16/replication_package/programs/solvemodel.m

function Z=solvemodel(x, targs, vers, show)

% % This function solves the model, given (1) values for the parameters that
% don't change, which are global variables, (2) values for "calibrated"
% parameters, over which the fsolve function is searching, and which are
% passed to this function as the vector 'x'. 
% 
% The second argument, 'targs,' gives values for the three statistics whose
% values are being targeted by fsolve as part of the calibration exercise. 
% The vector of values, 'Z,' that are returned by this function are the 
% squared residuals that fsolve seeks to set to zero.
%
% The variable 'vers,' determines which variables are being passed to the
% function in 'x' and which ones are fixed at the global values set in the
% main m-file (as specified below).
%
% For the variable 'show,' a value of 1 tells the function to print the
% values of the various statistics of interest.
%
% 

% parameters:
global pigrdpts pigrid pigriddiv yb yg beta b gam eta pkh lam alph pi0 kh sigeps A kr

% output or results that we want to be available globally:
global pe pf u e0 e1 w0 w1 tendist probmatch probg H_pi_dist pibar theta oneqhazrate twoqhazrate threeplushazrate hiresrate jfr meanvacdur 

if vers==1
    localpi0=pi0;
    localkh=kh;
    localsigeps=x(1);
    localA=x(2);
    localkr=x(3);
elseif vers==2
    localpi0=x(1);
    localkh=kh;
    localsigeps=sigeps;
    localA=x(2);
    localkr=x(3);
elseif vers==3
    localpi0=pi0;
    localkh=x(3);
    localsigeps=x(1);
    localA=x(2);
    localkr=kr;
end


% % Compute the cumulative distribution of pi, conditional on normally
% distributed signal (z). This must be re-computed every time the values
% of sigeps or pi0 change (which is every time that this function is
% called).

% Find value of z such that pi equals a particular value of pigriddiv
for j=1:pigrdpts+1
    zz(j)=fsolve(@(z) pigriddiv(j)-localpi0*exp(-(z-yg)^2/2/localsigeps^2)/(localpi0*exp(-(z-yg)^2/2/localsigeps^2)+(1-localpi0)*exp(-(z-yb)^2/2/localsigeps^2)), 0, optimoptions('fsolve','TolFun',1e-16,'Display','off')); 
end    

% Probabilities of getting a value of pi between two successive values of
% pigriddiv (the range surrounding a value of pigrid)
H_pi_dist(1,:)=localpi0*normcdf(zz(2:end), yg, localsigeps)+(1-localpi0)*normcdf(zz(2:end), yb, localsigeps)-(localpi0*normcdf(zz(1:end-1), yg, localsigeps)+(1-localpi0)*normcdf(zz(1:end-1), yb, localsigeps));
H_pi_dist(1,:)=H_pi_dist(1,:)/sum(H_pi_dist(1,:)); % normalize, to ensure elements sum to 1 exactly.


%% Solve Bellman equations

xx=1/2000; % adjustment factor for convergence of pe
pe=.2;     % initial value for pe

% Create/initiate value functions
S0old=zeros(1,length(pigrid));
S1old=zeros(1,length(pigrid));
Je0old=zeros(1,length(pigrid));
Je1old=zeros(1,length(pigrid));
Juold=0;
Ve0old=zeros(1,length(pigrid));
Ve1old=zeros(1,length(pigrid));
Vuold=0;
Vu=0;

difpe=1;
tol=1e-8;
while abs(difpe)>tol

    pf=localA^(1/(1-gam))*pe^((-gam)/(1-gam));

    difs=1;
    while difs>tol
        S0new=max(pigrid*(yg-yb)+yb-localkh-b+beta*(1-lam)*(alph*pigrid*(pkh*S1old(end)+(1-pkh)*S0old(end))+(1-alph)*(pkh*S1old+(1-pkh)*S0old))-...
            beta*pe*eta*H_pi_dist*S0old',0);

        S1new=max(pigrid*(yg-yb)+yb-b+beta*(1-lam)*(alph*pigrid*S1old(end)+(1-alph)*S1old)-...

            beta*pe*eta*H_pi_dist*S0old',0);
        difs=max(max(abs(S1old-S1new)),max(abs(S0old-S0new)));
        S1old=S1new;
        S0old=S0new;
    end       

    peold=pe;
    Ju=(-localkr+beta*pf*(1-eta)*H_pi_dist*S0new')/(1-beta);

    difpe=Ju;

    pe=peold*(1+difpe*xx);  % increase worker's finding rate if Ju>0

end

theta=pe/pf;
w0=b+eta*(pigrid*yg+(1-pigrid)*yb-localkh-b+localkr*theta);
w1=b+eta*(pigrid*yg+(1-pigrid)*yb-b+localkr*theta);

pibar=pigrid(sum(S0new==0));

%% Calculate statistics of interest, given the equilibrium pe, pf, and pibar from above

probmatch=H_pi_dist*(pigrid>pibar)';         % fraction of meetings with high enought pi to form/continue match



probg=(H_pi_dist/sum(H_pi_dist(pigrid>pibar)))*(pigrid.*(pigrid>pibar))'; % probability that matches are actually good (conditional on having been formed)

trans=zeros(3,3);                            % three states: unemployed, unknown type, known good matches

trans=[1-pe*probmatch lam+(1-lam)*alph*(1-probg) lam;...
        pe*probmatch (1-lam)*(1-alph) 0;...
        0 (1-lam)*alph*probg 1-lam];

