192 J
. Zhao Mathematical Social Sciences 37 1999 189 –204
˜ where x is the maximum solution. This converts the oligopoly game to a coalition
function form game in the a -core fashion as follows: G 5 N, v ? ,
6
h j
a a
and any core allocation of G is the a -core profit allocation for the original market.
a
In contrast, the b -core is defined by computing the reaction function of S, x z ,
S 2S
and its reaction profit function, p z , which are respectively the maximal solution
S 2S
function and the extreme value function in
i i
p z 5Max
O
p x , z 5
O
p x z , z
. 7
S 2S
S 2S
S 2S
2S x [Y
S S
i [S i [S
For each S [1, let
i
ˆ ˆ
ˆ v
S 5 Min p z uz
[ Y
5 p z
5
O
p x z , z
, 8
h j
b S
2S 2S
2S S
2S S
2S 2S
i [S
ˆ where z
is the minimum solution. This converts the oligopoly game to a coalition
2S
function form game in the b -core fashion: G 5 N, v ? .
9
h j
b b
It follows from v N 5v N and v S v S for S ±N that CoG CoG . Note
a b
a b
b a
that deriving the TU games 6 and 9 only requires the continuity of demand and cost functions, because all firms have finite capacities.
We are now ready to study the convexity of G and G .
a b
3. Main results
We shall focus our attention on linear industries, where the inverse demand function and cost functions become
i i
i i i
i i
i
¯ PX 5 a 2 bX, and C x 5 d 1 c x , x [ Y 5 [0, y ],
i i
where a .0, b .0; d 0 is i’s fixed cost, and c 0 is its marginal cost. The standard
i i
¯ symmetric Cournot industry is a special case in which d 50, c 5c, y 5 1` for all i.
Without loss of generality, we assume the negative slope of the inverse demand is 1 or b 51 Otherwise, simply scale up or down all the parameters.
1 n
1 n
1 n
¯ ¯
¯ Let y 5y , . . . y , c 5c , . . . ,c , and d 5d , . . . ,d denote respectively the
vectors of capacities, marginal and fixed costs, then a linear industry is defined by a
3n 11
¯ vector
ha, c, d, yj[R
. We first state two sufficient conditions on the parameter ha, c,
1
¯ d
, y j for convexity Lemma 2 and Theorem 2 and then establish the necessary and
sufficient condition for convexity Theorem 3.
3n 11 i
¯ ¯
Lemma 2. Consider any ha,c,d,y [R
. If Min hy ui[Nja, then the oligopoly TU
1
games G and G are both convex.
a b
J . Zhao Mathematical Social Sciences 37 1999 189 –204
193
i
¯ The lemma is proved in Section 4. Though the condition Min
hy ui[Nja is a strong assumption, it includes the standard symmetric or asymmetric Cournot industries as
special cases. As readers shall see, our main results are much more general than Lemma 2, which are obtained by using the following three assumptions:
Assumption 1. Weak Synergy: For each coalition S, its marginal cost and capacity are
i i
¯ ¯
respectively c 5Min hc ui[Sj and y 5o
y .
S S
i [S
Assumption 2. Capacity sufficiency: For each coalition S, its optimal choices in the a - and b -core fashions are both bounded by its capacity.
Assumption 3. No shut-down price: For each coalition S, its average variable cost is always less than or equal to the price.
Assumption 1 assumes that a coalition’s most efficient technology can be costlessly adopted by all firms in S. Thus, the contributions of all nonefficient members in S are
their capacities, and the contributions of the most efficient member are its capacity and its ability to reduce other member’s marginal cost. Assumption 2 assumes that a
coalition always has sufficiently large capacity. Assumption 3 rules out the possibility that some coalition might be forced to shut down. These can alternatively be given as the
next lemma:
Lemma 3. a Assumption 1 holds if and only if each coalition’s production set is equal
to
S i
i
˜ ¯
Y 5 x [ R
u
O
x
O
y ;
H J
S S
1 i [S
i [S
b Assumption 2 holds if and only if
i i
i
¯ ¯
a 2
O
y Min c 1 y ui [ N ;
h j
i [N
b Assumption 3 holds if and only if
i i
i
¯ ¯
Max c 2 y ui [ N a 2
O
y .
