Technical Appendix to Accompany “Electricity Market Mergers with Endogenous Forward Contracting” by David P. Brown and Andrew Eckert In this Technical Appendix, we provide a detailed characterization of the mixed nonlinear complementarity program used to solve the large-scale nonlinear EPEC. This analysis is based on the methods established in Facchinei and Kanzow (1997), Xian et al. (2004), and Hu and Ralph (2007). For numerical tractability and without loss of generality, we separate out each firm’s minimum stable generation (MSG) from its output decision and analyze each firm’s flexible output decision beyond its M SG ), bounded below by zero and above by its maximum capacity q M ax of its units that are MSG (qit it M SG ) to be the variable cost from firm i’s MSG output. dispatchable (beyond their MSG). Define Cit (qit
This adjusts the mixed complementarity conditions specified in (19) and (20) to: h i f MR M SG Pt (Qt ) + Pt0 (Qt ) qit + qit + qit − qit − Cit0 (qit ) − λit ≤ 0 max qit ≤ qit
A1
⊥
λit ≥ 0,
⊥
qit ≥ 0,
(48)
∀ i = 1, 2, 3, 4, 5.
(49)
Status Quo: No Merger
Consider the setting where there is no merger. First, we reformulate the nonlinear mixed complementarity conditions that characterize the solution to the second-stage spot market defined in (19) and (20). Second, we characterize the solution to each firm’s MPEC defined in (21) by representing the problem as a nonlinear program and deriving the corresponding Karush-Kuhn-Tucker (KKT) conditions. We introduce slack variables s1it and s2it such that the nonlinear mixed complementarity program (MCP) represented by (48) and (49) becomes: h i f MR M SG Pt (Qt ) + Pt0 (Qt ) qit + qit + qit − qit − Cit0 (qit ) − λit + s1it = 0;
(50)
max qit − qit + s2it = 0;
(51)
s1it ≥ 0
⊥
qit ≥ 0;
(52)
s2it ≥ 0
⊥
λit ≥ 0
∀ i = 1, 2, 3, 4, 5.
(53)
In the formulation of each firm’s first-stage MPEC defined in (21), conditions (52) and (53) represent complementarity constraints that are difficult to solve using nonlinear optimization algorithms. Following Facchinei and Kanzow (1997) and Xian et al. (2004), we use a nonlinear complementarity function that has the following property: ψ(a, b) =
p a2 + b2 − a − b
⇔
a≥0
⊥ b ≥ 0.
(54)
Applying (54) to (52) and (53) allows us to rewrite the spot market MCP in (50) - (53) as: h i f MR M SG Pt (Qt ) + Pt0 (Qt ) qit + qit + qit − qit − Cit0 (qit ) − λit + s1it = 0;
1
(55)
max qit − qit + s2it = 0; p (s1it )2 + (qit )2 − s1it − qit = 0; p (s2it )2 + (λit )2 − s2it − λit = 0
(56) (57) ∀ i = 1, 2, 3, 4, 5.
(58)
Having removed the MCP conditions from the second-stage spot market, we reformulate each firm’s MPEC defined in (21) as a mixed nonlinear program where they maximize their objective function, choosing all endogenous variables in the two-stage model (for additional details see Xian et al. (2004) and Hu and Ralph (2007)). Define qt = (q1t , q2t , q3t , q4t , q5t ), λt = (λ1t , λ2t , λ3t , λ4t , λ5t ), s1t = (s11t , s12t , s13t , s14t , s15t ), and s2t = (s21t , s22t , s23t , s24t , s25t ). In period t, the first-stage optimization problem of each firm i is represented as:
maximize
f qit ,qt ,λt ,s1t ,s2t
subject to
MR M SG M SG Pt (Qt ) qit + qit + qit − Cit (qit ) − Cit (qit ) f f,M ax qit ≤ qit
: ρit ≥ 0 h
i
f MR M SG 0 Pt (Qt ) + Pt0 (Qt ) qjt + qjt + qjt − qjt − Cjt (qjt ) − λjt + s1jt = 0
: ηijt
max qjt − qjt + s2jt = 0 q (s1jt )2 + (qjt )2 − s1jt − qjt = 0 q (s2jt )2 + (λjt )2 − s2jt − λjt = 0
: ijt : ψijt : ωijt ∀
j = 1, 2, 3, 4, 5; (59)
where
f qit
≤
f,M ax qit
represents the upper bound constraint on firm i’s forward position. Subsequently, the
Lagrangian function for a player i in MPEC (59) is: h i f,M ax f MR M SG M SG Lit = Pt (Qt ) qit + qit + qit − Cit (qit ) − Cit (qit ) + ρit qit − qit
+
5 X
h h i i f MR M SG 0 ηijt Pt (Qt ) + Pt0 (Qt ) qjt + qjt + qjt − qjt − Cjt (qjt ) − λjt + s1jt
j=1
+
5 X j=1
+
5 X
q 5 X max 2 2 (s1jt ) + (qjt ) − s1jt − qjt ijt qjt − qjt + s2jt + ψijt j=1
q 2 2 ωijt (s2jt ) + (λjt ) − s2jt − λjt .
