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
i
¯ Proof of Lemma 2. It follows from Min
hy ui[Nja that
j
¯ ¯
P0, y 5 a 2
O
y 0,
S D
2S j [
⁄ S
for any coalition S [1. That is, any nonempty complementary coalition N\S can drive
˜ the price down to zero or a negative level. By 12, 14 and 15, vS 50 for all S ±N.
This leads to the convexity of 14. Q.E.D
J . Zhao Mathematical Social Sciences 37 1999 189 –204
197
˜
Proof of Lemma 3. Part a is proved by replacing Y by Y in the profit-maximization
S S
problems of 5 and 7. For Part b, notice that a coalition’s optimal unconstrained total supply in both the a - and b -fashions is
1
j
¯ ]
¯ X 5
a 2
O
y 2 c ,
S D
S S
2
j [ ⁄ S
i i
¯ ¯
¯ where c 5Min
hc ui[Sj. Its capacity is sufficiently large if and only if X y 5o y ,
S S
S i [S
that is, 1
j i
] ¯
¯ a 2
O
y 2 c
O
y ,
S D
S
2
j [ ⁄ S
i [S
which is equivalent to
i i
¯ ¯
a 2
O
y c 1
O
y .
S i [N
i [S
The above expression holds for all S if and only if
i i
i
¯ ¯
a 2
O
y Min c 1 y ui [ N .
h j
i [N
For Part b, notice that a coalition does not shut down production if and only if price is always above the average variable cost, which equals its marginal cost. This is
equivalent to
j i
i
¯ ¯
¯ ¯
P0, y 5 a 2
O
y c or a 2
O
y c 2
O
y .
S D
2S S
S j [
⁄ S i [N
i [S
The above expression holds for all S if and only if
i i
i
¯ ¯
a 2
O
y Max c 2 y ui [ N . Q.E.D.
h j
i [N
Proof of Lemma 4. The claim is proved by observing that a coalition’s final supply in both the a - and b -core fashions is equal to
1
j
] ¯
X 5 a 2
O
y 2 c . Q.E.D.
S D
S S
2
j [ ⁄ S
˜ ˜
˜
Proof of Theorem 1. We need to show vS 1vT vS T for S T 55. Let
˜ ˜
˜ g 4 5 vS T 2 vS 1 vT ,
i
we need to show g 0. We first prove g 0 in the special case of c 5c for all i. By 15,
2 2
2
¯ ¯
¯ gu 5 u 2 y
2 u 2 y 2 u 2 y
,
2S T 2S
2T j
¯ ¯
where u 5a 2c, and y 5
o y . Note that g is a ‘‘U’’ shaped function whose
2S j [
⁄ S
i 2
2
¯ ¯
maximum is u 5 y 5 o
y because dg du 5g05 22,0, and the solution of g950
j [N
198 J
. Zhao Mathematical Social Sciences 37 1999 189 –204
¯ ¯
¯ is u 5 y. Without loss of generality, assume y y . Under Assumptions 1, 2 and 3, we
S T
have
j
¯ ¯
0 , X 5 a 2
O
y 2 c
Y
2 y ,
S D
S S
j [ ⁄ S
which is equivalent to ¯
¯ ¯
¯ y 2 y u y 1 y .
S S
Since g is concave, and
2 2
¯ ¯
¯ ¯
¯ gy 2 y 5 y 2 y 2 y . 0,
S T
T S
2 2
2
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯ ¯
gy 1 y 5 2y 1 y 2 2y 2 y 1 y 5 y [2y 2 y ] . 0,
S S
T S
S T
S T
S
¯ ¯
¯ ¯
we have gu .0 for all u [[y 2 y , y 1 y ]. Thus,
S S
˜ ˜
˜ v
S 1 vT vS T holds for S T 55 in industries with identical marginal cost.
Now consider nonidentical marginal costs. Note first that the above inequality holds if
j
all firms in S T have the same lowest marginal cost c 5c 5Min
hc u j[STj. Since
S T
˜ ˜
vS is a decreasing function of the marginal costs, the right-hand side vS T remains unchanged and the left-hand side terms both decrease when a member j’s marginal cost
j
is raised from c 5c to the true value c . By repeating this for all members, we see
S T
˜ ˜
˜ that vS 1vT vS T holds. This competes the proof. Q.E.D
Before we prove Theorems 2 and 3, we need to establish Claims 1 and 3. Claim 2 provides a direct proof for the special case of Theorem 2 with identical costs without
using the notations designed for Theorem 3. ¯
¯
Claim 1. For any coalitions S ,T, S ±T, let X and X be the optimal choices of S and T
S T
respectively. Then ¯
¯ a X 50
⇒ X 50;
T S
¯ ¯
¯ b X .0
⇒ X .X .0;
S T
S j
1
¯ ¯
¯
]
c X 5X 1 o
y 1c 2c ;
T S
j [T \S S
T 2
˜ ˜
d The new game G 5 hN, vj given by 14 is monotonic.
Part a says if the optimal choice of a larger coalition T is to shut down production, then so does the subcoalition S. Part b says if a coalition S produces a positive amount
of outputs and this coalition is enlarged, then the enlarged coalition shall also produce a positive larger amount.
¯
Proof of Claim 1. For Part a, X 50 requires
T j
¯ ¯
P0, y 5 a 2
O
y c .
S D
2T T
j [ ⁄ T
J . Zhao Mathematical Social Sciences 37 1999 189 –204
199
It follows from S ,T and c c that
S T
j j
¯ ¯
¯ P0, y
5 a 2
O
y a 2
O
y c c .
S D S
D
2S T
S j [
⁄ S j [
⁄ T
¯ Thus X 50.
