numerical analysis 0909
question 1: Write the “cournot” function to return the function value fval and the jacobian value fjac for the system of first-order conditions.

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%matlab code:
%create a function named "cournot"
function [fval, fjac] = cournot(q)

%set the parameters
c=[0.6;0.8];eta=1.6;e=-1/eta;

%define the fval, f here is first-order equations exactly.
fval =sum(q)^e+e*sum(q)^(e-1)*q-diag(c)*q;

%define the fjac
fjac =e*sum(q)^(e-1)*ones(2,2)+e*sum(q)^(e-1)*eye(2)+(e-1)*e*sum(q)^(e-2)*q*[1 1]-diag(c);

%main file
q=newton('cournot', [0.2;0.2]);

%get the answer
%[0.839567603529586;0.688796431162672]

qusetion 2: Write the “cournot2” function to return the same output as in E1, but using symbolic expressions in MATLAB

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%matlab code
%create a function named "cournot2"
function [fval, fjac] = cournot2(q)

%set the parameters
c=[0.6;0.8];eta=1.6;e=-1/eta;

%define the symbolic expressions
syms q1 q2;

%define the function f
f=[(q1+q2)^e+e*(q1+q2)^(e-1)*q1-c(1)*q1;(q1+q2)^e+e*(q1+q2)^(e-1)*q2-c(2)*q2];

%define the jacobian
J=jacobian(f,[q1,q2]);

%get the actual value of f and J at certain points
fval=double(subs(f,[q1 q2],[q(1) q(2)]));fjac=double(subs(J,[q1 q2],[q(1) q(2)]));

%main file
q=newton('cournot2',[0.2;0.2]);

%get the answer
%[0.839567603529586;0.688796431162672]

question 3: In your main file, write the algorithm to implement the newton method
matlab code:

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%main file
%set the tolerance
tol=1e-8;
%give the initial guess
q=[0.2;0.2];

%set the for loop
for it=1:100 
%get the fval and fjac from cournot2 
[fval,fjac]=cournot2(q); 
%use the newton method to update 
q=q-fjac\fval; 
%check whether the root is closer enough 
if norm(fval)<tol, break, end

end

%get the answer
%[0.839567603535660;0.688796431163001]

question 4: Solve the Cournot game, by considering a planner.

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%just derive the new first-order equations and put them into the code of cournot2
%create a function named "cournot3"
function [fval, fjac] = cournot3(q)

%set the parameters
c=[0.6;0.8];eta=1.6;e=-1/eta;

%define the symbolic expressions
syms q1 q2;

%define the function f
f=[(1+e)*(q1+q2)^e-c(1)*q1;(1+e)*(q1+q2)^e-c(2)*q2];

%define the jacobian
J=jacobian(f,[q1,q2]);

%get the actual value of f and J at certain points
fval=double(subs(f,[q1 q2],[q(1) q(2)]));fjac=double(subs(J,[q1 q2],[q(1) q(2)]));

%main file
q=newton('cournot3',[0.2;0.2]);

%get the answer
%[0.603825598888436;0.452869199166327]