(60)
j=1
We use (60) to characterize the KKT conditions for each player i. The Nash equilibrium of the twostage multi-firm forward-spot market model represented by the EPEC can be obtained by concatenating all of the KKT conditions for each player’s MPEC. Using (60), for each period t, the aggregated mixed nonlinear complementarity program is represented by the following conditions:
2
f qit :
qit :
− ηiit Pt0 (Qt ) − ρit = 0
∀
i = 1, 2, 3, 4, 5;
(61)
MR M SG Pt0 (Qt ) qit + qit + qit + Pt (Qt ) − Cit0 (qit ) h h i i f MR M SG + ηiit 2 Pt0 (Qt ) + Pt00 (Qt ) qit + qit + qit − qit − Cit00 (qit ) +
5 X
h h ii f MR M SG ηijt Pt0 (Qt ) + Pt00 (Qt ) qjt + qjt + qjt − qjt
j=1 j6=i
h i − 1 + iit + ψiit qit (s1it )2 + (qit )2 2 − 1 = 0 qjt :
Pt0 (Qt )
qit +
MR qit
+
M SG qit
+
5 X
∀ i = 1, 2, 3, 4, 5;
(62)
h h ii f MR M SG ηikt Pt0 (Qt ) + Pt00 (Qt ) qit + qit + qit − qit
k=1 k6=j
h h i i f MR M SG 00 + ηijt 2Pt0 (Qt ) + Pt00 (Qt ) qjt + qjt + qjt − qjt − Cjt (qjt ) i h − 1 + ijt +ψijt qjt (s1jt )2 + (qjt )2 2 − 1 = 0
∀ i, j = 1, 2, 3, 4, 5 with i 6= j; (63)
λjt :
h i − 1 − ηijt + ωijt λjt (s2jt )2 + (λjt )2 2 − 1 = 0
∀
i, j = 1, 2, 3, 4, 5;
(64)
s1jt :
h i − 1 ηijt + ψijt s1jt (s1jt )2 + (qjt )2 2 − 1 = 0
∀
i, j = 1, 2, 3, 4, 5;
(65)
s2jt :
i h − 1 ijt + ωijt s2jt (s2jt )2 + (λjt )2 2 − 1 = 0
∀
i, j = 1, 2, 3, 4, 5;
(66)
ηijt
f f,M ax qit ≤ qit
⊥ ρit ≥ 0 ∀ i = 1, 2, 3, 4, 5; (67) h i f MR M SG 0 : Pt (Qt )+Pt0 (Qt ) qjt + qjt + qjt − qjt −Cjt (qjt )−λjt +s1jt = 0 ∀ j = 1, 2, 3, 4, 5; (68)
ρit :
ijt : φijt : ωijt :
max qjt − qjt + s2jt = 0 q (s1jt )2 + (qjt )2 − s1jt − qjt = 0 q (s2jt )2 + (λjt )2 − s2jt − λjt = 0
∀
j = 1, 2, 3, 4, 5;
(69)
∀
j = 1, 2, 3, 4, 5;
(70)
∀
j = 1, 2, 3, 4, 5.
(71)
Using (54), we can rewrite the complementarity constraint in (67) as: q f,M ax f 2 f,M ax f (qit − qit ) + (ρit )2 − (qit − qit ) − ρit = 0.
