Description

Scarf's Activity Analysis Example

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

Rather than write the demand function explicitly (as in the GAMSLIB model
scarfmcp), we instead

max_x(c,h)  u_h(x)
   s.t    p'*x <= i_h, x >= 0

In this model we generate a new utility function
   u_h(x) = sum_c gamma(c,h) * x(c,h)**beta(c,h)

for which there is no closed form solution for the demand function
(but which is chosen to have the same soln as the scarf model)

Contributor: Michael Ferris, October 2010

Small Model of Type : EQUIL

Category : GAMS EMP library

Main file : scarfemp-altdem.gms

$title  Scarf's Activity Analysis Example - non-closed form demand function (SCARFEMP-ALTDEM,SEQ=52)

$onText

Scarf's Activity Analysis Example

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

Rather than write the demand function explicitly (as in the GAMSLIB model
scarfmcp), we instead

max_x(c,h)  u_h(x)
   s.t    p'*x <= i_h, x >= 0

In this model we generate a new utility function
   u_h(x) = sum_c gamma(c,h) * x(c,h)**beta(c,h)

for which there is no closed form solution for the demand function
(but which is chosen to have the same soln as the scarf model)

Contributor: Michael Ferris, October 2010
$offText

$call gamslib -q scarfmcp
$include scarfdata.inc

option limrow = 0, limcol = 0;

positive
variables

        p(c)    commodity price,
        y(s)    production,
        i(h)    income;

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

*       distinguish ces and cobb-douglas demand functions:

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

                sum(h$(esub(h) ne 1),
                (i(h)/sum(cc, alpha(cc,h) * p(cc)**(1-esub(h)))) *
                alpha(c,h) * (1/p(c))**esub(h)) +

                sum(h$(esub(h) eq 1),
                i(h) * alpha(c,h) / p(c));

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

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

model scarf / mkt.p, profit.y, income.i/;

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

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

*       fix the price of numeraire commodity:

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

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

solve scarf using mcp;

* Formulate an alternative version of the model which calibrate to the
* common benchmark point but assumes a alternative utility function for
* which demand functions are unavailable in closed form.

parameter beta(c,h)  Elasticity terms
          gamma(c,h) Utility function share parameters;

option seed = 101;
beta(c,h) = uniform(0.5,1.5);

positive variables x(c,h) Consumption demand by commodity and household;

x.l(c,h) = ((i.l(h)/sum(cc, alpha(cc,h) * p.L(cc)**(1-esub(h)))) *
           alpha(c,h) * (1/p.L(c))**esub(h))$(esub(h)<>1) +
           (i.l(h) * alpha(c,h) / p.l(c))$(esub(h)=1);

gamma(c,h)$x.l(c,h) = p.l(c) / (beta(c,h) * x.l(c,h)**(beta(c,h)-1));

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

objdef(h)..
  z(h) =e= sum(c$gamma(c,h), gamma(c,h)*x(c,h)**beta(c,h));

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

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

model scarfemp /objdef, budget, mktcons, profit, income/;

* set benchmark starting point for emp model
* budget.m(h) = 1;

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

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

* determine good starting point for consumption variables
p.l(c) = 1;
y.l(s) = 1;
i.fx(h) = sum(c, p.l(c) * e(c,h));

solve scarfemp 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 scarfemp using emp;

display y.l;