h j
i [N i
¯ Remark 1. In symmetrical case c 5c, y 5y, all i , Assumptions
1 –3 become a 2 c n 1 1 y a 2 c 2 [2 n 2 1],
h j
where a –c n 11 is a firm ’s Cournot supply, and a–c 2 is the monopoly supply. In
other words , a firm’s capacity is assumed to be above its Cournot equilibrium supply,
but bounded from above by 2 n 21 of the monopoly supply . Thus, under these
assumptions , capacities are assumed to be just right, not too large and not too small.
For each S [1, let
194 J
. Zhao Mathematical Social Sciences 37 1999 189 –204
i
c 5 Min c ui [ S
10
h j
S
denote the marginal cost of the most efficient firm in S, and
j
¯ ¯
P0, y 5 a 2
O
y 11
S D
2S j [
⁄ S
¯ denote the price when the outsiders chose y
and the coalition produces zero. Given the
2S
¯ outside choice y
, a coalition’s unconstrained optimal total supply is
2S
¯ if P0, y
, c .
2S S
¯ 1
X 5 12
j S
] ¯
¯ a 2
O
y 2 c if P0, y
c .
5 6
S D
S 2S
S
2
j [ ⁄ S
i
¯ Plugging 12 into
o p x , y
at y 5y , we obtain the equivalence between the
i [S S
2S 2S
2S
a- and b-cores:
Lemma 4. Under Assumptions 1 –3, a coalition’s profits in the a- and b-core fashions
are equal and given by:
2
1
j i
] ¯
v S 5 v S 5 v S 5
a 2
O
y 2 c 2
O
d . 13
S D
a b
S
4
j [ ⁄ S
i [S
For future developments, it is useful to introduce a new TU game ˜
˜ G 5
hN, vj 14
˜ by removing the fixed costs in 13. That is, for each S in the game G, its payoff is
2
1
i j
˜ ]
¯ v
S 5 vS 1
O
d 5 a 2
O
y 2 c .
15
S D
S
4
i [S j [
⁄ S
Since 14 is obtained from 13 by adding an additive game, the convexity is unharmed. Therefore, we only need to study the game 14. A game G 5
hN, vj is superadditive if for any S T 55, vS 1vT vS T . We first show that the oligopoly
game 14 is superadditive.
Theorem 1. Under Assumptions
1, 2 and 3, the oligopoly TU game 14 satisfies ˜
˜ ˜
v S 1 vT vS T
for all S, T [1 and S T 50. In order to define our key conditions, we introduce the following notations and
definitions given by 16–19. Let V 5 S, T, i
uS , T , N, i [ NT and c 2 c . c 2 c
16
h j
S S i
T T i
h j h j
denote those firms and the associated coalitions whose marginal cost exhibits strict
J . Zhao Mathematical Social Sciences 37 1999 189 –204
195
6
supermodularity. This can alternatively be interpreted as those firm whose ability of reducing a coalition’s marginal cost decreases as the coalition expands i.e. c 2c
is
S S
hi j
reduced to c 2c when S is enlarged to T . Thus, the elements in V are the potential
T T
hi j
factors that might destroy the convexity of 14, because marginal cost enters vS as negative terms. As shown in the next theorem, the game 14 is convex when there are
no such potential damaging factors V 55.