(72)
f f f f f Define qtf = (q1t , q2t , q3t , q4t , q5t ) ρt = (ρ1t , ρ2t , ρ3t , ρ4t , ρ5t ), ηt = (η11t , η12t , η13t , η14t , η15t , η21t , ..., η55t ),
t = (11t , 12t , 13t , 14t , 15t , 21t , ..., 55t ), ψt = (ψ11t , ψ12t , ψ13t , ψ14t , ψ15t , ψ21t , ..., ψ55t ), and ωt = (ω11t , ω12t , ω13t , ω14t , ω15t , ω21t , ..., ω55t ). The concatenated KKT conditions characterized in (61) - (66), (68) - (71), and (72) represent a “square” constrained nonlinear system where the number of conditions equals the number of endogenous variables (qt , λt , s1t , s2t , qtf , ρt , ηt , t , ψt , ωt ) for each period t. To implement this large-scale EPEC, we 3
use the semi-smooth Levenberg-Marquardt algorithm on the KNITRO solver using the GAMS software. To reduce the likelihood that the equilibrium to this constrained nonlinear system is a local optimum, we use the multistart algorithm to solve this EPEC 50 times for all 8,760 hours in our sample. The best equilibrium solution is returned.
A2
Merger
It is without loss of generality to assume that firm’s 1 and 2 merge. We treat the new merged firm (M ) f as a multi-product firm that chooses quantities (q1t , q2t ) in the spot market and qM t forward contracts in
the first-stage. The second-stage spot market choice for the multi-product merged firm, taking P f and f qM t as given, can be characterized by:
" maximize
q1t ≥0,q2t ≥0
subject to
Pt (Qt )
2 X
# M SG
MR qit + qit + qit
f f M SG M SG − qM ) − C2t (q1t ) − C2t (q2t ) + P f qM t − C1t (q1t ) − C1t (q1t t
i=1 M ax q1t ≤ q1t
: λ1t
M ax q2t ≤ q2t
: λ2t . (73)
Using (73) and introducing slack variables s1it and s2it for both i = 1, 2, the merged firm’s optimal output choice satisfies: " Pt (Qt ) +
Pt0 (Qt )
2 X
# qit +
MR qit
+
M SG qit
−
f qM t
− Cit0 (qit ) − λit + s1it = 0;
(74)
i=1
qit −
max qit
+ s2it = 0;
s1it ≥ 0
⊥
s2it ≥ 0
(75)
qit ≥ 0; ⊥
(76)
λit ≥ 0
∀ i = 1, 2.
(77)
Applying (54) to (76) and (77) allows us to rewrite the spot market MCP in (74) - (77) as: " Pt (Qt ) + Pt0 (Qt )
2 X
# M SG
MR qit + qit + qit
−
f qM t
− Cit0 (qit ) − λit + s1it = 0;
(78)
i=1 max qit − qit + s2it = 0; p (s1it )2 + (qit )2 − s1it − qit = 0; p (s2it )2 + (λit )2 − s2it − λit = 0
(79) (80) ∀ i = 1, 2.
(81)
The second-stage spot market optimum conditions for the non-merging firms i = 3, 4, 5 are analogous to those characterized in (55) - (58). Having characterized the lower-level problem as a MCP for the merged and non-merged firms, we formulate the MPEC for the merged firm and the non-merged firms as a mixed nonlinear program choosing all endogenous variables in the two-stage model. Using (55) - (58) and (78) - (81), the first-stage
4
optimization problem of each non-merging firm i = 3, 4, 5 is represented as:
maximize
f qit ,qt ,λt ,s1t ,s2t
MR M SG M SG Pt (Qt ) qit + qit + qit − Cit (qit ) − Cit (qit ) f f,M ax qit ≤ qit
: ρit ≥ 0 2 X f MR M SG 0 Pt (Qt ) + Pt0 (Qt ) qjt + qjt + qjt − qM t − C1t (q1t ) − λ1t + s11t = 0 : ηi1t
subject to
j=1
2 X f MR M SG 0 Pt (Qt ) + Pt0 (Qt ) qjt + qjt + qjt − qM t − C2t (q2t ) − λ1t + s12t = 0 : ηi2t j=1
h
i f MR M SG 0 Pt (Qt ) + Pt0 (Qt ) qjt + qjt + qjt − qjt − Cjt (qjt ) − λjt + s1jt = 0 ∀
: ηijt
j = 3, 4, 5;
max qjt − qjt + s2jt = 0 q (s1jt )2 + (qjt )2 − s1jt − qjt = 0 q (s2jt )2 + (λjt )2 − s2jt − λjt = 0
: ijt : ψijt : ωijt ∀
j = 1, 2, 3, 4, 5; (82)
where
f qit
≤
f,M ax qit
represents the upper bound constraint on firm i’s forward position.
