%This matlab file shows that EXAMPLE 2A in the article is an equilibrium of the model.

%Community-schooling alternatives:
%urban,public
%urban,private
%suburban,public


%In example 2a (urban-trap equilibrium):
%- Households with income 5-55 live in the urban area and use the local public school, i.e. choose urban,public
%- Households with income 60-75 choose urban,private
%- Households with income y=80 are just indifferent between urban,private and suburban,public. 
%   A fraction y80ur of them chooses urban,private; the rest, y80sub, chooses suburban,public.
%- Households with income 85-100 choose suburban,public.



clear

%parameters
%utility function

sigma    =    1.71;
delta    =    0.0026;

%income distribution

mu       =    3.663;       %mean parameter
sig      =     .82;       %variance parameter


I        =   20;           %number of discrete income classes
int      =    5;           %size of intervals between income classes

for i=1:I
y(i)     =  (int*i);
end

%f (? in the paper) is the normalizing constant which assures that l(i) add up to 1. It is necessary because we use a truncated income distribution.
%l(i) (l is ? in the paper) stands for the mass of households in income class i; L(i) stands for the mass of households in income class i and below.


f        =   1/logncdf(y(I),mu,sig);

for i=1:I
if i<2
l(i)     =   f*logncdf(y(i),mu,sig);
L(i)     =   f*logncdf(y(i),mu,sig);
else
l(i)     =   f*logncdf(y(i),mu,sig)-f*logncdf(y(i-1),mu,sig);
L(i)     =   f*logncdf(y(i),mu,sig);

end
end

yp      =   transpose(y);

%total income = mean income = ymean


ymean   =   l*yp


%size of communities

Hu    =    0.91
Hs    =    0.09


%cost of construction ch

ch = 2.5


%vector of public policies and housing prices
Eu   =   0.74953299819283
Es   =   2.59522570994683
tu   =   0.24961778562079
ts   =   0.83897991133674
phu  =   ch
phs  =   3.09331090635042
pu   =   phu*(1+tu)
ps   =   phs*(1+ts)


%income of median voters
mvu=25
mvs=90


%1/ RATIONAL CHOICES

%First, we compute the utility each alternative provides to households with each level of income.
%For those households who opt for private education, we also compute their demand for private education

for i=1:20    
    vu(i) = (1/(1-sigma))*(((y(i)-pu)^(1-sigma))+(delta*(Eu^(1-sigma))));
    
    if y(i)-ps<=0
        vs(i) = -999;
    else
        vs(i) = (1/(1-sigma))*((y(i)-ps)^(1-sigma)+(delta*(Es^(1-sigma))));
    end
    
    xu(i) = ((y(i)-pu)*(delta^(1/sigma)))/(1+(delta^(1/sigma)));
    
    wu(i) = (1/(1-sigma))*((y(i)-pu)^(1-sigma)*(1+delta^(1/sigma))^sigma);   


%Second, for each level of income, we establish the alternative which maximizes a household utility. 

    
    
%tolerance
tol = 10^(-10);
    
    z1(i) = vu(i) - wu(i);
    z2(i) = vu(i) - vs(i);
    z3(i) = vs(i) - wu(i);
    
    absz1(i) = abs(z1(i));   
    absz2(i) = abs(z2(i));
    absz3(i) = abs(z3(i));
    
if absz1(i)>tol
    if z1(i)>0
        if absz2(i)>tol
            if z2(i)>0
                sprintf('households with income y=%g prefer "urban,public"',y(i))
            end
            if z2(i)<0
            sprintf('households with income y=%g prefer "suburban,public"',y(i))
            end
        else
            sprintf('households with income y=%g are indifferent between "urban public" and "suburban,public"',y(i))
        end
    end
    if z1(i)<0
        if absz3(i)>tol
            if z3(i)>0
                sprintf('households with income y=%g prefer "suburban,public"',y(i))
            end
            if z3(i)<0
                sprintf('households with income y=%g prefer "urban,private"',y(i))            
                sprintf('demand for private schooling is %g',xu(i))
            end
        else
            sprintf('households with income y=%g are indifferent between "suburban,public" and "urban,private"',y(i))            
            sprintf('demand for private schooling is %g',xu(i))
        end
    end
else
        if absz2(i)>tol
            if z2(i)>0
                sprintf('households with income y=%g are indifferent between "urban,public" and "urban,private"',y(i))
                sprintf('demand for private schooling is %g',xu(i))
            end
            if z2(i)<0
                sprintf('households with income y=%g prefer "suburban,public"',y(i))
            end
        else
            sprintf('households with income y=%g are indifferent between "urban public", "urban,private" and "suburban,public"',y(i))
            sprintf('demand for private schooling is %g',xu(i))
        end            
end
end


