function [ar_k_con,ar_l_con]=con_kl(varargin)

close all; 

bl_profile = false;      % Switch off profile if running a tester/calling from another function 

if(bl_profile) 

profile off;

profile on;

end 

addpath(genpath('/Users/sidhantkhanna/Documents/GitHub/BKS modified/')); 

if ~isempty(varargin) 

    [ar_a_v,ar_z_v,ar_a,ar_z ... 

        fl_ahi,fl_zhi, fl_phi,fl_theta,fl_alpha,fl_w,fl_r,fl_delta,... 

        fl_kappa,it_agridno,it_zgridno, ... 

        ] = varargin{:}; 

   % Display matrix and graphs of constrained k and l (for every combination of a and z)

    % if testing the index below, print out graph, and also show each

    % iteration of the first order taylor approximation. if bl_state_test

    % is true, then test specific state iteration to arrive at \bar{k}.

    bl_print           = false; 

    bl_state_test      = false; 

    it_test_a_idx      = 1;         

    it_test_z_idx      = 1; 

    bl_saveimg         = false;  % True if images are to be saved, false if you want to publish them 

    it_k_n             = 20000;                            % capital grid points to create capital grid and search for  

else

    fl_ahi             = 50;

    fl_zhi             = 7;

    it_agridno         = 5;

    it_zgridno         = 6;

    ar_a               = linspace(0,fl_ahi,it_agridno);    % Sample asset grid

    ar_z               = linspace(0.2,fl_zhi,it_zgridno);  % Sample entrepreneurial productivity grid

    %ar_a              = 50;                               % for testing single a value  

    %ar_z              = 1;                                % for testing single z value 

    [a_m,z_m]          = meshgrid(ar_a,ar_z);              % Meshed matrix of assets and entrepreneurial productivity

                                                           % grid (size for a_m = it_zgridnoxit_agridno, z_m =it_zgridnoxit_agridno)

    ar_a_v             = a_m(:);                           % Meshed-vectorised asset grid

    ar_z_v             = z_m(:);                           % Meshed-vectorised entrepreneurial productivity grid

    it_k_n             = 20000;                            % capital grid points to create capital grid and search for 

                                                           % constrained k   

    fl_phi             = 0.1;

    fl_alpha = 0.4;

    fl_theta = 0.79 - fl_alpha;

    fl_delta = 0.06;

    fl_kappa = 2;

    [fl_r,fl_w,fl_ahi] = ...

        deal(0.04,1.8,100);

    %% Testing Controls

    bl_print           = true;                             % Print matrix of optimal k with graphs

    bl_state_test      = true;                             % Plotting taylor method graphs for root finding

    it_test_a_idx      = 2;                                % a index for drawing graph or root search of k using TS approx

    it_test_z_idx      = 4;                                % z index for drawing graph or root search of k using TS approx

    bl_saveimg         = false; 

end 

%% Parameter controls

it_max_iter_taylor     = 7;                 % No of iterations for taylor series approximation for constraint function 

%% Initialize Matrices

fl_R              = fl_r + fl_delta; 

mt_k_con          = ones(it_agridno,it_zgridno);    % Matrix for storing constrained k , for each combination on a and z 

mt_l_con          = ones(it_agridno,it_zgridno); 

fl_version    = 3; 

% version 1 = numerical method, version 2 = vectorised, version 3 = taylor

% series approximation for loop - has both vectorised and unvectorised

% versions

fl_vectorised = true; % for version 3 - loop or vectorised 

%% Version 1 - Numerical Method

if(fl_version == 1) 

    for i = 1:it_agridno*it_zgridno

        fl_a    = ar_a_v(i);

        fl_z    = ar_z_v(i);

        [ar_k_con(i),ar_l_con(i)] = conkl_numerical(fl_a,fl_z,fl_r,fl_w,fl_alpha,fl_theta,fl_kappa,fl_delta,fl_phi,fl_R);

    end

    ar_k_con = ar_k_con';                 % transposing to get same order as input vector (it_agridnoxit_zgridno x 1)

    ar_l_con = ar_l_con';                 % transposing to get same order as input vector (it_agridnoxit_zgridno x 1)

    mt_k_con = reshape(ar_k_con, [it_zgridno,it_agridno]);

    mt_l_con = reshape(ar_l_con, [it_zgridno,it_agridno]);

    mt_k_con = mt_k_con';                   % getting to matrix of dimension it_agridnoxit_zgridno

    mt_l_con = mt_l_con';                   % getting to matrix of dimension it_agridnoxit_zgridno

