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REPRO-Bench / 107 /replication_package /Code /MATLAB /compute_efficiency.m
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function results = compute_efficiency(commit_models,no_commit_models, ...
no_p_commit_models)
% compute efficiency of behavior change & other outcomes from the
% commitment contract and two piece-rate incentives
n = length(commit_models);
% compute average change in attendance from the contract
delta_a_Cs = [];
avg_delta_a_C = 0;
for i = 1:n
avg_delta_a_C = avg_delta_a_C + (commit_models(i).mu .* ...
(commit_models(i).avg_att - no_p_commit_models(i).avg_att));
delta_a_Cs = [delta_a_Cs, commit_models(i).avg_att - no_p_commit_models(i).avg_att];
end
results.avg_delta_a_C = avg_delta_a_C;
% compute average period 1 incremental welfare from the contract
avg_delta_surplus = 0;
for i = 1:n
avg_delta_surplus = avg_delta_surplus + (commit_models(i).mu .* ...
(commit_models(i).t1_utility - no_p_commit_models(i).t1_utility));
end
results.avg_delta_surplus = avg_delta_surplus;
% compute the surplus just for those who took up the contract
mu_commit = 0;
for i = 1:n
mu_commit = mu_commit + commit_models(i).mu;
end
results.commit_delta_surplus = avg_delta_surplus / mu_commit;
% compute average change in expected benefits from the contract
avg_delta_benefits = 0;
for i = 1:n
avg_delta_benefits = avg_delta_benefits + ...
(commit_models(i).mu .* commit_models(i).b .* ...
(commit_models(i).avg_att - no_p_commit_models(i).avg_att));
end
results.avg_delta_benefits = avg_delta_benefits;
% compute average change in expected costs from the contract
avg_delta_costs = 0;
for i = 1:n
avg_delta_costs = avg_delta_costs + (commit_models(i).mu .* ...
(commit_models(i).avg_costs - no_p_commit_models(i).avg_costs));
end
results.avg_delta_costs = avg_delta_costs;
% compute efficiency of behavior change from the commitment contract
results.V_E = results.avg_delta_benefits - results.avg_delta_costs;
% compute the optimal r
syms r
eqns_n = []; % placeholder for numerator
eqns_d = []; % placeholder for denominator
for k = 1:n
eqns_n = [ eqns_n; commit_models(k).mu .* (1 - commit_models(k).beta) .* ...
commit_models(k).b .* commit_models(k).beta .* ...
commit_models(k).f(commit_models(k).beta .* (commit_models(k).b + r)) ];
eqns_n = [ eqns_n; no_commit_models(k).mu .* (1 - no_commit_models(k).beta) .* ...
no_commit_models(k).b .* no_commit_models(k).beta .* ...
no_commit_models(k).f(no_commit_models(k).beta .* (no_commit_models(k).b + r)) ];
eqns_d = [ eqns_d; commit_models(k).mu .* (commit_models(k).beta .^ 2) .* ...
commit_models(k).f(commit_models(k).beta .* (commit_models(k).b + r)) ];
eqns_d = [ eqns_d; no_commit_models(k).mu .* (no_commit_models(k).beta .^ 2) .* ...
no_commit_models(k).f(no_commit_models(k).beta .* (no_commit_models(k).b + r)) ];
end
r_opt = vpasolve(sum(eqns_n) ./ sum(eqns_d) == r, r);
results.r_opt = double(r_opt);
% compute outcomes under r_opt
results.U_E_r_opt = compute_incentive_efficiency(results.r_opt, ...
commit_models, no_commit_models);
[results.delta_surplus_r_opt, results.delta_benefits_r_opt, ...
results.delta_costs_r_opt, results.delta_att_r_opt] = ...
compute_incentive_outcomes(results.r_opt, ...
commit_models, no_commit_models);
% find the r that generates the same expected change in attendance
syms r
eqns = [];
for k = 1:n
eqns = [ eqns; (commit_models(k).mu .* ...
(commit_models(k).F(commit_models(k).beta .* ...
(commit_models(k).b + r)) ...
- commit_models(k).F(commit_models(k).beta .* ...
commit_models(k).b))) ];
eqns = [ eqns; (no_commit_models(k).mu .* ...
(no_commit_models(k).F(no_commit_models(k).beta .* ...
(no_commit_models(k).b + r)) ...
- no_commit_models(k).F(no_commit_models(k).beta .* ...
no_commit_models(k).b))) ];
end
r_att = vpasolve(commit_models(1).T .* sum(eqns) == avg_delta_a_C, r);
results.r_att = double(r_att);
% compute outcomes under r_att
results.U_E_r_att = compute_incentive_efficiency(results.r_att, ...
commit_models, no_commit_models);
[results.delta_surplus_r_att, results.delta_benefits_r_att, ...
results.delta_costs_r_att, results.delta_att_r_att] = ...
compute_incentive_outcomes(results.r_att, ...
commit_models, no_commit_models);
end
%% Helper functions
function U_E = compute_incentive_efficiency(r, commit_models, ...
no_commit_models)
% compute the efficiency of behavior change under incentive r
N = length(commit_models);
U_E = 0;
inc = [r 0];
for j = 1:N
commit_models(j) = commit_models(j).compute_incentive_behavior(inc);
U_E = U_E + commit_models(j).mu .* ...
(commit_models(j).net_surplus_r(1) - ...
commit_models(j).net_surplus_r(2));
no_commit_models(j) = no_commit_models(j).compute_incentive_behavior(inc);
U_E = U_E + no_commit_models(j).mu .* ...
(no_commit_models(j).net_surplus_r(1) - ...
no_commit_models(j).net_surplus_r(2));
end
end
function [delta_surplus, delta_benefits, delta_costs, delta_a_r] = ...
compute_incentive_outcomes(r, commit_models, no_commit_models)
% compute the expected change in agent surplus, benefits, and costs
% under incentive r
N = length(commit_models);
delta_surplus = 0;
delta_benefits = 0;
delta_costs = 0;
delta_a_r = 0;
inc = [r 0];
for j = 1:N
commit_models(j) = commit_models(j).compute_incentive_behavior(inc);
no_commit_models(j) = no_commit_models(j).compute_incentive_behavior(inc);
% expected change in agent surplus
delta_surplus = delta_surplus + commit_models(j).mu .* ...
(commit_models(j).surplus_r(1) - ...
commit_models(j).surplus_r(2));
delta_surplus = delta_surplus + no_commit_models(j).mu .* ...
(no_commit_models(j).surplus_r(1) - ...
no_commit_models(j).surplus_r(2));
% expected change in health benefits
delta_benefits = delta_benefits + commit_models(j).mu .* ...
(commit_models(j).benefits_r(1) - ...
commit_models(j).benefits_r(2));
delta_benefits = delta_benefits + no_commit_models(j).mu .* ...
(no_commit_models(j).benefits_r(1) - ...
no_commit_models(j).benefits_r(2));
% expected change in costs
delta_costs = delta_costs + commit_models(j).mu .* ...
(commit_models(j).costs_r(1) - ...
commit_models(j).costs_r(2));
delta_costs = delta_costs + no_commit_models(j).mu .* ...
(no_commit_models(j).costs_r(1) - ...
no_commit_models(j).costs_r(2));
% expected change in attendance
delta_a_r = delta_a_r + commit_models(j).mu .* ...
commit_models(j).change_att_r(1);
delta_a_r = delta_a_r + no_commit_models(j).mu .* ...
no_commit_models(j).change_att_r(1);
end
end