Nash-Cournot(mcp) ----> Stackelberg(mpec), still Nash-Cournot output.

[Vegard Skonseng Bjerketvedt]

Trying to extend my Nash-Cournot model (mcp) to a Stackelber model (mpec).  

In my Stackelberg model firm 2 act as leader, 1 and 3 are followers. 

The object function of the model is firm 2`s (leaders) protfit function. 
The constraints and parameters are identical to the Nash-Cournot. 

KKT conditions for firm 1 and 3 (followers) are same in both Nash-Cournot
and Stackelberg (mpec) sub-game. 

My output for both models are equal. Do someone know why my "Stackelberg-model" 
acts as a Nash-Cournot? 

x(i,b) in the Nash-Cournot: 
x(1,1) = 21 000 x(1,2) = 12 200 

x(2,1) = 17 000 x(2,2) = 0 

x(3,1) = 18 000 x(3,2) = 0

wanted output for the Stackelberg model: 

x(1,1) = 21 000 x(1,2) = 16 000 

x(2,1) = 17 000 x(2,2) = 0 

x(3,1) = 18 000 x(3,2) = 0 

Both codes below.

 
This is the Nash-Cournot model(mcp): 
- 3 firms with production and transmission constraints 

*Nash-Cournot

*Linear inverse demand function with 3 price-making firms

*Transmission contraint

Sets

i firms /i1, i2, i3 /

n node /n1, n2, n3 /

b production tech /b1, b2 /

alias(i,j)

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) peocess capacities

          b1      b2

i1       21000    16000

i2       17000    22000

i3       18000    14000

TABLE T(i,n) peocess capacities

          n1      n2       n3

i1       1000000  1000000  1000000

i2       1000000  1000000  1000000

i3       1000000  1000000  1000000

PARAMETERS alpha(n) inverse demand intercept node n

/n1 2.5, n2 2.5, n3 2.5/ ;

PARAMETERS beta(n) inverse demand slope node n

/n1 0.00005, n2 0.00005, n3 0.00005/ ;

POSITiVE VARIABLES

x(i,b) production by i with tech b

z(i,n) sold fom firm i to node n

q(n) demand quant in node n

gamma(i,b) dual of capacity constraint

omega(i,n) dual til transmisjons constraint

theta(i) dual

;

EQUATIONS

Cap(i,b) Production capacity

Trans(i,n) Transsmision capacity

Salg(i) Sold

KKTx(i,b) KKTcondition (MR=MC)

KKTz(i,n) KKTcondition

Demand(n) demand in node n

;

Cap(i,b).. - x(i,b) + K(i,b) =g= 0 ;

Trans(i,n).. - z(i,n) + T(i,n) =g= 0 ;

Salg(i).. - sum(n,z(i,n)) + sum(b,x(i,b)) =g= 0 ;

KKTx(i,b).. C(i,b) + gamma(i,b) - theta(i) =e= 0 ;

KKTz(i,n).. - alpha(n) + beta(n)*q(n) + beta(n)*z(i,n) + omega(i,n) + theta(i) =e= 0 ;

Demand(n).. q(n) - sum(i,z(i,n)) =e= 0 ;

MODEL

Nash /Cap.gamma, Trans.omega, Salg.theta, KKTx.x, KKTz.z, Demand.q/ ;

Stackelberg: 

*Stackelberg

Sets

i firms /i1, i2, i3 /

ifol(i) following firms /i1, i3/

b production tech /b1, b2/

n node /n1, n2, n3 /

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) peocess capacities

          b1      b2

i1       21000    16000

i2       17000    22000

i3       18000    14000

TABLE T(i,n) peocess capacities

          n1      n2       n3

i1       1000000  1000000  1000000

i2       1000000  1000000  1000000

i3       1000000  1000000  1000000

PARAMETERS alpha(n) inverse demand intercept node n

/n1 2.5, n2 2.5, n3 2.5/ ;

PARAMETERS beta(n) inverse demand slope node n

/n1 0.00005, n2 0.00005, n3 0.00005/ ;

POSITIVE VARIABLES

x2(b) production by leader from b

z2(n) sold by leder i n

x(ifol, b) production by i from tech b

z(ifol, n) sold av ifol i n

q demanded quantity

gamma(ifol, b) dual of capacity cont for followers

omega(ifol, n) dual of tran cont for followers

theta(ifol) dual til kan ikke selge mer en produsert

;

VARIABLES

NegStackObj negative of Stack objtive

;

EQUATIONS

StackObjDef Stackelberg objective defination

Capacity2(b) upper limmint cap generation for b

Trans2(n) upper limint trans

Salg2 kan ikke selge mer en produsert

Capacity(ifol, b) upper limmit

Trans(ifol, n) upper lim trans

Salg(ifol) kan ikke selge mer en produsert

MarkClear market clearing

KKTx(ifol,b) kktx followers

KKTz(ifol,n) kktz followers

;

StackObjDef.. sum(b, C('i2', b)*x2(b)) - sum(n,(alpha(n) - beta(n)*q(n))*z2(n))

              - NegStackObj =e= 0 ;

Capacity2(b).. x2(b) - K('i2', b) =l= 0 ;

Trans2(n).. z2(n) - T('i2', n) =l= 0;

Salg2.. sum(n,z2(n)) - sum(b,x2(b)) =l= 0;

Capacity(ifol, b).. - x(ifol, b) + K(ifol, b) =g= 0 ;

Trans(ifol, n).. - z(ifol, n) + T(ifol, n) =g= 0 ;

Salg(ifol).. - sum(n, z(ifol,n)) + sum(b,x(ifol,b)) =g= 0 ;

MarkClear(n).. q(n) - sum(ifol, z(ifol, n)) - z2(n) =e= 0 ;

KKTx(ifol,b).. C(ifol,b) + gamma(ifol,b) - theta(ifol) =e= 0 ;

KKTz(ifol,n).. - alpha(n) + beta(n)*q(n) + beta(n)*z(ifol,n) + omega(ifol,n)

               + theta(ifol) =e= 0 ;

MODEL Stackelberg

/StackObjDef, Capacity2, Trans2, Salg2, Capacity.gamma, Trans.omega, Salg.theta,

MarkClear, KKTx.x, KKTz.z/ ;

Stackelberg.optfile = 1 ;

SOLVE Stackelberg minimizing NegStackObj using mpec ;