exc2x2emp-dem.gms : pure exchange model (ie no production)

**Description**

Model from Ermoliev et al, On Convergence of SJM Model uses explicit demand functions Computational issues with x2('2') = 0 at solution gams exc2x2 --sol=mcp1 This sets up the standard Arrow Debreu model and solves using MCP gams exc2x2 --sol=mcp2 This sets up the Ermoliev aggregate embedded model, solved using MCP gams exc2x2 --sol=neg This does Negishi iterations on the Ermoliev embedded model - ie sequence of NLP's trying to get to fixed point on "budget" constraint. Contributor: Michael Ferris, October 2010

**Small Model of Type : ** EQUIL

**Category :** GAMS EMP library

**Main file :** exc2x2emp-dem.gms

```
$title pure exchange model (ie no production) (EXC2X2EMP-DEM,SEQ=58)
$ontext
Model from Ermoliev et al, On Convergence of SJM
Model uses explicit demand functions
Computational issues with x2('2') = 0 at solution
gams exc2x2 --sol=mcp1
This sets up the standard Arrow Debreu model and solves using MCP
gams exc2x2 --sol=mcp2
This sets up the Ermoliev aggregate embedded model, solved using MCP
gams exc2x2 --sol=neg
This does Negishi iterations on the Ermoliev embedded model - ie
sequence of NLP's trying to get to fixed point on "budget"
constraint.
Contributor: Michael Ferris, October 2010
$offtext
$if NOT set sol $set sol mcp1
set i /1*2/;
alias(i,j);
parameter w1(j) 'endowment of player 1' /
1 1
2 1 /,
w2(j) 'endowment of player 2' /
1 1 /;
variables obj1, obj2, obj3;
positive variables p(j), x1(j), x2(j);
equations defobj1, defobj2, defobj3, budget1, budget2, norm;
defobj1.. obj1 =e= x1('2');
budget1..
sum(j, p(j) * x1(j)) =l= sum(j, w1(j)*p(j));
defobj2.. obj2 =e= sqrt(x2('1')*x2('2'));
budget2..
sum(j, p(j) * x2(j)) =l= sum(j, w2(j)*p(j));
* market player
defobj3.. obj3 =e= sum(j, p(j)*(x1(j) + x2(j) - w1(j) - w2(j)));
norm..
sum(j, p(j)) =e= 1;
* The combined KKT forms a complementarity system
* solution is p = (0,1), x1 = (x11,1), x2 = (x21,0)
* with 0 <= x11 + x21 <= 2
* model does not satisfy gross substitutability assumption
* so tatonment process fails
file myinfo /'%emp.info%'/;
put myinfo 'equilibrium';
put / 'max obj1 x1 defobj1 budget1';
put / 'max obj2 x2 defobj2 budget2';
putclose / 'max obj3 p defobj3 norm ';
model nashemp /defobj1,defobj2,defobj3,budget1,budget2,norm/;
* protect function evaluations
* x1.lo(j) = 1e-6; x2.lo(j) = 1e-6;
x1.l(j) = 1; x2.l(j) = 1;
p.l(j) = 1;
budget1.m = 1; budget2.m = 1; norm.m = 1;
$if %sol% == mcp1 solve nashemp using emp;
$ontext
* Following is the complementarity model generated
equations d_x1(j), d_x2(j), d_p(j), Nbudget1, Nbudget2, Nnorm;
positive variables x1(j), x2(j), p(j), lambda1, lambda2;
free variables mu;
d_x1(j)..
-1$sameas(j,'2') + lambda1*p(j) =g= 0;
* rewrite budget as negative due to "max" problem
Nbudget1..
sum(j, w1(j)*p(j)) =g= sum(j, p(j) * x1(j));
* fix this constraint by multiplying through by x2(j)
d_x2(j)..
-0.5*sqrt(x2('1')*x2('2')) + lambda2*p(j)*x2(j) =g= 0;
* -0.5*sqrt(x2('1')*x2('2'))/x2(j) + lambda2*p(j) =g= 0;
Nbudget2..
sum(j, w2(j)*p(j)) =g= sum(j, p(j) * x2(j));
d_p(j)..
- x1(j) - x2(j) + w1(j) + w2(j) + mu =g= 0;
Nnorm..
1 =e= sum(j, p(j));
model nashmcp /d_x1.x1,Nbudget1.lambda1,d_x2.x2,Nbudget2.lambda2,d_p.p,Nnorm.mu/;
lambda1.l = budget1.m; lambda2.l = budget2.m; mu.l = norm.m;
nashmcp.iterlim = 0;
solve nashmcp using mcp;
$offtext
* new variables for negishi version
variable utility, wVar(i) 'Negishi weights computed endogenously';
equations
utilDef 'utility in Negishi version'
prodLimit(j)
budget(i) 'set income = expenditure';
utilDef .. utility =e=
wVar('1') * log(x1('2'))
+ wVar('2') * 0.5 * sum(j, log(x2(j)));
budget(i) .. wVar(i) =e=
sum(j, (w1(j)$sameas(i,'1') + w2(j)$sameas(i,'2')) * p(j));
prodLimit(j) ..
x1(j) + x2(j) =l= w1(j) + w2(j);
model endogW / utilDef, prodLimit, budget /;
model fixedW 'basic NLP model - fixed Negishi weights' / utilDef, prodLimit /;
put myinfo '* negishi model';
put / 'dualVar p prodLimit';
putclose / 'dualEqu budget wVar';
$if not %sol% == mcp2 $goto skipmcp2
wVar.l(i) = 1;
* add a numeraire, since the weights are unique only
* in a relative sense
wVar.fx('1') = 1;
solve endogW maximizing utility using emp;
wVar.fx(i) = wVar.l(i);
solve fixedW maximizing utility using nlp;
$label skipmcp2
$if not %sol% == neg $goto skipneg
set iters / iter1 * iter60 /;
scalar
err / 1 /,
m 'control the Negishi weight adjustment' / .95 /;
parameter
wBarI(i,iters),
errI(iters);
wVar.fx(i) = 1/card(i);
x1.lo(j) = 1e-6; x2.lo(j) = 1e-6;
loop {iters$[err > 1e-6],
wBarI(i,iters) = wVar.l(i);
solve fixedW using nlp maximizing utility;
wVar.fx(i) = (1-m)*wVar.l(i) +
m*(w1(i)$sameas(i,'1') + w2(i)$sameas(i,'2')) * prodlimit.m(i);
err = sum(i, abs(wVar.l(i) - wBarI(i,iters)));
errI(iters) = err;
};
* set prices for reporting purposes
p.l(j) = prodLimit.m(j);
display wBarI, errI;
$label skipneg
display x1.l, x2.l, p.l;
```