1 Model Lockdown Calibration
Taking advantage of snw_calibrate_lockdown_c from the PrjOptiSNW Package. This function finds the proportional discount to current utility in one period to account for aggregate reducntions in consumption during lockdown.
1.1 Lock Down Impact on Consumption
"To calibrate the drop in marginal utility, we estimate that 10.9 percent of the goods that make up the consumer price index become highly undesirable, or simply unavailable, during the pandemic: food away from home, public transportation including airlines, and motor fuel. As we use a coefficient of risk aversion equal to one, we simply multiply utility from consumption during the period of the epidemic by a factor of 0.891"
There is one MIT shock period in which households face a one-period change in the current utility of consumption due to lock down. Solve the model and evaluate the effects on aggregate consumption (during the MIT shock period) with different proportional adjustments on consumption given differing intertemperal preference assumptions.
1.2 Graphical Illustraiton for Gamma=2 (docdense) and Gamma=1 (dense)
Solved Gamma=2 with docdense (81x5 shocks), and Gamma=1 with dense (7x5) shocks. Did not have time to resolve Gamma=1 with docdense.
Solved for gamma equals to 2, results visualized below.
% Percentage C drop desired
fl_c_drop_percent = 0.109;First, grid of proportional one period current utility shifts:
% Grid of proportional one period current utility shifts.
[fl_invbtlock_min, fl_invbtlock_max, it_invbtlock_points] = deal(0, 1, 20);
st_grid_type = 'grid_powerspace';
mp_grid_control = containers.Map('KeyType', 'char', 'ValueType', 'any');
mp_grid_control('grid_powerspace_power') = 1.5;
[ar_fl_invbtlock] = ff_saveborr_grid(...
fl_invbtlock_min, fl_invbtlock_max, it_invbtlock_points, ...
st_grid_type, mp_grid_control);
ar_fl_invbtlock = 1-ar_fl_invbtlock;
% display
% disp(ar_fl_invbtlock);Second, stored aggregate consumption values from docdense:
% From gamma=2 docdense
ar_fl_cons_mean_betaedu_innerwgt_gamma2_docdense = [...
1.0466,1.0434,1.0374,1.0293,...
1.0191,1.0067,0.99214,0.97543,...
0.95642,0.93482,0.90995,0.88078,...
0.847,0.80734,0.76126,0.70507,...
0.635, 0.54343, 0.40755, 0.00013065];
% From gamma=1 dense
ar_fl_cons_mean_betaedu_innerwgt_gamma1_dense = [...
1.2517,1.2467,1.237,1.2228,...
1.2031,1.1783,1.1489,1.1147,...
1.0754,1.0311,0.98169,0.92291,...
0.85528,0.77933,0.69408,0.5964,...
0.47976, 0.34416, 0.18672, 0.00013642];Third, polynomial fit (4th order) between the 3rd and 14th point:
% Generate polynomial fit coefficients
ft_polynomial_gamma2_docdense = polyfit(ar_fl_invbtlock(3:13), ...
ar_fl_cons_mean_betaedu_innerwgt_gamma2_docdense(3:13), 4);
ft_polynomial_gamma1_dense = polyfit(ar_fl_invbtlock(3:13), ...
ar_fl_cons_mean_betaedu_innerwgt_gamma1_dense(3:13), 4);
% Evaluate polynomial fits
ar_fl_cons_mean_polyfit_gamma2_docdense = ...
polyval(ft_polynomial_gamma2_docdense, ar_fl_invbtlock)';
ar_fl_cons_mean_polyfit_gamma1_dense = ...
polyval(ft_polynomial_gamma1_dense, ar_fl_invbtlock)';Fourth, find which proportional current utility change in 2020 matches a 10.9 percent drop in aggregate consumption in 2020:
% Identify the exact point along polynomial where c drops by 10.9 percent
% Gamma = 2, docdense
ar_fl_invbtlock_interpgrid = linspace(0.50, 0.95, 100000);
ar_fit_reduce_grid = 1 - ...
polyval(ft_polynomial_gamma2_docdense, ar_fl_invbtlock_interpgrid)...