Using (55) - (58) and (78) - (81), the first-stage optimization problem of the merged firm is represented as: maximize
f qM t ,qt ,λt ,s1t ,s2t
subject to
Pt (Qt ) f qM t
2 X
MR M SG M SG M SG qit + qit + qit − C1t (q1t ) − C1t (q1t ) − C2t (q1t ) − C2t (q2t )
i=1 f,M ax qM t
: ρM t ≥ 0 2 X f MR M SG 0 Pt (Qt ) + Pt0 (Qt ) qjt + qjt + qjt − qM t − C1t (q1t ) − λ1t + s11t = 0 : ηM 1t ≤
j=1
2 X f MR M SG 0 Pt (Qt ) + Pt0 (Qt ) qjt + qjt + qjt − qM t − C2t (q2t ) − λ1t + s12t = 0 : ηM 2t j=1
h i f MR M SG 0 Pt (Qt ) + Pt0 (Qt ) qjt + qjt + qjt − qjt − Cjt (qjt ) − λjt + s1jt = 0 ∀
: ηM jt
j = 3, 4, 5;
max qjt − qjt + s2jt = 0 q (s1jt )2 + (qjt )2 − s1jt − qjt = 0 q (s2jt )2 + (λjt )2 − s2jt − λjt = 0
: M jt : ψM jt : ωM jt ∀
j = 1, 2, 3, 4, 5; (83)
where
f qM t
≤
f,M ax qM t
represents the upper bound constraint on the merged firm’s forward position. 5
Subsequently, the Lagrangian function for a player i = 3, 4, 5 in MPEC (82) and the merged firm M in MPEC (83) is: h i f,M ax f MR M SG M SG Lit = Pt (Qt ) qit + qit + qit − Cit (qit ) − Cit (qit ) + ρit qit − qit
+
2 X
2 X f MR M SG 0 ηijt Pt (Qt ) + Pt0 (Qt ) qjt + qjt + qjt − qM t − Cjt (qjt ) − λjt + s1jt
j=1
+
5 X
j=1
h h i i f MR M SG 0 ηijt Pt (Qt ) + Pt0 (Qt ) qjt + qjt + qjt − qjt − Cjt (qjt ) − λjt + s1jt
j=3
+
5 X
q 5 X max 2 2 (s1jt ) + (qjt ) − s1jt − qjt ijt qjt − qjt + s2jt + ψijt
j=1
+
5 X
j=1
q 2 2 ωijt (s2jt ) + (λjt ) − s2jt − λjt ;
(84)
j=1
LM t = Pt (Qt )
2 X
MR M SG M SG M SG qit + qit + qit − C1t (q1t ) − C1t (q1t ) − C2t (q1t ) − C2t (q2t )
i=1
i h f f,M ax − q + ρM t qM Mt t +
2 X
2 X f MR M SG 0 ηM jt Pt (Qt ) + Pt0 (Qt ) qjt + qjt + qjt − qM t − Cjt (qjt ) − λjt + s1jt
j=1
+
5 X
j=1
h h i i f MR M SG 0 ηM jt Pt (Qt ) + Pt0 (Qt ) qjt + qjt + qjt − qjt − Cjt (qjt ) − λjt + s1jt
j=3
+
5 X
q 5 X max 2 2 (s1jt ) + (qjt ) − s1jt − qjt M jt qjt − qjt + s2jt + ψijt
j=1
+
5 X
j=1
q 2 2 (s2jt ) + (λjt ) − s2jt − λjt . ωM jt
(85)
j=1
We use (84) and (85) to characterize the KKT conditions for the merged and non-merged firms. The Nash equilibrium of the two-stage multi-firm forward-spot market model represented by the EPEC can be obtained by concatenating all of the KKT conditions for each player’s MPEC. Using (84) and (85), for each period t, the aggregated mixed nonlinear complementarity program is represented by the following conditions: f qM t :
