%  share.m
%  Pareto optimal allocations in static setting with power certainty equivalent 
%  Details:  two agents, potentially diff risk aversion parameters 
%  Notation:  ci^(1-alphai)/(1-alphai)
format compact
clear all 
clf
maxit = 20;
eps = 1.e-10;
FontSize = 12;
disp('Allocations for 2-agent risk-sharing problem')
disp('********************************************')
disp(' ')

%  parameters
alpha1 = 1.0;
alpha2 = 2.0;
%lambda = 1;

%  Output and probs
y = 2*[0.8 0.9 1 1.1 1.2]';
y = 2*[0.5 0.75 1 1.25 1.5]';
p = [1 1 1 1 1]'/5;
ybar = sum(p.*y);

%  Initializations
theta1 = 1;
theta2 = 1; 
c = y/2;        
f = theta1*c.^(-alpha1) - theta2*(y-c).^(-alpha2);
%  Newton iterations 
for it = 1:maxit
    fp = -alpha1*theta1*c.^(-alpha1-1) - alpha2*theta2*(y-c).^(-alpha2-1);
%    fp2 = (theta1*(c+eps).^(-alpha1) - theta2*(y-c-eps).^(-alpha2) - f)./eps;
%    [c f fp]
    cnext = c - f./fp;
    fnext = theta1*cnext.^(-alpha1) - theta2*(y-cnext).^(-alpha2);
    fnorm = max(abs(f-fnext));
    cnorm = max(abs(c-cnext));
    if fnorm < eps, break, end 
    f = fnext;
    c = cnext;
end 
disp('Converged on iteration')
it 
disp('Norms should be zero') 
[fnorm cnorm] 

c1 = c;
c2 = y - c1;
figure(1)
plot(c1,c2,'*')
%axis equal
axis([0 2 0 2])
xlabel('Consumption of Agent 1','FontSize',FontSize)
ylabel('Consumption of Agent 2','FontSize',FontSize)
hold on 

%  log-linear approximation
c1grid = [0.1:0.05:1.9]';
c2grid = c1grid.^(alpha1/alpha2);
plot(c1grid,c2grid,'-')
plot(c1grid,c1grid,':') 

print -deps riskshare1fig1.eps


%  Allocation rule (log-lin approx) 
ygrid = [1:0.05:3]';
c1grid = (ygrid/2).^(2*alpha2/(alpha1+alpha2));

figure(2)
plot(ygrid,c1grid,'-')
hold on
plot(y,c1,'*')
plot(y,y/2,':') 
ylabel('Consumption of Agent 1','FontSize',FontSize)
xlabel('Aggregate Endowment','FontSize',FontSize)

print -deps riskshare1fig2.eps

return