% steady state
mtr=eye(3)-trans;
mtr(3,1:3)=1;
invmtr=inv(mtr);
erg=invmtr(:,3);
u=erg(1);                   % unemployment
e0=erg(2);                  % matches of unknown type
e1=erg(3);                  % matches known to be good


% % Calculate the job tenure distribution
% --Allow mass of one unit to enter and the trace the realized tenure 
% distribution of that cohort 
% --This is equivalent to looking at the point-in-time distribution if one 
% unit is allowed to enter each period. The whole distribution would simply
% scale up or down with the size of the mass of entrants.

tendist=[0;1;0]; 
trans1=trans;
trans1(:,1)=0;              % don't allow any exits from unemployment after first period, so this just tracks the initial mass of entrants
for i=2:2000                % 2000 periods is 2000/52 (roughly 40) years
    tendist(:,i)=trans1*tendist(:,i-1) ; 
end

% fqseps: All separations among jobs newly created in the quarter=(start in first week of quarter and separate within 12
% periods)+(start in second week of quarter and separate within 11
% periods)+(start in third week of quarter and separate within 10)+...etc

fqseps=13*sum(tendist(2:3,1))-sum(tendist(2:3,14))-sum(tendist(2:3,13))-...
    sum(tendist(2:3,12))-sum(tendist(2:3,11))-sum(tendist(2:3,10))-...
    sum(tendist(2:3,9))-sum(tendist(2:3,8))-sum(tendist(2:3,7))-...
    sum(tendist(2:3,6))-sum(tendist(2:3,5))-sum(tendist(2:3,4))-...
    sum(tendist(2:3,3))-sum(tendist(2:3,2));

% empspells: (Employment among all tenure categories at start of
% quarter)+(new starts in 2nd week)+(new starts in third week)+...+(new
% starts in 13th week)

empspells=sum(sum(tendist(2:3,:)))+12*sum(tendist(2:3,1)); 

sqrate=fqseps/empspells;  % incidence rate of one-quarter spells (q_1 in paper)

hires=13*sum(tendist(2:3,1));

% % First quarter hazard rate: among those who enter a job in a given
% quarter, what fraction have separated by the start of the next quarter?
oneqhazrate=fqseps/hires;

% Note: we could have used either tendist or scaledtendist in the above
% calculations for sqrate and fqhazrate. Whether they are scaled or not 
% doesn't matter much so long as the numerators and denominators are scaled
% equivalently.

% 2 quarter hazard rate

twoqseps=sum(sum(tendist(2:3,2:14)))-sum(sum(tendist(2:3,15:27)));
twoqhazrate=twoqseps/sum(sum(tendist(2:3,2:14)));

% 3+ quarters separation rate

threeplusseps=sum(sum(tendist(2:3,15:end)))-sum(sum(tendist(2:3,28:end)));
threeplushazrate=threeplusseps/sum(sum(tendist(2:3,15:end)));

% Quarterly hires rate (# hires in quarter divided by total employment spells)
hiresrate=hires/empspells;

% Monthly job-finding rate
jfr=pe*probmatch+(1-pe*probmatch)*pe*probmatch +(1-pe*probmatch)^2*pe*probmatch+(1-pe*probmatch)^3*pe*probmatch; % weekly frequency: "month"==4 weeks

% % Mean vacancy duration in days
% meanvacdur=7*(1/pf/probmatch);

% Mean vacancy duration in weeks
meanvacdur=(1/pf/probmatch);

% print output 

if show==1
%     fid=fopen('results.txt','w');
%     fprintf(fid,'Unemployment rate: \t\t\t\t %g\r', u);
%     fprintf(fid,'Monthly job-finding rate:\t\t %g\r', jfr);
%     fprintf(fid,'Ave. vacancy duration (weeks):\t %g\r', 1/pf/probmatch);
%     fprintf(fid,'Prob(good|match):\t\t\t\t %g\r', probg);
%     fprintf(fid,'1-H(pi^n):\t\t\t\t\t\t %g\r', probmatch);
%     fprintf(fid,'Worker meeting rate:\t\t\t\t %g\r', pe);
%     fprintf(fid,'First quarter hazard rate:\t\t %g\r',oneqhazrate);
%     fprintf(fid,'Second quarter hazard rate:\t\t %g\r',twoqhazrate);
%     fprintf(fid,'3+ quarter hazard rate:\t\t\t %g\r',threeplushazrate);
%     fprintf(fid,'Hires rate:\t\t\t\t\t\t %g\r',hiresrate);
%     fclose(fid);
%     type results.txt

% print results to be cut-and-pasted into tables in Latex
    fid=fopen('results2.txt','w');
    fprintf(fid,'& %.3f & %.3f & %.3f & %.3f & %.3f & %.2f & %.3f', oneqhazrate, twoqhazrate, threeplushazrate, hiresrate, jfr, meanvacdur, u);
    fclose(fid);
    type results2.txt

    
end

Z=[(log(targs(1))-log(probg))^2; (log(targs(2))-log(jfr))^2; (log(targs(3))-log(meanvacdur))^2];