Duopoly Example

This example is taken from the book “Complementarity Modeling in Energy Markets.”

We have two firms i producing q(i). The total production Qtot is then defined by adding up these two quantities. Because of say transmission capacity or available drilling riggs, there is a limit on the total production Qtot. The cost function of each firm is linear: Cost(i) = γ(i)*q(i). The demand price P is determined by an inverse demand function of the form P=α-β*Qtot.

The two firms each face an optimization problem: maximize profit = revenue – cost. To combine the optimization problem of both agents we form the first-order conditions (KKT conditions) and combine these in one model. For details see the book.

A “scalar” model is given in the appendix of the book. The model does not use GAMS indexing so it is merely using GAMS as a calculator. Here is a GAMSified version:

$ontext    duopoly example    Reference:       Steven A. Gabriel, Antonio J. Conejo, J. David Fuller,       Benjamin F. Hobbs, Carlos Ruiz       Complementarity Modeling in Energy Markets       Springer 2012 $offtext set    i /firm1,firm2/; parameters    gamma(i)  'unit cost' /firm1 1, firm2 2/    beta      'coefficient inverse demand function' /5/    alpha     'coefficient inverse demand function' /10/    Qmax      'upper limit on total quantity produced' / 1.06 / ; positive variables    q(i)        'production at each firm'    lambda_MAX  'dual of MaxQ constraint'    P           'price' ; free variable    Qtot  'total production: automatic positive -- q(i) is positive' ; equations    FOCfirm(i)  'first-order conditions: derivative of lagrangian wrt q(i)'    MaxQ        'max total production restriction'    QDef        'definition of Qtot (total production)'    Demand      'inverse demand function' ; FOCFirm(i)..   0 =g= P - beta*q(i) - gamma(i) - lambda_MAX; MaxQ..         Qmax =g= Qtot; QDef..         Qtot =e= sum(i, q(i)); Demand..       P =g= alpha - beta*Qtot; model problem2 /        FOCfirm.q        MaxQ.lambda_MAX        QDef.QTot        Demand.P /; solve problem2 using mcp; parameter results(*,*,*); $macro report(a) \ results(a, 'unit cost', i)       = gamma(i);      \ results(a, 'q',         i)       = q.l(i);        \ results(a, 'q',         'total') = Qtot.l;        \ results(a, 'q',         'max')   = Qmax;          \ results(a, 'p',         'total') = P.l;           \ results(a, 'dual',      'max')   = lambda_MAX.l;  \ display results; report('base case') * experiments: * make firm 2 very expensive gamma('firm2') = 6; solve problem2 using mcp; report('firm2 expensive') * make firm 2 very cheap gamma('firm2') = .5; solve problem2 using mcp; report('firm2 cheap')

The results look like:

----     75 PARAMETER results                                  firm1       firm2       total         max base case      .unit cost       1.000       2.000 base case      .q               0.630       0.430       1.060       1.060 base case      .p                                       4.700 base case      .dual                                                0.550 firm2 expensive.unit cost       1.000       6.000 firm2 expensive.q               0.900                   0.900       1.060 firm2 expensive.p                                       5.500 firm2 cheap    .unit cost       1.000       0.500 firm2 cheap    .q               0.480       0.580       1.060       1.060 firm2 cheap    .p                                       4.700 firm2 cheap    .dual                                                1.300

Notes:

• In the base case the capacity constraints is binding and we have a dual.
• In the second case the capacity constraint is no longer binding and the dual is zero. Furthermore, firm 2 is too expensive to  make it worthwhile to produce anything (the unit cost is larger than the price).
• If we make firm 2 very cheap we see that firm 1 stays in business. The capacity constraint effectively puts a floor on the price, and at this price firm 1 is still able to produce.
• In the model, both firms share a single dual λ in the FOC equation.
• This model is easily extended to more firms (no equations need to be changed), in which case we have a oligopoly.
• Qtot is a free variable and we match it to a =e= constraint. We could also make it a positive variable. In that case we could make QDef a =g= equation.