−
2 X
ηM it Pt0 (Qt ) − ρM t = 0;
(86)
i=1
6
f qit :
q1t :
− ηiit Pt0 (Qt ) − ρit = 0 " 2 # X 0 MR M SG 0 Pt (Qt ) qit + qit + qit + Pt (Qt ) − C1t (q1t )
∀
i = 3, 4, 5;
(87)
i=1
" + ηM 1t 2 Pt0 (Qt ) + Pt00 (Qt )
" 2 X
# M SG
MR qit + qit + qit
#
f 00 − qM t − C1t (qit )
i=1
"
"
+ ηM 2t 2 Pt0 (Qt ) + Pt00 (Qt )
2 X
## M SG
MR qit + qit + qit
f − qM t
i=1
+
5 X
h h ii f MR M SG ηM jt Pt0 (Qt ) + Pt00 (Qt ) qjt + qjt + qjt − qjt
j=3
i h − 1 + M 1t + ψM 1t q1t (s11t )2 + (q1t )2 2 − 1 = 0 and; MR M SG Pt0 (Qt ) qit + qit + qit " + ηi1t 2 Pt0 (Qt ) + Pt00 (Qt )
" 2 X
# M SG
MR qit + qit + qit
−
f qM t
−
f qM t
(88)
# 00 − C1t (q1t )
i=1
"
"
2 Pt0 (Qt )
+ ηi2t
+
Pt00 (Qt )
2 X
## qit +
MR qit
+
M SG qit
i=1
+
5 X
h h ii f MR M SG ηijt Pt0 (Qt ) + Pt00 (Qt ) qjt + qjt + qjt − qjt
j=3
i h − 1 + i1t + ψi1t q1t (s11t )2 + (q1t )2 2 − 1 = 0 " q2t :
Pt0 (Qt )
2 X
∀ i = 3, 4, 5;
(89)
# MR M SG 0 qit + qit + qit + Pt (Qt ) − C2t (q1t )
i=1
" + ηM 1t
2 Pt0 (Qt )
+
Pt00 (Qt )
" 2 X
## qit +
MR qit
+
M SG qit
−
f qM t
i=1
" + ηM 2t 2 Pt0 (Qt ) + Pt00 (Qt )
" 2 X
# M SG
MR qit + qit + qit
#
f 00 − qM t − C2t (qit )
i=1
+
5 X
h h ii f MR M SG ηM jt Pt0 (Qt ) + Pt00 (Qt ) qjt + qjt + qjt − qjt
j=3
h i − 1 + M 2t + ψM 2t q2t (s12t )2 + (q2t )2 2 − 1 = 0 and; MR M SG Pt0 (Qt ) qit + qit + qit " + ηi1t 2 Pt0 (Qt ) + Pt00 (Qt )
"
2 X i=1
7
## M SG
MR qit + qit + qit
f − qM t
(90)
" + ηi2t 2 Pt0 (Qt ) + Pt00 (Qt )
" 2 X
# M SG
MR qit + qit + qit
−
f qM t
# 00 − C2t (q2t )
i=1
+
5 X
h h ii f MR M SG ηijt Pt0 (Qt ) + Pt00 (Qt ) qjt + qjt + qjt − qjt
j=3
i h − 1 + i2t + ψi2t q2t (s12t )2 + (q2t )2 2 − 1 = 0 qkt :
Pt0 (Qt )
" 2 X
∀ i = 3, 4, 5;
(91)
# MR M SG qit + qit + qit
i=1 2 X
+
" ηM jt
Pt0 (Qt )
+
Pt00 (Qt )
" 2 X
j=1
## qit +
MR qit
+
M SG qit
−
f qM t
i=1
h h i i f MR M SG 00 + ηM kt 2Pt0 (Qt ) + Pt00 (Qt ) qkt + qkt + qkt − qkt − Ckt (qkt ) +
5 X
h h ii f MR M SG + qjt − qjt ηM jt Pt0 (Qt ) + Pt00 (Qt ) qjt + qjt
j=3 j6=k
i h − 1 + M kt + ψM kt qkt (s1kt )2 + (qkt )2 2 − 1 = 0
∀ k = 3, 4, 5 and;
(92)
MR M SG Pt0 (Qt ) qit + qit + qit " " 2 ## 2 X X f MR M SG + ηijt Pt0 (Qt ) + Pt00 (Qt ) qit + qit + qit − qM t j=1
i=1
i i f 00 M SG MR (qkt ) − Ckt − qkt + qkt + ηikt 2Pt0 (Qt ) + Pt00 (Qt ) qkt + qkt h
+
5 X
h
h h ii f MR M SG ηijt Pt0 (Qt ) + Pt00 (Qt ) qjt + qjt + qjt − qjt
j=3 j6=k
