harker.gms : Models of Spatial Competition
A spatial equilibrium model is used to demonstrate different
ways of modeling market behavior. Competitive and Monopolistic markets
are modeled using consumers' and producers' surplus. A Cournot-Nash
Oligopoly model is solved by a Diagonalization or Jacobi algorithm.
Reference:
- Harker, P T, Alternative Models of Spatial Competition. Operations Research 34, 3 (1986), 410-425.
Small Model of Type: NLP
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$Title Models of Spatial Competition (HARKER,SEQ=85)
$Ontext
A spatial equilibrium model is used to demonstrate different
ways of modeling market behavior. Competitive and Monopolistic markets
are modeled using consumers' and producers' surplus. A Cournot-Nash
Oligopoly model is solved by a Diagonalization or Jacobi algorithm.
Network structure:
one two three demand and supply points
/ \ \ /
/ \ \ /
four -- five -- six transport nodes
linear demand function: demand(price) = (rho - price)/eta
linear marginal cost function: cost(supply) = alpha + 2*beta*supply
transport cost: cost(t(ij)) = kappa(ij)*t(ij) + nu(ij)*t(ij)^3
Harker, P T, Alternative Models of Spatial Competition. Operations
Research 34, 3 (1986), 410-425.
$Offtext
Sets n nodes / one, two, three, four, five, six /
l(n) regions / one, two, three /
Alias (l,lp),(n,np)
Table coefs(l,*) demand and supply data
alpha beta rho eta
one 1.0 .5 19 .2
two 2.0 .4 27 .01
three 1.5 .3 30 .3
Table pairs(n,np,*) transport data
kappa nu
one.four 1 .5
one.five 2 .2
two.six 3 .3
three.six 1 .4
four.one 2 .3
four.five 1 .1
four.six 1 .1
five.one 3 .5
five.four 2 .2
five.six 1 1.0
six.two 2 .25
six.three 2 .2
six.four 1 .9
six.five 3 .8
Set arc(n,np) active arcs; arc(n,np) = pairs(n,np,"kappa");
Positive
Variables t(n,np) transport
tt(l,lp) notional o-d flows
d(n) demand
s(n) supply
Variables obj objective value
Equations bal supply demand balance
sbal(l) supply balance
dbal(l) demand balance
nbal(n) node balance
in(l) inflow balance
objdef objective definition
objoli objective definition oligopoly ;
Scalars pm product market type
tm transport market type;
bal.. sum(l, d(l)) =e= sum(l, s(l)) ;
nbal(n).. s(n)$l(n) + sum(arc(np,n), t(arc)) =e= d(n)$l(n) + sum(arc(n,np), t(arc));
objdef.. obj =e= sum(l, coefs(l,"rho")*d(l) - pm*coefs(l,"eta")*sqr(d(l)) )
- sum(l, coefs(l,"alpha")*s(l) + coefs(l,"beta")*sqr(s(l)) )
- sum(arc, pairs(arc,"kappa")*t(arc)
+ tm*pairs(arc,"nu")*power(t(arc),3) );
Model hark / bal, nbal, objdef /;
Parameters rep1 transport summary
rep2 supply demand and price summary;
s.l(l) = 25; d.l(l)=25;
* 1. classical spatial price equilibrium: perfectly competitive
* producers and suppliers facing average cost pricing
* of transportation:
pm = .5; tm = 1/3; Solve hark maximizing obj using nlp;
rep1(arc, "cspe2") = t.l(arc);
rep2("supply",l,"cspe2") = s.l(l);
rep2("demand",l,"cspe2") = d.l(l);
rep2("price ",l,"cspe2") = coefs(l,"rho") - coefs(l,"eta")*d.l(l);
* 2. monopoly pricing equilibrium in which the firm owns both
* means of production and distribution network (hence, marginal
* cost pricing prevails at both the factory and the railhead):
pm = 1; tm = 1; Solve hark maximizing obj using nlp;
rep1(arc, "monop1") = t.l(arc);
rep2("supply",l,"monop1") = s.l(l);
rep2("demand",l,"monop1") = d.l(l);
rep2("price ",l,"monop1") = coefs(l,"rho") - coefs(l,"eta")*d.l(l);
* 3. monopoly pricing equilibrium in which the firm uses the
* distribution network with average cost pricing:
pm = 1; tm = 1/3; Solve hark maximizing obj using nlp;
rep1(arc, "monop2") = t.l(arc);
rep2("supply",l,"monop2") = s.l(l);
rep2("demand",l,"monop2") = d.l(l);
rep2("price ",l,"monop2") = coefs(l,"rho") - coefs(l,"eta")*d.l(l);
* 4. multi-producer oligopoly model with average cost pricing
* of transportation links:
pm = 1; tm = 1/3;
* Additional equation required to solve the a Cournot-Nash
* Oligopoly model by a Diagonalization or Jacobi algorithm.
*
* note the use of d.l(l)-tt.l(lp,l) which holds the values of the
* previous iteration
Variable tt(l,lp) notional o-d flows
Equations sbal(l) supply balance
dbal(l) demand balance
in(l) inflow balance
objoli objective definition oligopoly ;
sbal(l).. s(l) =e= sum(lp, tt(l,lp));
dbal(l).. d(l) =e= sum(lp, tt(lp,l));
in(l).. d(l) =e= tt(l,l) + sum(arc(n,l), t(arc));
objoli.. obj =e= sum(l, coefs(l,"rho")*d(l) - pm*coefs(l,"eta")*
sum(lp, (d.l(l)-tt.l(lp,l)+tt(lp,l))*tt(lp,l)) )
- sum(l, coefs(l,"alpha")*s(l) + coefs(l,"beta")*sqr(s(l)) )
- sum(arc, pairs(arc,"kappa")*t(arc)
+ tm*pairs(arc,"nu")*power(t(arc),3) );
model harkoli / nbal, sbal, dbal, in, objoli / ;
set iter iteration count / iter1*iter20 /;
parameters objold previous objective, irep(iter,n,np) iteration summary;
option irep:8:1:2;
tt.l(l,lp)=1; Option limcol=0,limrow=0;
objold=0; harkoli.objval=1;
loop(iter$(abs(objold-harkoli.objval) > 1e-5),
objold = harkoli.objval;
Solve harkoli maximizing obj using nlp;
harkoli.solprint=%solprint.Quiet%;
irep(iter,arc) = t.l(arc) );
Display irep;
rep1(arc, "oligop") = t.l(arc);
rep2("supply",l,"oligop") = s.l(l);
rep2("demand",l,"oligop") = d.l(l);
rep2("price ",l,"oligop") = coefs(l,"rho") - coefs(l,"eta")*d.l(l);
Display rep1, rep2;
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