./ar_fl_cons_mean_polyfit_gamma2_docdense(1);
[fl_mingap, it_minidx] = min(abs(ar_fit_reduce_grid-fl_c_drop_percent));
fl_fit_best_reduce = ar_fl_invbtlock_interpgrid(it_minidx);
fl_lockdown_u_prop_reduction_gamma2_docdense = (1-fl_fit_best_reduce)*100;
st_reduction_gamma2_docdense = ['current util in 2020 multiply by ', ...
num2str(fl_fit_best_reduce), ...
' for 10.9 percent drop in C'];
% Gamma = 1, dense
ar_fl_invbtlock_interpgrid = linspace(0.50, 0.95, 100000);
ar_fit_reduce_grid = 1 - ...
polyval(ft_polynomial_gamma1_dense, ar_fl_invbtlock_interpgrid)...
./ar_fl_cons_mean_polyfit_gamma1_dense(1);
[fl_mingap, it_minidx] = min(abs(ar_fit_reduce_grid-fl_c_drop_percent));
fl_fit_best_reduce = ar_fl_invbtlock_interpgrid(it_minidx);
fl_lockdown_u_prop_reduction_gamma1_dense = (1-fl_fit_best_reduce)*100;
st_reduction_gamma1_dense = ['current util in 2020 multiply by ', ...
num2str(fl_fit_best_reduce), ...
' for 10.9 percent drop in C'];Third, graphical illustration:
for it_graph_type=1:2
if (it_graph_type == 1)
st_title_add = 'Zoomed in';
else
st_title_add = 'All consumption drop levels';
end
for it_gamma=2:1:2
figure();
hold on;
if (it_graph_type == 1 && it_gamma == 1)
ar_fl_cons_mean_betaedu_innerwgt_select = ...
ar_fl_cons_mean_betaedu_innerwgt_gamma1_dense(1:12);
ar_fl_cons_mean_polyfit_select = ...
ar_fl_cons_mean_polyfit_gamma1_dense(1:12);
st_gamma = 'Gamma = 1 (crra Parameter, 7x5), (2020 no Trump check, xi=0, b=1)';
fl_lockdown_u_prop_reduction = fl_lockdown_u_prop_reduction_gamma1_dense;
st_reduction = st_reduction_gamma1_dense;
elseif (it_graph_type == 1 && it_gamma == 2)
ar_fl_cons_mean_betaedu_innerwgt_select = ...
ar_fl_cons_mean_betaedu_innerwgt_gamma2_docdense(1:12);
ar_fl_cons_mean_polyfit_select = ...
ar_fl_cons_mean_polyfit_gamma2_docdense(1:12);
st_gamma = 'Gamma = 2 (crra Parameter, 81x5), (2020 no Trump check, xi=0, b=1)';
fl_lockdown_u_prop_reduction = fl_lockdown_u_prop_reduction_gamma2_docdense;
st_reduction = st_reduction_gamma2_docdense;
elseif (it_graph_type == 2 && it_gamma == 1)
ar_fl_cons_mean_betaedu_innerwgt_select = ...
ar_fl_cons_mean_betaedu_innerwgt_gamma1_dense;
ar_fl_cons_mean_polyfit_select = ...
ar_fl_cons_mean_polyfit_gamma1_dense;
st_gamma = 'Gamma = 1 (crra Parameter, 7x5), (2020 no Trump check, xi=0, b=1)';
fl_lockdown_u_prop_reduction = fl_lockdown_u_prop_reduction_gamma1_dense;
st_reduction = st_reduction_gamma1_dense;
elseif (it_graph_type == 2 && it_gamma == 2)
ar_fl_cons_mean_betaedu_innerwgt_select = ...
ar_fl_cons_mean_betaedu_innerwgt_gamma2_docdense;
ar_fl_cons_mean_polyfit_select = ...