h i − 1 + ikt + ψikt qkt (s1kt )2 + (qkt )2 2 − 1 = 0 qit :
∀ i, k = 3, 4, 5 with i 6= k;
(93)
MR M SG Pt0 (Qt ) qit + qit + qit + Pt (Qt ) − Cit0 (qit ) h h i i f MR M SG + ηiit 2 Pt0 (Qt ) + Pt00 (Qt ) qit + qit + qit − qit − Cit00 (qit ) +
5 X
h h ii f MR M SG ηijt Pt0 (Qt ) + Pt00 (Qt ) qjt + qjt + qjt − qjt
j=3 j6=i
+
2 X
ηijt
2 X f MR M SG Pt0 (Qt ) + Pt00 (Qt ) qjt + qjt + qjt − qM t
j=1
j=1
h i − 1 + iit + ψiit qit (s1it )2 + (qit )2 2 − 1 = 0
8
∀ i = 3, 4, 5;
(94)
λjt :
h i − 1 − ηM jt + ωM jt λjt (s2jt )2 + (λjt )2 2 − 1 = 0
λkt :
h i − 1 − ηikt + ωikt λkt (s2kt )2 + (λkt )2 2 − 1 = 0
s1jt :
i h − 1 ηM jt + ψM jt s1jt (s1jt )2 + (qjt )2 2 − 1 = 0
s1kt :
h i − 1 ηikt + ψikt s1kt (s1kt )2 + (qkt )2 2 − 1 = 0
s2jt :
h i − 1 M jt + ωM jt s2jt (s2jt )2 + (λjt )2 2 − 1 = 0
s2kt :
i h − 1 ikt + ωikt s2kt (s2kt )2 + (λkt )2 2 − 1 = 0
ρit :
f f,M ax qit ≤ qit
⊥ "
ηM jt :
Pt (Qt )+Pt0 (Qt )
2 X
∀ ∀
ρit ≥ 0
j = 1, 2, 3, 4, 5;
(97)
i = 3, 4, 5 and k = 1, 2, 3, 4, 5; (98) ∀
∀
(95)
i = 3, 4, 5 and k = 1, 2, 3, 4, 5; (96) ∀
∀
j = 1, 2, 3, 4, 5;
j = 1, 2, 3, 4, 5;
(99)
i = 3, 4, 5 and k = 1, 2, 3, 4, 5; ∀
i = M, 3, 4, 5;
(100) (101)
# f 0 MR M SG q1t + qit + qit − qM t −Cjt (qjt )−λjt +s1jt = 0
∀
j = 1, 2;
i=1
(102) ηijt : ijt : φijt : ωijt :
Pt (Qt )+Pt0 (Qt )
h i f MR M SG 0 qjt + qjt + qjt − qjt −Cjt (qjt )−λjt +s1jt = 0
max qjt − qjt + s2jt = 0 q (s1jt )2 + (qjt )2 − s1jt − qjt = 0 q (s2jt )2 + (λjt )2 − s2jt − λjt = 0
∀
j = 3, 4, 5;
(103)
∀
j = 1, 2, 3, 4, 5;
(104)
∀
j = 1, 2, 3, 4, 5;
(105)
∀
j = 1, 2, 3, 4, 5.
(106)
Using (54), we can rewrite the complementarity constraint in (101) as: q f,M ax f 2 f,M ax f (qit − qit ) + (ρit )2 − (qit − qit ) − ρit = 0.
(107)
The concatenated KKT conditions characterized in (86) - (100), (102) - (107) represent a “square” constrained nonlinear system where the number of conditions equals the number of endogenous variables for each period t. To implement this large-scale EPEC, we use the semi-smooth Levenberg-Marquardt algorithm on the KNITRO solver using the GAMS software. To reduce the likelihood that the equilibrium to this constrained nonlinear system is a local optimum, we use the multistart algorithm to solve this EPEC 50 times for all 8,760 hours in our sample. The best equilibrium solution is returned. This approach is carried out for all possible two firm mergers.
9