%% Version 2 - Vectorised solution  

elseif (fl_version == 2) 

    ar_ks1 = linspace(0, 5*fl_ahi, it_k_n);  % Grid of capital to search for roots of constraint equation (order = 1 X it_k_n) 

    ar_ks  = fliplr(ar_ks1);                   % flipping array to pick up the second root (non-zero root)

    f_lmax = @(z,k) (fl_w./((k.^fl_alpha).*fl_theta.*z)).^(1/(fl_theta - 1)); % max l for a given constrained k

    f_con_concave = @(a,z,k) fl_phi.*((k.^fl_alpha).*z.*(f_lmax(z,k)).^fl_theta - fl_w.*(f_lmax(z,k)))-(1+fl_r)*fl_kappa.*ones(numel(a),it_k_n)+(1+fl_r).*(a).*ones(1,it_k_n); 

    % concave part of the constraint function

    f_con_straight = @(a,z,k) (1-fl_phi).*(1-fl_delta).*ones(numel(a),1).*k+fl_R.*ones(numel(a),1).*k;

    % straight line part of the constraint function

    f_con = @(a,z,k) f_con_concave(a,z,k) - f_con_straight(a,z,k); % constraint function

    f1 = f_con(ar_a_v, ar_z_v, ar_ks); 

    % f1 evaulates the constraint for each combination of a,z and k. Order of f1 = ((it_agridnoxit_zgridno) X it_k_n)

    % for i=1:it_agridno*it_zgridno

    %    f1(i,it_k_n)=-1000;

    % end

    f1(:,it_k_n)  = -1000;  % assigning a large absolute value to the constraint at k=0 so as to

                           % remove the root k=0

    [val, idx]    = min(abs((f1)'));  

   % for each combi of a,z, root of the constraint equation is the k for 

   % which the value of the constraint function is closest to zero. 

   % min function on a matrix minimizes over each column, min (NXM) is of

   % order (1XM). As f1 is of the order ((it_agridnoxit_zgridno) X it_k_n), 

   % dimension of val and idx = (1 X (it_agridnoXit_zgridno)). Since we need to minimize f1 over

   % k for each combination of a and z, and dim f1=((it_agridnoxit_zgridno) X it_k_n),

   % dim f1' = (it_k_n x (it_agridnoxit_zgridno)), dim min(abs((f1)'))= 1 X (it_agridnoxit_zgridno), 

   % dim val , idx = 1 X (it_agridnoxit_zgridno)                                                                   

   [val2,idx2]    = max(((f1)'));     % store the max value of f1(k) for each combination of a,z

    ar_k_con      = ar_ks(idx);    %  storing constrained k values, dim = 1 X (it_agridnoxit_zgridno)   

    % Analysing cases where the only root of constraint equation is k=0 in

    % the range of k's we are checking

    % There are 2 subcases:

    % 1) k = 0 is the only root and max of f is obtained at k = 0 ( all other

    % f's are negative in the range of k's )

    % 2) k = 0 is not the only root, but only root in the given range of k's

    % we are checking. The actual constrained k is very high in this case,

    % so assuming the optimal would surely be less than the constrained k

    for i=1:it_agridno*it_zgridno 

        if (ar_k_con(i)==ar_ks1(2))

            if (val2(i)<=0)

                ar_k_con(i)=0;  % Case 1)

            else

                ar_k_con(i)=1000000; % Case 2)

            end

        end

    end

    % Limitation of this method is for case 2), if optimal k was greater than

    % constrained k, those cases you are not sure about (not sure if those

    % cases exist)

    ar_k_con = ar_k_con'; % transposing to get same order as input vector (it_agridnoxit_zgridno x 1)

    ar_l_con = (fl_w./(fl_theta.*ar_z_v.*ar_k_con.^(fl_alpha))).^(1/(fl_theta-1));

    mt_k_con = reshape(ar_k_con, [it_zgridno,it_agridno]);

    mt_l_con = reshape(ar_l_con, [it_zgridno,it_agridno]);

    mt_k_con = mt_k_con';           % getting to matrix of dimension it_agridnoxit_zgridno

    mt_l_con = mt_l_con';            % getting to matrix of dimension it_agridnoxit_zgridno

%% Version 3 - Taylor Series Method - For loop

elseif (fl_version == 3) 

    if(~fl_vectorised) 

        % Solving constraint k for each combination of a and z one by one.

