Solving N non-linear equations using fsolve. How do I pass these equations into my function without typing them out individually?

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Ethan Goode 2020-6-29

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评论： Celestine Siameh 2021-8-21

Hello all,

I am currently working with the Eaton-Kortum Trade Model in MATLAB. In this model we have N countries, and wish to solve 2N + N^2 non-linear equations for equilibrium outcomes. I am working currently on an example with four countries, which means I will need to use fsolve to solve 24 equations. I understand that I could type all 24 equations individually, but what happens when we allow N to grow in the model (to better reflect what the world looks like)? If I wanted to consider trade between 10 countries I would have to type 120 equations seperatley! Luckily these equations take one of three forms.

N of the equations take the form: (gam.*((sum(Ti.*(dni.*(w(i).^(beta)).*(p(i))).^(1-beta)).^(-theta)).^(-1./theta))) - p(n);

N^2 of the equations take the form: Ti.*(gam.*dni.*(w(i).^(beta))*(p(i).^(1-beta))*(1./p(i))).^(theta) - (x(i));

N of the equations take the form: ((beta).*(sum(Ti.*(gam.*dni.*(w(i).^(beta))*(p(i).^(1-beta))*(1./p(i))).^(theta)*x(i)))) - (w(i).*Li)

Where our unknowns are w's, p's, and x's and everything else is given.

Is there a way for me to iteratively feed these equations into fsolve?

For example:

F(1) = (gam.*((sum(Ti.*(dni.*(w(i).^(beta)).*(p(i))).^(1-beta)).^(-theta)).^(-1./theta))) - p(1);
F(2) = (gam.*((sum(Ti.*(dni.*(w(i).^(beta)).*(p(i))).^(1-beta)).^(-theta)).^(-1./theta))) - p(2);
F(3) = (gam.*((sum(Ti.*(dni.*(w(i).^(beta)).*(p(i))).^(1-beta)).^(-theta)).^(-1./theta))) - p(3);
F(4) = (gam.*((sum(Ti.*(dni.*(w(i).^(beta)).*(p(i))).^(1-beta)).^(-theta)).^(-1./theta))) - p(4);

If there is not a way to do what I suggest how should I attempt to implement this?

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Ethan Goode 2020-6-30

在 MATLAB Online 中打开

@ Walter Roberson

Could you provide a little more commentary?

Here is a valid way to solve my problem:

%I write a function s.t.

function F= ekmodelv2(x)
Ti= 1;
Li= 1;
dni = 2;
theta = 4;
gam = 1;
beta = 0.5;
%p(1:4) = x(1:4) These are the four price equations.
%x(1:16) = x(5:20) These are the sixteen trade share equations.
%w(1:4) = x(21:24) These are the four wage equations.
F(1) = (gam.*(((Ti.*(dni.*(x(21).^(beta))*(x(1).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(22).^(beta))*(x(2).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(23).^(beta))*(x(3).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(24).^(beta))*(x(4).^(beta))).^(-theta)))))-x(1);
F(2) = (gam.*(((Ti.*(dni.*(x(21).^(beta))*(x(1).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(22).^(beta))*(x(2).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(23).^(beta))*(x(3).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(24).^(beta))*(x(4).^(beta))).^(-theta)))))-x(2);
F(3) = (gam.*(((Ti.*(dni.*(x(21).^(beta))*(x(1).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(22).^(beta))*(x(2).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(23).^(beta))*(x(3).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(24).^(beta))*(x(4).^(beta))).^(-theta)))))-x(3);
F(4) = (gam.*(((Ti.*(dni.*(x(21).^(beta))*(x(1).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(22).^(beta))*(x(2).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(23).^(beta))*(x(3).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(24).^(beta))*(x(4).^(beta))).^(-theta)))))-x(4);
F(5) = Ti.*(gam.*dni.*(x(21).^(beta))*(x(1).^(1-beta))*(1./x(1))).^(theta) - (x(5));
F(6) = Ti.*(gam.*dni.*(x(21).^(beta))*(x(1).^(1-beta))*(1./x(2))).^(theta) - (x(6));
F(7) = Ti.*(gam.*dni.*(x(21).^(beta))*(x(1).^(1-beta))*(1./x(3))).^(theta) - (x(7));
F(8) = Ti.*(gam.*dni.*(x(21).^(beta))*(x(1).^(1-beta))*(1./x(4))).^(theta) - (x(8));
F(9) = Ti.*(gam.*dni.*(x(22).^(beta))*(x(2).^(1-beta))*(1./x(1))).^(theta) - (x(9));
F(10) = Ti.*(gam.*dni.*(x(22).^(beta))*(x(2).^(1-beta))*(1./x(2))).^(theta) - (x(10));
F(11) = Ti.*(gam.*dni.*(x(22).^(beta))*(x(2).^(1-beta))*(1./x(3))).^(theta) - (x(11));
F(12) = Ti.*(gam.*dni.*(x(22).^(beta))*(x(2).^(1-beta))*(1./x(4))).^(theta) - (x(12));
F(13) = Ti.*(gam.*dni.*(x(23).^(beta))*(x(3).^(1-beta))*(1./x(1))).^(theta) - (x(13));
F(14) = Ti.*(gam.*dni.*(x(23).^(beta))*(x(3).^(1-beta))*(1./x(2))).^(theta) - (x(14));
F(15) = Ti.*(gam.*dni.*(x(23).^(beta))*(x(3).^(1-beta))*(1./x(3))).^(theta) - (x(15));
F(16) = Ti.*(gam.*dni.*(x(23).^(beta))*(x(3).^(1-beta))*(1./x(4))).^(theta) - (x(16));
F(17) = Ti.*(gam.*dni.*(x(24).^(beta))*(x(4).^(1-beta))*(1./x(1))).^(theta) - (x(17));
F(18) = Ti.*(gam.*dni.*(x(24).^(beta))*(x(4).^(1-beta))*(1./x(2))).^(theta) - (x(18));
F(19) = Ti.*(gam.*dni.*(x(24).^(beta))*(x(4).^(1-beta))*(1./x(3))).^(theta) - (x(19));
F(20) = Ti.*(gam.*dni.*(x(24).^(beta))*(x(4).^(1-beta))*(1./x(4)).^(theta) - (x(20)));
F(21) = ((x(5).*x(21).*Li)+(x(6).*x(22).*Li)+(x(7).*x(23).*Li)+(x(8).*x(24).*Li))...
    -(x(21).*Li);
F(22) = ((x(9).*x(21).*Li)+(x(10).*x(22).*Li)+(x(11).*x(23).*Li)+(x(12).*x(24).*Li))...
    -(x(22).*Li);
F(23) = ((x(13).*x(21).*Li)+(x(14).*x(22).*Li)+(x(15).*x(23).*Li)+(x(16).*x(24).*Li))...
    -(x(23).*Li);
F(24) = ((x(17).*x(21).*Li)+(x(18).*x(22).*Li)+(x(19).*x(23).*Li)+(x(20).*x(24).*Li))...
    -(x(24).*Li);
end
%And then,
x0 = ones(1, 24);
[x, fval] = fsolve(@ekmodelv2, x0);

