$title General Equilibrium Variant of the von Thunen Model (VONTHMCP,SEQ=136)

$onText
General Equilibrium Variant of the von Thunen Model.

Rowse, Mackinnon, Samuelson, and von Thunen, General Equilibrium Variant
of the von Thunen Model.

Keywords: mixed complementarity problem, general equilibrium model, von Thunen model
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Set
   r 'regions'     / r1*r12 /
   c 'commodities' / wheat,  rice,  corn,  barley /
   h 'households'  / worker, owner, porter        /;

Scalar
   ltot 'labor endowment'     / 30 /
   trns 'transport endowment' / 20 /;

Parameter
   d(r)       'distance to region r'
   a(r)       'area of region r'
   t(c)       'transport requirement'
   phi(c)     'cost function scale parameter'
   beta(c)    'cost function share parameter'
   alpha(*,h) 'demand function share';

d(r) = 5*(2*ord(r) - 1);
a(r) = 2*3.1415*d(r);

Table misc(*,*)
            wheat  rice   corn   barley  leisure
   phi      1      2      3        4
   beta     0.9    0.7    0.5      0.3
   t        0.015  0.006  0.004    0.01
   worker   0.2    0.3    0.1      0.3       0.1
   owner    0.3    0.3    0.2      0.2
   porter   0.6    0.2    0.1      0.1          ;

phi(c)     = misc("phi",c);
beta(c)    = misc("beta",c);
t(c)       = misc("t",c);
alpha(c,h) = misc(h,c);
alpha("leisure",h) = misc(h,"leisure");

Positive Variable
   cst(r,c) 'unit cost function'
   y(r,c)   'output level'
   ip(r)    'intervention purchase'
   pk(r)    'rental rate'
   wp       'porter wage'
   wl       'worker wage'
   p(c)     'market price';

Equation
   def_cst(r,c)
   prf_y(r,c)
   pkbnd(r)
   mkt_k(r)
   mkt_t
   mkt_l
   mkt_g(c);

def_cst(r,c)..  phi(c)*cst(r,c) =g= (wl/beta(c))**beta(c)*(pk(r)/(1 - beta(c)))**(1 - beta(c));

prf_y(r,c)..    cst(r,c) + t(c)*d(r)*wp =g= p(c);

pkbnd(r)..      pk(r) =g= 0.0001*wl;

mkt_k(r)..      a(r) - ip(r) =g= sum(c, y(r,c)*(1 - beta(c))*cst(r,c))/pk(r);

mkt_t..         trns =g= sum((r,c), t(c)*d(r)*y(r,c));

mkt_l..         wl*ltot  =g= sum((r,c), y(r,c)*beta(c)*cst(r,c)) + alpha("leisure","porter")*ltot;

mkt_g(c)..      p(c)*sum(r, y(r,c)) =g= alpha(c,"porter")*trns*wp
                                     +  alpha(c,"worker")*ltot*wl
                                     +  alpha(c,"owner")*sum(r, pk(r)*(a(r) - ip(r)));

p.lo(c)  = 0.1;
pk.lo(r) = 0.001;

Model vonthun / def_cst.cst, prf_y.y, pkbnd.ip, mkt_k.pk, mkt_t.wp, mkt_l.wl, mkt_g.p /;

cst.l(r,c) = 1;
y.l(r,c)   = 1;
ip.l(r)    = 0;
pk.l(r)    = 1;
wp.l       = 1;
wl.l       = 1;
p.l(c)     = 1;

* use the labor wage as numeraire:
wl.fx = 1;

solve vonthun using mcp;

display y.l, cst.l;