ar_fl_cons_mean_polyfit_gamma2_docdense;
st_gamma = 'Gamma = 2 (crra Parameter, 81x5), (2020 no Trump check, xi=0, b=1)';
fl_lockdown_u_prop_reduction = fl_lockdown_u_prop_reduction_gamma2_docdense;
st_reduction = st_reduction_gamma2_docdense;
end
ar_fl_invbtlock_select = ar_fl_invbtlock(1:length(ar_fl_cons_mean_betaedu_innerwgt_select));
ar_x = 1-ar_fl_invbtlock_select;
ar_y = -(1-ar_fl_cons_mean_betaedu_innerwgt_select/ar_fl_cons_mean_betaedu_innerwgt_select(1));
ar_y_poly = -(1-ar_fl_cons_mean_polyfit_select/ar_fl_cons_mean_polyfit_select(1));
ar_x = ar_x*100;
ar_y = ar_y*100;
ar_y_poly = ar_y_poly*100;
% Points scatter
scatter(ar_x, ar_y, 300, [57 106 177]./255, 'd');
% Points polynomial approximate
pl_poly = plot(ar_x, ar_y_poly);
pl_poly.Color = [83 81 84]./255;
pl_poly.LineStyle = '-';
pl_poly.LineWidth = 2;
% Actual lines linearly connected
line = plot(ar_x, ar_y);
line.Color = [57 106 177]./255;
line.LineStyle = '--';
line.LineWidth = 3;
% X-axis for -10.9 percent
yline0 = yline(-10.9);
yline0.HandleVisibility = 'off';
yline0.Color = [204 37 41]./255;
yline0.LineStyle = '--';
yline0.LineWidth = 3;
% Y-axis for -10.9 percent along the polynomial
xline0 = xline(fl_lockdown_u_prop_reduction);
xline0.HandleVisibility = 'off';
xline0.Color = [204 37 41]./255;
xline0.LineStyle = '--';
xline0.LineWidth = 3;
% labeling
title({'Current Utility Lock Down',...
st_reduction, st_gamma, ...
st_title_add});
ylabel('Aggregate C Percent Reduction in Lockdown Year');
xlabel('Lock Down Current Utility Percent/Proportional Reduction');
grid on;
end
end
Fourth, tabular display for Gamma 2:
%% Table Dispaly
% Generate Table
tb_show = array2table([ar_x,ar_y',ar_y_poly']);
it_num_rows = length(ar_x);
% Generate Row and Column Names
cl_col_names = {'Lock Down Util Perc Reduce', 'Agg C Percent Reduc Lockdown Yr', 'Agg C Percent Reduc Polynomial Fit'};
cl_row_names = strcat('lockdown_', string((1:it_num_rows)));
tb_show.Properties.VariableNames = matlab.lang.makeValidName(cl_col_names);
tb_show.Properties.RowNames = matlab.lang.makeValidName(cl_row_names);
disp(tb_show);
LockDownUtilPercReduce AggCPercentReducLockdownYr AggCPercentReducPolynomialFit
______________________ __________________________ _____________________________
lockdown_1 0 0 0
lockdown_2 1.2075 -0.30575 -0.29672
lockdown_3 3.4152 -0.87904 -0.86414
lockdown_4 6.2741 -1.653 -1.643
lockdown_5 9.6596 -2.6276 -2.6234
lockdown_6 13.5 -3.8123 -3.8041
lockdown_7 17.746 -5.2035 -5.1879
lockdown_8 22.362 -6.8001 -6.7823
lockdown_9 27.322 -8.6165 -8.6029
lockdown_10 32.601 -10.68 -10.679
lockdown_11 38.183 -13.057 -13.058
lockdown_12 44.051 -15.844 -15.817
lockdown_13 50.193 -19.071 -19.067
lockdown_14 56.596 -22.861 -22.966
lockdown_15 63.25 -27.264 -27.728
lockdown_16 70.147 -32.632 -33.635
lockdown_17 77.277 -39.327 -41.055
lockdown_18 84.634 -48.077 -50.447
lockdown_19 92.21 -61.06 -62.385
lockdown_20 100 -99.988 -77.569