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3_4_2_6.gms
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3_4_2_6.gms
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* -----------------------------------------------------------------------
* This code implements the example discussed in Section 3 of the book
* "Complementarity Modeling in Energy Markets" by Steven A. Gabriel et al.
* using EMP framework.
*
* Example from Section 3.4.2.6 - Nash-Cournot Model of Three Firms
* Results roughly map to Table 3.2. There may be typos in the book.
* -----------------------------------------------------------------------
$ONEMPTY
SET i "firms" / i1, i2, i3 /;
SET b "production technology" / b1, b2 /;
ALIAS(i,ii);
ALIAS(b,bb);
* -----------------------------------------------------------------------
* Define MACRO to store the result in the following PARAMETERs.
* -----------------------------------------------------------------------
SET fs "firm related solution" / revenue, varcost, surplus /;
SET os "other solution" / q, price, conspayment, conssurplus, totsurplus, socialwelfare /;
PARAMETER firm_x(*,i,b) Level values of a solution x;
PARAMETER firm_sol(*,fs,i) Firm related solution;
PARAMETER ot_sol(*,os) Other solution;
$MACRO STORE(name) \
firm_x(name,i,b) = x.l(i,b); \
ot_sol(name,'q') = sum(i, sum(b, x.l(i,b))); \
ot_sol(name,'price') = alpha - beta*ot_sol(name,'q'); \
ot_sol(name,'conspayment') = ot_sol(name,'q')*ot_sol(name,'price'); \
ot_sol(name,'conssurplus') = alpha*ot_sol(name,'q') \
- 0.5*beta*ot_sol(name,'q')**2 \
- ot_sol(name,'conspayment'); \
firm_sol(name,'revenue',i) = ot_sol(name,'price')*sum(b, x.l(i,b)); \
firm_sol(name,'varcost',i) = sum(b, C(i,b)*x.l(i,b)); \
firm_sol(name,'surplus',i) = firm_sol(name,'revenue',i) \
- firm_sol(name,'varcost',i); \
ot_sol(name,'totsurplus') = sum(i, firm_sol(name,'surplus',i)); \
ot_sol(name,'socialwelfare') = ot_sol(name,'conssurplus') \
+ ot_sol(name,'totsurplus')
* -----------------------------------------------------------------------
* Parameters and variables to define problems.
* -----------------------------------------------------------------------
TABLE C(i,b) variable costs
b1 b2
i1 0.55 0.81
i2 0.62 1.25
i3 0.78 1.35
;
TABLE K(i,b) process capacities
b1 b2
i1 21000 16000
i2 17000 22000
i3 18000 14000
;
PARAMETER alpha "inverse demand intercept" / 2.5 /;
PARAMETER beta "inverse demand slope" / 0.0000166666667 /;
POSITIVE VARIABLE x(i,b) "production by firm i from process b";
POSITIVE VARIABLE q "demand quantity";
VARIABLE obj(i) "objective value of firm i";
* Capacity constraint
x.up(i,b) = K(i,b);
file empinfo / '%emp.info%' /;
* -----------------------------------------------------------------------
* Define and solve a Nash game where all firms are price-takers.
* -----------------------------------------------------------------------
EQUATION objdef_cournot_nash(i) "objective definition of firm i for Cournot-Nash game";
EQUATION demand "market demand";
objdef_cournot_nash(i)..
obj(i)
=E=
sum(b, C(i,b)*x(i,b) - (alpha - beta*sum((ii,bb), x(ii,bb)))*x(i,b));
MODEL CournotNash / objdef_cournot_nash /;
PUT empinfo 'equilibrium';
loop(i,
PUT / 'min', obj(i);
loop(b, PUT x(i,b));
PUT objdef_cournot_nash(i);
);
PUTCLOSE empinfo;
* -----------------------------------------------------------------------
* The contents of this empinfo file look like this:
* -----------------------------------------------------------------------
* equilibrium
* min obj('i1') x('i1','b1') x('i1','b2') objdef_cournot_nash('i1')
* min obj('i2') x('i2','b1') x('i2','b2') objdef_cournot_nash('i2')
* min obj('i3') x('i3','b1') x('i3','b2') objdef_cournot_nash('i3')
* -----------------------------------------------------------------------
* initial starting point
x.l(i,b) = 0;
SOLVE CournotNash using emp;
STORE('CournotNash');
* -----------------------------------------------------------------------
* Format and display solution in LST file.
* -----------------------------------------------------------------------
OPTION firm_x:0:1:2;
OPTION firm_sol:0:1:2;
DISPLAY firm_x, firm_sol;