%2/ HOUSING MARKET EQUILIBRIUM: 

%For housing markets to be in equilibrium, the mass of households living in each community must equal its housing capacity. 
%Because Hs+Hu is equal to 1, it suffices to show that Ns=Hs:

%For Ns=Hs, the following conditions must hold:
%Condition (2a) the mass of households with income 85-100 must be smaller than or equal to Hs.
%Condition (2b) the mass of households with income 80-100 must be greater than or equal to Hs.
%y80sub measures the mass of households with income 80 who choose suburbs,public.
%y80ur measures the mass of households with income 80 that choose urban,private.


if L(100/int)-L(80/int)<=Hs             %condition 2a
    if L(100/int)-L(75/int)>=Hs         %condition 2b
        sprintf('housing market in equilibrium')
    end
end

y80sub  =   Hs-(L(100/int)-L(80/int));

y80ur   =   l(80/int)-y80sub;

Nu      =   L(75/int)+y80ur
Ns      =   L(100/int)-L(80/int)+y80sub
 


%3/ VOTING EQUILIBRIUM

%In our setting, schooling and community choices are committed when voting takes place. Therefore, voters take the population of their community, 
%the distribution of students across public and private schools and the price of housing as given when they vote. Moreover, voters know 
%their local government budget constraint (LGBC), which establishes a one-to-one relationship among tax rates and per student expenditures.

%Preferences are therefore single-peaked at the voting stage. For households choosing a private school the peak is at t = 0. 
%Since they do not benefit from expenditures on public education, their peak is at the tax rate which maximizes their level of disposable
%income. 

%Households using the local public school in community j, in turn, have their peak at the tax rate, t_j, that maximizes:

                %u(y - phj * (1 + tj), tj * phj * Nj / nj))
                
%where the LGBC has been used to establish the relationship among the tax rate and the level of expenditure, Nj is the total mass of households 
%living in community j, and nj is the mass of students who attend the local public school in community j. Because u(b,x) is strictly concave 
%this problem has a unique solution and these households preferences are also single-peaked. Moreover, it can easily be shown that this peak 
%is increasing in income. It is given by the expressions:
%urban area:  tu*     =     (mvu-phu)/(phu+(((delta*Nu/nu)^(-1/sigma))*(phu*Nu/nu)))
%suburbs:     ts*     =     (mvs-phs)/(phs*(1+delta^(-1/sigma)))
%where mvj stands for community j median voter level of income and phj is community j houring rent. Given the LGBC:
%ubrban area: Eu*     =      tu*phu*(Nu/nu);
%suburbs:     Es*     =      ts*phs.


%Given all this, suppose the median voter prefers t*. For t* to be a voting equilibrium in a community, the mass of households who would vote for this 
%tax rate against any strictly higher (lower) alternative tax rate satisfying the LGBC must be greater than or equal to half the size of the community. 
%That is to say, the mass of households in the income class which contains the median voter plus the mass of those who prefer a lower (higher) tax rate must equal Nj/2.
%Thus, this is what we now check for both communities:


% 3.1. Suburbs

if y80sub+l(85/int)+l(90/int)>=l(95/int)+l(100/int)
    if l(100/int)+l(95/int)+l(90/int)>=y80sub+l(85/int)
        sprintf('median voter in suburbs has income 90')
        tscheck = (mvs-phs)/(phs*(1+delta^(-1/sigma)))
        escheck = tscheck*phs
    end
end
    

%3.2. Urban area

%By Epple and Romano (1996) the "ends against the middle" property holds in a voting equilibrium in a mixed community.
%That is to say, a voting equilibrium "is characterized by a balancing of a middle-income coalition preferring higher 
%public expenditure against a coalition of high- and low-income households preferring lower expenditure." (Epple and Romano, 1996, p. 322). 


if y80ur+l(60/int)+l(65/int)+l(70/int)+l(75/int)+L(25/int)>=l(30/int)+l(35/int)+l(40/int)+l(45/int)+l(50/int)+l(55/int)
    if l(30/int)+l(35/int)+l(40/int)+l(45/int)+l(50/int)+l(55/int)+l(25/int)>=y80ur+l(60/int)+l(65/int)+l(70/int)+l(75/int)+L(4)
        sprintf('median voter in urban area has income 25')
        nu   =   L(55/int)
        tucheck = (mvu-phu)/(phu+(((delta*Nu/nu)^(-1/sigma))*(phu*Nu/nu)))
        eucheck = tucheck*phu*(Nu/nu)
    end
end