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