        % The constraint function has a linear part and a curved part.

        % Starting at any point, the the curved part is approximated to a straight line with a taylor series expansion

        % The intersection of the first approximation and the linear part

        % is the first guess of k. At the first guess of k, approximate the

        % curve part again as a stright line, then find its intersection

        % with a straight line for the next guess of k. Continue doing that

        % until it_max_iter_taylor iterations (you converge to the correct k in less than those no. of iterations 

        syms k w alph theta kappa r phi delta R

        for i = 1:it_agridno

            for j = 1:it_zgridno

                % Defining the functions using symbols only

                f_lmax             = (w/((k^alph)*theta*ar_z(j)))^(1/(theta - 1));  % optimal l to constrained k

                f_con_concave      = phi*((k^alph)*ar_z(j)*(f_lmax)^theta - w*(f_lmax))-(1+r)*kappa+(1+r)*(ar_a(i));

                % Concave part of constraint function of k

                f_con_straight     = (1-phi)*(1-delta)*k+R*k;

                % Straight line part of constraint function of k

                % Substituting parameters in the constraint function

                f_con_straight_sub = subs(f_con_straight,{w,alph,theta,kappa,r,phi,delta,R},{fl_w,fl_alpha fl_theta,fl_kappa,fl_r,fl_phi,fl_delta, fl_R});

                f_con_concavej_k   = subs(f_con_concave, {w,alph,theta,kappa,r,phi,delta,R}, {fl_w,fl_alpha fl_theta,fl_kappa,fl_r,fl_phi,fl_delta, fl_R});

                m  = 1;   % no if iteration for guess of k            

                kj = 1;   % First guess of k

                % * a = constant term in taylor approximate of the concave part of constraint function

                % * b = coefficient of k in taylor approximate of the concave part of constraint function

                % * c = 0 (constant term of straight line part of constraint - staight line function chosen that way)

                % * d = coefficient of k of straight line part of constraint 

                if (bl_state_test)

                    ar_k_store = zeros(1, it_max_iter_taylor);

                    % Store all values of guesses of k

                end

                if (fl_phi == 0) % assuming kappa = 0, id a = 0 and phi = 0, opti k = 0

                mt_k_con(i,j) = ((1+fl_r)*(ar_a(i)- fl_kappa))/(1-fl_delta+fl_R);

                mt_l_con(i,j) = (fl_w/((mt_k_con(i,j)^fl_alpha)*fl_theta*ar_z(j)))^(1/(fl_theta - 1));

                    else

                % bl_solution_exits = false;

                s=0; % s increases value when a<0, case where there is no intersection of the 2 straight lines

                while (m < it_max_iter_taylor)

                    f_con_concavej        = double(subs(f_con_concavej_k, {k}, {kj})); % substituting first guess of k

                    % computing coefficients a and b of taylor series approximate

                    f_con_concave_diff_k  = diff(f_con_concave,k);

                    f_con_concave_diff_kj = double(subs(f_con_concave_diff_k, {k,w,alph,theta,kappa,r,phi,delta,R}, {kj,fl_w,fl_alpha fl_theta,fl_kappa,fl_r,fl_phi,fl_delta, fl_R}));

                    d = (1-fl_phi)*(1-fl_delta)+ fl_R;

                    a = double(f_con_concavej-f_con_concave_diff_kj*kj); 

                    b = double(f_con_concave_diff_kj);

                    if (b>d) % Case where slope of tangent to curve greater 

                        % than that of straight line), first guess needs to

                        % a large no.  

