+b*(1-xi))
+                        [inc_umemp,earn_unemp]=snw_hh_individual_income(j,a,eta,educ,...
+                            theta, r, agrid, epsilon, eta_H_grid, SS, Bequests, bequests_option,...
+                            xi,b);
+                        % We assume spousal income NOT impcated by (xi+b*(1-xi))
+                        spouse_inc_unemp=snw_hh_spousal_income(j,educ,kids,earn_unemp,SS(j,educ), jret);
+                        % Total household income and earnings under unemployment
+                        % under earn_unemp_tot, note UI = 0, b = 0
+                        % xi=0 means no wage earnings in 2009
+                        earn_unemp_tot = earn_unemp*(xi)+(married-1)*spouse_inc_unemp*exp(eta_S_grid(eta));
+
+                        % 4. Collect results to mn Matrixes
+                        mn_earn_tot(j,a,eta,educ,married,kids) = earn_tot*(1 - fl_unemp_edu_age);
+                        mn_earn_unemp(j,a,eta,educ,married,kids) = (earn_unemp*(xi))*fl_unemp_edu_age;
+                        mn_earn_unemp_tot(j,a,eta,educ,married,kids) = earn_unemp_tot*fl_unemp_edu_age;
+                        mn_earn_unemp_weighted(j,a,eta,educ,married,kids) = ...
+                            fl_unemp_edu_age*earn_unemp_tot + (1 - fl_unemp_edu_age)*earn_tot;
+
+                    end
+                end
+            end
+        end
+    end
+end
+
+%% Compute weighted values, weight with mass by states
+mn_earn_tot_wgted = Phi_true.*mn_earn_tot;
+mn_earn_unemp_wgted = Phi_true.*mn_earn_unemp;
+mn_earn_unemp_tot_wgted = Phi_true.*mn_earn_unemp_tot;
+mn_earn_unemp_weighted_wgted = Phi_true.*mn_earn_unemp_weighted;
+
+%% Compute b: The Ratio of UI Benefits to Wage is 2.1 percent in 2009
+% 1. we know the total wages, household head and household spouse
+fl_total_wage = sum(mn_earn_unemp_weighted_wgted,'all');
+
+% 2. what is the total b spending?
+% fl_ratio_ui_benefits_to_wage = 0.021;
+fl_total_b_spending = fl_total_wage*fl_ratio_ui_benefits_to_wage;
+
+% 3. Total wage earning by household head unemployed (with unemployment duration xi)
+fl_total_wage_unemp_hhhead = sum(mn_earn_unemp_wgted, 'all');
+
+% 4. Wages lost for unemployed household head
+fl_total_wage_unemp_hhhead_lost = (fl_total_wage_unemp_hhhead/xi)*(1-xi);
+
+% 5. Compute the b, share of lost income recovered parameter.
+fl_b_calibrated_by_ui_share = fl_total_b_spending/fl_total_wage_unemp_hhhead_lost;
+
+% Gather statistics
+mp_stats_wage_ui_spending = containers.Map('KeyType', 'char', 'ValueType', 'any');
+mp_stats_wage_ui_spending('fl_total_wage') = fl_total_wage;
+mp_stats_wage_ui_spending('fl_total_b_spending') = fl_total_b_spending;
+mp_stats_wage_ui_spending('fl_total_wage_unemp_hhhead') = fl_total_wage_unemp_hhhead;
+mp_stats_wage_ui_spending('fl_total_wage_unemp_hhhead_lost') = fl_total_wage_unemp_hhhead_lost;
+mp_stats_wage_ui_spending('fl_b_calibrated_by_ui_share') = fl_b_calibrated_by_ui_share;
+
+%% Timing and Profiling End
+if (bl_timer)
+    tm_end = toc(tm_start);
+    st_complete = strjoin(...
+        ["Completed SNW_calibrate_2009", ...
+         ['SNW_MP_PARAM=' char(mp_params('mp_params_name'))], ...
+         ['SNW_MP_CONTROL=' char(mp_controls('mp_params_name'))], ...
+         ['time=' num2str(tm_end)] ...
+        ], ";");
+    disp(st_complete);
+end
+
+%% Print
+if (bl_print_calibrate_2009)
+    ff_container_map_display(mp_stats_wage_ui_spending, 9, 9);
+    if (bl_print_calibrate_2009_verbose)
+        mp_outcomes = containers.Map('KeyType', 'char', 'ValueType', 'any');
+        mp_outcomes('mn_earn_tot_wgted') = mn_earn_tot_wgted;
+        mp_outcomes('mn_earn_unemp_wgted') = mn_earn_unemp_wgted;
+        mp_outcomes('mn_earn_unemp_tot_wgted') = mn_earn_unemp_tot_wgted;
+        mp_outcomes('mn_earn_unemp_weighted_wgted') = mn_earn_unemp_weighted_wgted;
+        ff_container_map_display(mp_outcomes, 9, 9);
+    end
+end
+
+%% Return
+varargout = cell(nargout,0);
+for it_k = 1:nargout
+    if (it_k==1)
+        ob_out_cur = fl_b_calibrated_by_ui_share;
+    elseif (it_k==2)
+        ob_out_cur = mp_stats_wage_ui_spending;
+    elseif (it_k==3)
+        ob_out_cur = mn_earn_tot_wgted;
+    elseif (it_k==4)
+        ob_out_cur = mn_earn_unemp_wgted;
+    elseif (it_k==5)
+        ob_out_cur = mn_earn_unemp_tot_wgted;
+    elseif (it_k==6)
+        ob_out_cur = mn_earn_unemp_weighted_wgted;
+    end
+    varargout{it_k} = ob_out_cur;
+end
+
+end

PrjOptiSNW/doc/calibrate/preamble.yml

@@ -45,3 +45,28 @@
         in_header: '../../../hdga.html'
   always_allow_html: true
   urlcolor: blue
+
+- file: snwx_calibrate_2009_b
+  title: "UI Benefit Unemployment Lost Wage Recovery Parameter b Calibration"
+  titleshort: "UI Benefit Unemployment Lost Wage Recovery Parameter b Calibration"
+  description: |
+    The ratio of UI benefits to wages and salary is 2.1 percent in 2009. 0 < xi < 1 governs the duration of unemployment shock for those unemployed. This equals to 0.532 in 2009 (xi = 0 no wages earned).
+    We solve for total wage earnings from unemployed and employed in 2009, for employed, same as under steady-state. For unemployed, they lose (1 - xi) share of the wage they would otherwise have earned. Our unemployment probability in 2009 is conditional on age and edu groups (SNW_UNEMP_2008.m) computed based on rectiilnear restriction.
+    We know total UI amount (multiply its share of total "Wages and salary" by total "wages and salary". We know how much wage was lost due to xi. The ratio of these two levels is b, which is the parameter that is the share of lost-wage recovered.
+  core :
+    - package: PrjOptiSNW
+      code: |
+        [snw_calibrate_2009_b()](https://github.com/FanWangEcon/PrjOptiSNW/blob/master/PrjOptiSNW/calibrate/snw_calibrate_2009_b.m)
+  date: 2021-11-26
+  date_start: 2021-11-26
+  output:
+    pdf_document:
+      pandoc_args: '../../../_output_kniti_pdf.yaml'
+      includes:
+        in_header: '../../../preamble.tex'
+    html_document:
+      pandoc_args: '../../../_output_kniti_html.yaml'
+      includes:
+        in_header: '../../../hdga.html'
+  always_allow_html: true
+  urlcolor: blue