$title Simple 2 x 2 x 2 General Equilibrium Model (TWO3MCP,SEQ=131)

$onText
Simple 2 x 2 x 2 General Equilibrium Model.

Shoven, J, and Whalley, J, Applied G.E. Models. Journal of Economic
Literature 22 (1984).

Keywords: mixed complementarity problem, general equilibrium model
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Set
   f 'factors'    / labor, capital /
   s 'sectors'    / mfrs,  nonmfrs /
   h 'households' / rich,  poor    /;

Alias (s,ss), (f,ff);

* demand function parameters.
Parameter sigmac(h) / rich 1.5, poor 0.75 /;

Table alpha(s,h)
             rich  poor
   mfrs       0.5   0.3
   nonmfrs    0.5   0.7;

Table e(f,h)
             rich  poor
   labor             60
   capital     25      ;

* production function parameters.
Parameter phi(s) / mfrs 1.5, nonmfrs 2.0 /;

Table delta(f,s) 'factor share coefficients'
             mfrs  nonmfrs
   labor      0.6      0.7
   capital    0.4      0.3;

Parameter
   sigma(s) 'elasticities of factor substitution' / mfrs 2.0, nonmfrs 0.5 /
   tshr(h)  'share of tax revenue'
   t(f,s)   'ad-valorem tax rates';

tshr(h) = 0;
t(f,s)  = 0;

Positive Variable
   W(f) 'factor price'
   P(s) 'commodity price'
   Y(s) 'production level'
   I(h) 'income';

Equation
   fmkt(f)   'factor market'
   cmkt(s)   'commodity market'
   profit(s) 'zero profit'
   income(h) 'income equation';

fmkt(f)..
   sum(h, e(f,h)) =g=  sum(s, Y(s)*phi(s)**(sigma(s) - 1)
                   *  (delta(f,s)*(sum(ff, delta(ff,s)**sigma(s)
                   *  (W(ff)*(1 + t(ff,s)))**(1 - sigma(s)))
                   ** (1/(1 - sigma(s)))/phi(s))
                   /  (W(f)*(1 + t(f,s))))**sigma(s));

cmkt(s)..
   Y(s) =g= sum(h, (i(h)/sum(ss, alpha(ss,h)*P(ss)**(1 - sigmac(h))))*alpha(s,h)*(1/P(s))**sigmac(h));

profit(s)..
   sum(f, delta(f,s)**sigma(s)*(W(f)*(1 + t(f,s)))**(1 - sigma(s)))**(1/(1 - sigma(s)))/phi(s) =g= P(s);

income(h)..
   I(h) =g= sum(f, E(f,h)*W(f)) + tshr(h)
         *  sum((s,f),  t(f,s)*W(f)*Y(s)*phi(s)**(sigma(s) - 1)
                     * (delta(f,s)*(sum(ff, delta(ff,s)**sigma(s)
                     * (W(ff)*(1 + t(ff,s)))**(1 - sigma(s)))
                    ** (1/(1 - sigma(s)))/phi(s))/(W(f)*(1 + t(f,s))))**sigma(s));

Model jel / fmkt.W, cmkt.P, profit.Y, income.I /;

* compute solution for this dimension problem:
W.lo(f) = 0.0001;
P.lo(s) = 0.0001;
W.l(f)  = 1;
P.l(s)  = 1;
Y.l(s)  = 10;
I.l(h)  = sum(f, W.l(f)*E(f,h));

* solve the reference case:
W.fx("labor") = 1;
solve jel using mcp;

* apply tax in test problem:
tshr("rich") = 0.4;
tshr("poor") = 1 - tshr("rich");
t("capital","mfrs") = 0.5;
solve jel using mcp;