This solves 24 equations for 24 unknowns. This code, in my opinion, is like opening a paint can with a battle axe. I typed out every single equation. I can build a cell array of function handles, but it is unclear to me how to then use fsolve the system of non linear equations. I do not have a lot of experience using fsolve in MATLAB. Previously, I have only ever needed to use it to solve three equations at most. How should I go about generalizing this to solve n equations in your opinion?

Walter Roberson 2020-6-30

在 MATLAB Online 中打开

F(1) = (gam.*(((Ti.*(dni.*(x(21).^(beta))*(x(1).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(22).^(beta))*(x(2).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(23).^(beta))*(x(3).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(24).^(beta))*(x(4).^(beta))).^(-theta)))))-x(1);
F(2) = (gam.*(((Ti.*(dni.*(x(21).^(beta))*(x(1).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(22).^(beta))*(x(2).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(23).^(beta))*(x(3).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(24).^(beta))*(x(4).^(beta))).^(-theta)))))-x(2);
F(3) = (gam.*(((Ti.*(dni.*(x(21).^(beta))*(x(1).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(22).^(beta))*(x(2).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(23).^(beta))*(x(3).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(24).^(beta))*(x(4).^(beta))).^(-theta)))))-x(3);
F(4) = (gam.*(((Ti.*(dni.*(x(21).^(beta))*(x(1).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(22).^(beta))*(x(2).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(23).^(beta))*(x(3).^(beta))).^(-theta)))...
    + ((Ti.*(dni.*(x(24).^(beta))*(x(4).^(beta))).^(-theta)))))-x(4);

The only difference I can see between those four is what is being subtracted at the very end. I re-checked several times, but that is the only difference I can find. If I have not missed anything, then the only way those can be true is if x(1) and x(2) and x(3) and x(4) are all equal to each other. If that is what is desired, then just force them to be equal by substitution.

Ethan Goode 2020-7-1

在 MATLAB Online 中打开

This is a first attempt at solving the model with all of the interesting variables constant across the n countries. This is why the only difference between many equations is what is being subtracted at the very end. In the end, I will want to make dni, Ti, and even Li vary. So, I don't want to let all the x's equal each other (even though letting matlab solve the system gives you exactly that) in the code.

Here is what I have tried

function F = price_equations(x)
%%Put parameters here
Ti= 1;
Li= 1;
dni = 2; 
theta = 4;
gam = 1;
beta = 0.5;
n = 4;
for i = 1:n
    for j = 1:n
        for k = n+1:n+n^2
        F(i,1) = {@(x) (gam.*(sum(Ti.*(dni.*((x(k).^(beta).*(x(j).^(1-beta)))).^(-theta)))))-x(i)};  
        end
    end
end

My understanding is that the above command will give me a nx1 cell with each cell containing a function as defined above.

Similarly I repreat this in seperate files for the other types of functions. I'll put them below just to keep everything well documented.

function F = wage_equations(x)
%%Put parameters here
Ti= 1;
Li= 1;
dni = 2; 
theta = 4;
gam = 1;
beta = 0.5;
n = 4;
for i = (n + n^2+1) : (2*n + n^2)
    for k =(n+n^2+1) : (2*n + n^2)
        for j = n+1:n + n^2
     F(i, 1) = {@(x)(sum(x(j).*x(k).*Li))-(x(i).*Li)};
        end
    end
end

And,

function F = tradeshare_equations(x)
%%Put parameters here
Ti= 1;
Li= 1;
dni = 2; 
theta = 4;
gam = 1;
beta = 0.5;
n = 4;
for i = n+1:n + n^2
    for j = (n+n^2+1) : (2*n + n^2)
        for k = 1:n
    F(i, 1)= {@(x) Ti.*(gam.*dni.*(x(j).^(beta))*(x(k).^(1-beta))*(1./x(l))).^(theta) - (x(i))};
        end
    end
end

Walter Roberson 2021-8-21

It looks to me as if perhaps the poster deleted at least one of their comments, so I am not sure whether the code ended up working for them.

Celestine Siameh 2021-8-21

Thanks Walter. Let me check with him then. Hello Ethan Goode, did the code worked for you.

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Solving N non-linear equations using fsolve. How do I pass these equations into my function without typing them out individually?

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Solving N non-linear equations using fsolve. How do I pass these equations into my function without typing them out individually?

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