                        k_jplus1=kj*100;

                    elseif (a<0)   % case where there is no intersection of the 2 straight lines

                        k_jplus1 = kj;

                        s=s+1;

                    else

                    k_jplus1 = (a/(d-b)); % intersection of taylor approximate and straight line part of the constraint 

                    end

              % Displaying coefficients of straight line and tangent to

              % curved part of the constraint, and each guess of k

                    if (i == it_test_a_idx && j == it_test_z_idx && bl_state_test)

                        st_test_iter_info_one = ['iter:', num2str(m), ', kj:', num2str(kj)];

                        st_test_iter_info_two = ['y-interp for lin approx of concave:', num2str(a)];

                        st_test_iter_info_thr = ['slope for lin approx of concave:', num2str(b)];

                        st_test_iter_info_fou = ['slope for origin linear line:', num2str(d)];

                        disp(st_test_iter_info_one);

                        disp(st_test_iter_info_two);

                        disp(st_test_iter_info_thr);    

                        disp(st_test_iter_info_fou);

                        ar_k_store(m) = kj; % storing all k's which were guessed until convergence to root

                    end  

                    % updating

                    kj = k_jplus1; % next guess of k

                    m  = m+1; % updating iteration for next guess of k

                end

                %if(~bl_solution_exits)

                 %   display ('constraint equation has no positive solution for capital');

                %return;                    

                %else

                mt_k_con(i,j) = kj; % updating constrained k for each a and z

                mt_l_con(i,j) = (fl_w/((kj^fl_alpha)*fl_theta*ar_z(j)))^(1/(fl_theta - 1));

                if (s>0)

                    mt_k_con(i,j)=0;

                    mt_l_con(i,j)=0;

                end  

                % updating optimal l to correponding constrained k for each a and z

                bl_state_test = true;

                %% Graph Curve and line Intersection

                if (i== it_test_a_idx && j == it_test_z_idx && bl_state_test)

                    figure(1);

                    hold on;

                    %fl_k_max = min(0.001, kj + kj/10);

                    fl_k_max = 120;

                    fplot(f_con_concavej_k, [0, fl_k_max]);

                    fplot(f_con_straight_sub, [0, fl_k_max]);

                    for it_m = 1:1:it_max_iter_taylor

                        fl_kj_cur = ar_k_store(it_m);

                        line([real(fl_kj_cur) real(fl_kj_cur)],[0 fl_kj_cur+1], 'LineStyle', '--'); 

                        % -- lines are guesses of k

                    end

                    line([real(kj) real(kj)],[0 kj+1]);

                    % final solution k    

                    grid on;

                    grid minor;

                    if(bl_saveimg)

                    saveas(gcf, '/Users/sidhantkhanna/Documents/GitHub/BKS modified/code/Firms/figures/con_kl/taylor.png')

                    end

                end                

            end

            end

        end

        mt_k_con1 = mt_k_con';

        mt_l_con1 = mt_l_con';

        ar_k_con = mt_k_con1(:);                    % matrix unlayered one column under another

        ar_l_con = mt_l_con1(:);                    % matrix unlayered one column under another

%% Version 3 - Taylor Series Method - Vectorised Version 

    elseif(fl_phi==0) 

         ar_k_con =((1+fl_r).*(ar_a_v- fl_kappa.*ones(numel(ar_a_v),1)))./(1-fl_delta+fl_R);

         ar_l_con =(fl_w./(fl_theta.*ar_z_v.*ar_k_con.^(fl_alpha))).^(1/(fl_theta-1));

         ar_k_con(ar_k_con<0) = 0;

         ar_l_con(ar_l_con<0) = 0;

          mt_k_con=reshape(ar_k_con, [it_zgridno,it_agridno]);

    mt_l_con=reshape(ar_l_con, [it_zgridno,it_agridno]);

    mt_k_con=mt_k_con';

    mt_l_con=mt_l_con';

         else 

        ar_kj = ones(it_agridno*it_zgridno, 1); % initial guess of k 

        m     = 1; 

        s = zeros(it_agridno*it_zgridno, 1); 

        s1 = zeros(it_agridno*it_zgridno, 1); 

        while (m < it_max_iter_taylor) 

        f_lmax = (fl_w./((ar_kj.^fl_alpha)*fl_theta.*ar_z_v)).^(1/(fl_theta - 1)); 

