$title pure exchange model (ie no production) (EXC2X2EMP,SEQ=57)

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Model from Ermoliev et al, On Convergence of SJM
In this version, replace utility maximization with demand functions
This is similar to scarfemp-dual.gms except there is no production

Rather than form the MCP explicitly,
we instead use the KKT of the dual form optimization problem:
   max_p  sum (h, i(h) * log(expend_h(p))) - p'*sum(h, endow(.,h))
   s.t    sum(., p(.)) = 1, p >= 0

coupled with the income complementarity relationship.

Here expend_h is the expenditure function defined by:
   expend_h(p)  = min_c  p'*c   s.t.  u_h(c) >= 1

In this model, player 1 has u_1(x1) x1('2')
while player 2 has u_2(x2) = sqrt(x2('1')*x2('2'))

Contributor: Michael Ferris, October 2010
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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 /;

* Following is the complementarity model generated

equations objdef, income(j), norm;
positive variables p(j);
free variables z, w(j);

* Same as scarfmcp model except production "z" = 0
* expenditure functions are e_1 (p) = p2 and e_2(p) = 2 sqrt(p1*p2) resp

objdef..
  z =e=
  sum(j, w(j)*(log(p('2')$sameas(j,'1') + log(2*sqrt(p('1')*p('2')))$sameas(j,'2'))))

  - sum(j, p(j)*(w1(j) + w2(j)));

income(j)..
  w(j) =e= sum(i, p(i)*(w1(i)$sameas(j,'1') + w2(i)$sameas(j,'2')));

norm..
  sum(j, p(j)) =e= 1;

model nashemp /objdef,income,norm/;

file myinfo /'%emp.info%'/;
putclose myinfo 'dualequ income w';

p.l(j) = 1;
w.l(j) = 0.5;
solve nashemp using emp max z;

* now form problem explicitly
* move p away from zero to allow evaluation of d_p
p.lo(j) = 1e-8;

equations d_p(j), Nnorm;
free variables mu;

d_p(j)..
  (-w('1')/p(j))$sameas(j,'2')
  - 0.5*w('2')/p(j)
  + w1(j) + w2(j) + mu =g= 0;

Nnorm..
  1 =e= sum(j, p(j));

model nash /income.w,d_p.p,Nnorm.mu/;

nash.iterlim = 0;
mu.l = norm.m;
solve nash using mcp;

parameter x1(j), x2(j);
x1(j) = 1$sameas(j,'2')*w.l('1')/p.l(j) ;
x2(j) = 0.5*w.l('2')/p.l(j);

display x1, x2;