Description

Scarf's Activity Analysis Example

Scarf, H, and Hansen, T, The Computation of Economic Equilibria.
Yale University Press, 1973.

Contributor: Michael Ferris, October 2010

Small Model of Type : EQUIL

Category : GAMS EMP library

Main file : scarfemp-dem.gms

$title  Scarf's Activity Analysis Example (SCARFEMP-DEM,SEQ=51)

$onText

Scarf's Activity Analysis Example

Scarf, H, and Hansen, T, The Computation of Economic Equilibria.
Yale University Press, 1973.

Contributor: Michael Ferris, October 2010
$offText

$call gamslib -q scarfmcp
$include scarfdata.inc

option limrow = 0, limcol = 0;

positive
variables
        y(s)    production,
        x(c,h)  Marshallian demand,
        p(c)    commodity price,
        i(h)    income;

equations
        mkt(c)          commodity market,
        profit(s)       zero profit,
        income(h)       income index;

mkt(c)..        sum(s, a(c,s) * y(s)) + sum(h, e(c,h)) =g=
                sum(h, x(c,h));

profit(s)..     -sum(c, a(c,s) * p(c)) =g= 0;

income(h)..     i(h) =g= sum(c, p(c) * e(c,h));

p.l(c) = 1;
i.l(h) = sum(c, p.l(c) * e(c,h));

p.lo(c)  = 0.00001$(smax(h, alpha(c,h)));
y.lo(s) = 0;

*       fix the price of numeraire commodity:

i.fx(h)$(ord(h) eq 1) = i.l(h);

variables z(h);
equations objdef(h), budget(h);

* distinguish ces and cobb-douglas demand functions:
* following works for Cobb Douglas functions (esub(h) = 1)

objdef(h)..
  z(h) =e=
  (1/rho(h))*log(sum(c$alpha(c,h), lambda(c,h)*x(c,h)**rho(h)))$(esub(h) ne 1)

  + sum(c$alpha(c,h), alpha(c,h)*log(x(c,h)))$(esub(h) eq 1);

budget(h)..
  sum(c, p(c)*x(c,h)) =l= i(h);

model scarf /objdef, budget, mkt, profit, income/;

* For Cobb Douglas
* esub(h) = 1;

* Set initial values for production (y)
x.l(c,h) = 1;
* x.l(c,h) = 0.1;
x.lo(c,h) = 0.00001$alpha(c,h);
x.fx(c,h)$(not alpha(c,h)) = 0;

file myinfo / '%emp.info%' /;
put myinfo / 'equilibrium';
put / 'vi income i';
put / 'vi mkt p';
put / 'vi profit y';
loop(h,
  put / 'max' z(h);
  loop(c, put / x(c,h) );
  put / objdef(h) budget(h);
);
putclose /;

* determine good starting point for consumption variables
i.fx(h) = i.l(h);
p.fx(c) = p.l(c);
y.fx(s) = y.l(s);
solve scarf using emp;

p.lo(c)  = 0.00001$(smax(h, alpha(c,h)));
p.up(c) = inf;
y.lo(s) = 0; y.up(s) = inf;
solve scarf using emp;

* now solve for real
i.lo(h) = 0; i.up(h) = inf;
i.fx(h)$(ord(h) eq 1) = i.l(h);
solve scarf using emp;

display y.l;