Theorem 2. Under Assumptions 1, 2 and 3, the oligopoly TU game 14 of a linear
industry is convex if V 55.
i j
Remark 2. A special case of V 55 is c 5c for all i, j all firms have identical
marginal costs. Because this case allows firms to have different fixed costs and different capacities
, and allows a coalition to make positive profits, it is more general than the condition of Lemma 2. This special case can be established without using the complex
notations designed for the general condition , and this is provided as Claim 2 in Section
4. Now assume V ±5. For any S, T, i [V, let
2 2
2 2
i j
¯ ¯
fS, T, i 5 c 2 c 2 c 2 c
1 2y
O
y 1 c 2 c
S D
S S i
T T i
S T
h j h j
j [T S j
j
¯ ¯
1 2
O
y c 2 c 2
O
y c 2 c ,
17
F G
S S i
T T i
h j h j
j [ ⁄ S, j ±i
j [ ⁄ T, j ±i
fS, T, i ]]]]]]]]
FS, T, i 5 ,
18 2[c 2 c
2 c 2 c ]
S S i
T T i
h j h j
v 5 Min FS, T, i uS, T, i [ V .
19
h j
The next lemma shows that the above Co-value is positive.
Lemma 5. If Assumptions 1 –3 hold, then v .0.
The above v -value turns out to be critical in the convexity of oligopoly games:
Theorem 3. Under Assumptions 1, 2 and 3, the oligopoly TU game G 5G in a linear
a b
industry with V ±5 is convex if and only if a v. In other words, in an industry with V ±5 and with no firm too small nor too large, the
oligopoly TU game is convex if and only if the intercept of the inverse demand function
6
As shown in Part i of Claim 3 in Section 4, a coalition’s marginal cost exhibits supermodularity i.e. c
2c c 2c for S ,T and i [
⁄ T . Thus, an element S, T, i in V implies that their marginal cost
S hi j
S T
hi j T
exhibits strict supermodularity or c 2c ,c
2c for S ,T and i [
⁄ T.
S hi j
s T
hi j T
196 J
. Zhao Mathematical Social Sciences 37 1999 189 –204
is less than or equal to the v -value of 19. Since marginal costs and capacities are both asymmetric, the condition ‘‘a v ’’ fully characterizes the convexity of linear industries.
Remark 3. In Theorem 3, the value v can be understood as an index or measurement
for scale economies of or increasing returns to coalition size . If there is not enough
increasing returns to coalition size, v will fall below the intercept a , therefore the game
will be nonconvex . If there are enough increasing returns to coalition size, v will be
above the intercept a , therefore the game will be convex. The case of V 55 can also be
interpreted this way , because the minimum over an empty set is defined as 1` and thus
satisfies a v. Theorem 3 is illustrated in the following example, which are computed by using the
¯ data
ha, c, d, yj. It first shows if ‘‘av’’ fails to hold, the game is not convex. Then it shows that the game becomes convex when the condition ‘‘a v ’’ holds.
Example. Consider a general linear industry with three firms. The inverse demand function and the cost functions are respectively:
1 2
3 1
1 1
1 2
2 2
2
P 5 7 2 x 1 x 1 x ; C x 5 4x , x [ [0, 1.3]; C x 5 2.25x , x
3 3
3 3
[ [0, 1.3]; and C x 5 2.25x , x [ [0, 1.3].
Let v1, 25v 1, 25v 1, 2 denote the profit for S 5 h1, 2j as defined in 15. Using
a b
similar notations, the profits for each coalition are: v150.04, v25v351.1556, v1, 25v1, 35v2, 352.9756, v1, 2, 355.6406. For S 5
h1, 2j, T5h1, 3j, v
S 1 vT 5 5.9512 . vS T 1 vS T 5 5.6806, the game is therefore not convex. This is so because a 57.v 56.6907, which violates
the condition ‘‘a v.’’ Now let the intercept be decreased to a 56.65, and all other parameters remain the
same this will not change the value v, the payoffs become: v150.0006, v25 v
350.81, v1, 25v1, 35v2, 352.4026, v1, 2, 354.84. For any S and T, it can be verified that vS 1vT vS T 1vS T . The game becomes convex now,
because ‘‘a v ’’ is satisfied: a 56.65,v 56.6907.
4. Proofs