        f_con_concave = fl_phi*((ar_kj.^fl_alpha).*ar_z_v.*(f_lmax).^fl_theta - fl_w.*(f_lmax))-(1+fl_r)*fl_kappa.*ones(it_agridno*it_zgridno,1)+(1+fl_r).*(ar_a_v); 

        f_con_concave_diff = fl_phi.*(fl_alpha.*ar_kj.^(fl_alpha - 1).*ar_z_v.*((fl_w./(ar_kj.^fl_alpha.*fl_theta.*ar_z_v)).^(1/(fl_theta - 1))).^fl_theta - (fl_alpha.*ar_kj.^fl_alpha.*fl_w.*((fl_w./(ar_kj.^fl_alpha.*fl_theta.*ar_z_v)).^(1/(fl_theta - 1))).^(fl_theta - 1).*(fl_w./(ar_kj.^fl_alpha.*fl_theta.*ar_z_v)).^(1/(fl_theta - 1) - 1))./(ar_kj.^(fl_alpha + 1).*(fl_theta - 1)) + (fl_alpha.*fl_w^2.*(fl_w./(ar_kj.^fl_alpha.*fl_theta.*ar_z_v)).^(1/(fl_theta - 1) - 1))./(ar_kj.^(fl_alpha + 1).*fl_theta.*ar_z_v.*(fl_theta - 1))); 

        d = ((1-fl_phi)*(1-fl_delta)+ fl_R).*ones(it_agridno*it_zgridno, 1); 

        a = (f_con_concave-f_con_concave_diff.*ar_kj); 

        b = f_con_concave_diff; 

        % find k_{j+1} for everyone

        ar_k_jplus1 = (a./(d-b)); 

        % ar_k_jplus1_dup = ar_k_jplus1;

        % replace if lin approximate for slope is greater than linear slope

        ar_k_jplus1((b>d)) = ar_kj(b>d)*100; 

  %     s1(b>d) = s1(b>d)+1;

        ar_k_jplus1((a<0) & (b<d)) = ar_kj(a<0 & b<d); 

        s(a<0 & b<d) = s(a<0 & b<d)+1; 

        ar_kj = ar_k_jplus1; 

        m = m+1; 

        end 

        ar_k_con = ar_kj; 

        ar_l_con = (fl_w./(fl_theta.*ar_z_v.*ar_k_con.^(fl_alpha))).^(1/(fl_theta-1)); 

        ar_l_con1 = ar_l_con; 

        ar_k_con(s>0) = 0; 

        ar_l_con(s>0) = 0; 

 %      ar_k_con(s1>0) = ar_kj(s1>0);

 %      ar_l_con(s1>0) = ar_l_con1(s1>0);

    mt_k_con=reshape(ar_k_con, [it_zgridno,it_agridno]); 

    mt_l_con=reshape(ar_l_con, [it_zgridno,it_agridno]); 

    mt_k_con=mt_k_con'; 

    mt_l_con=mt_l_con'; 

    end 

end 

%% Print matrix for constrained k and l

if(bl_print) 

    disp('mt_k_con');

    disp(mt_k_con);

    disp('mt_l_con');

    disp(mt_l_con);

figure (2)

surf(ar_z, ar_a, mt_k_con);

title('K constraint given a and z');

xlabel('a');

ylabel('z');

zlabel('K constraint');

view(125,35);

if(bl_saveimg)

saveas(gcf, '/Users/sidhantkhanna/Documents/GitHub/BKS modified/code/Firms/figures/con_kl/Kconst_az.png')

end

figure (3)

surf(ar_z, ar_a, mt_l_con);

title('L constraint given a and z');

xlabel('a');

ylabel('z');

zlabel('L with K constrained');

view(125,35);

if(bl_saveimg)

saveas(gcf, '/Users/sidhantkhanna/Documents/GitHub/BKS modified/code/Firms/figures/con_kl/Lconst_az.png')

end

end 

if(bl_profile) 

profile off;

profile viewer;

st_file_name = '/Users/sidhantkhanna/Documents/GitHub/BKS modified/code/Profile/Firms/con_kl';

profsave(profile('info'), st_file_name);

end