S
Part b follows directly from c and c c . Part c can be obtained by rearranging
S T
¯ the formula for X .
T
˜ ˜
The game 14 is monotonic if vS vT for any S ,T. Part d follows directly from ˜
Parts a, b, and the formula for vS given by 15. Q.E.D
i j
Claim 2. Under Assumptions 1–3, 14 is convex if c 5c for all i ±j.
2
1
2 j
¯ ˜
] ¯
Proof of Claim 2. By 12 and 15,vS 5 X 5 a 2
o y 2 c for any S [1. We
S D
S S
4
j [ ⁄ S
˜ ˜
shall show that the inequality 2 holds in the game G 5 hN, v. That is, we shall show
that for any coalitions S ,T ,N, and firm i [N\T,
˜ ˜
˜ ˜
v S i 2 vS vT i 2 vT .
˜ It follows from the above formula for vS , Part c of Claim 1, and symmetrical
marginal costs that
2 2
¯ ¯
¯ ¯
¯ ¯
˜ ˜
v S i 2 vS 5 X
2 X 5 X 1 X X
2 X
S i S
S i S
S i S
h j h j
h j
1 1
i i
¯ ] ¯
] ¯
F G
5 2X 1 y 1 c 2 c
y 1 c 2 c
S S
S i S
S i h j
h j
2 2
1 1
i i
¯ ]
] ¯ ¯
S D
5 2X 1
y y .
S
2 2
Similarly, 1
1
2 2
i i
¯ ¯
¯ ˜
˜ ]
] ¯ ¯
S D
v T i 2 vT 5 X
2 X 5 2X 1
y y .
T i T
T h j
2 2
¯ ¯
The above two expressions and the fact that X X lead to
T S
˜ ˜
˜ ˜
v S i 2 vS vT i 2 vT .
Thus the TU game 14 is convex. Q.E.D
Proof of Theorem 2. It follows from Assumptions 2 and 3, 12 and 15 that
2
1
2 j
¯ ˜
] ¯
v S 5 X 5
a 2
O
y 2 c
S D
S S
4
j [ ⁄ S
for any coalition S [1. Similarly as in the proof of Claim 2, we shall show that for any coalitions S ,T ,N, and any firm i [N\T,
˜ ˜
˜ ˜
v S i 2 vS vT i 2 vT .
200 J
. Zhao Mathematical Social Sciences 37 1999 189 –204
This will lead to the convexity of 14. As already shown in the proof of Claim 2, we have
1 1
i i
¯ ˜
˜ ]
] ¯ ¯
F G
v S i 2 vS 5
2X 1 y 1 c 2 c
y 1 c 2 c
S S
S i S
S i h j
h j
2 2
1 1
i i
¯ ˜
˜ ]
] ¯ ¯
F G
v T i 2 vT 5
2X 1 y 1 c 2 c
y 1 c 2 c .
T T
T i T
T i h j
h j
2 2
It follows from V 55 that S, T, i [ ⁄ V. By Part i of Claim 3, we have
c 2 c 5 c 2 c
.
S S i
T T i
h j h j
¯ ¯
The previous three expressions and the fact that X X lead to
T S
˜ ˜
˜ ˜
v S i 2 vS vT i 2 vT .
This finishes the proof. Q.E.D
j
Claim 3. Let c 5Min
hc u j[Sj denote the marginal cost of the most efficient firm in a
S
coalition S. Then for any coalitions S ,T ,N, and any firm i [N\T, the following three inequalities hold:
i c 2c c 2c
0;
S S
hi j T
T hi j
2 2
2 2
ii c 2c 2c 2c
c 1c [c 2c
2c 2c ];
S S
hi j T
T hi j
T T
hi j S
S hi j
T T
hi j 2
2 2
2
iii c 2c 2c 2c
0.
S S
hi j T
T hi j
Proof of Claim 3. Part iii follows from i and ii. We first prove part i. If S 55,
j 7
then c 5Min hc u j[5j51`, and Part i holds. Thus, we only need to prove it for
S i
S ±5. Given this, only the following three possible cases can happen. Case 1. c ,c
T i
i i
c . Case 2. c c c . Case 3. c c ,c . In case 1, c 5c
5c , thus i follows
S T
S T
S S
hi j T
hi j
from c c ; In case 2, c 5c , and c 5c
, thus c 2c 5c 2c
50, and
T S
S S
hi j T
T hi j
S S
hi j T
T hi j
i
i holds; In case 3, c 2c 50, and c 2c
5c 2c .0, thus c 2c .c 2
T T
hi j S
S hi j
S S
S hi j
T
c 50.
T hi j
2 2
2 2
Part ii follows from the equation c 2 c 2 c 2 c
5 c 1 c c 2
S S i
T T i
S S i
S h j
h j h j
c 2 c 1 c
c 2 c and the facts that c 1c
c 1c , and c 2
S i T
T i T
T i S
S hi j
T T
hi j S
h j h j
h j
c 0. Q.E.D
S hi j
Proof of Lemma 5. There is nothing to prove if V 55. Now suppose
V 5 S, T, i uS , T , N, i [ NT, and c 2 c
. c c ±
5.
h j
S S i
T S i h j
h j
For any given S, T, i [V,
7
By Lemma I and Theorem 1, 14 is convex if 2 holds for S ±5, because 14 is superadditive. Thus, Theorem 3 can alternatively be proved by using Theorem 1 and by only using the properties of Claim 3 for
S ±5.
J . Zhao Mathematical Social Sciences 37 1999 189 –204
201
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
F G
S S i
T T i
h j h j
j [ ⁄ S, j ±i
j [ ⁄ T, j ±i
2 2
2 2
i j
¯ ¯
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
1
O
y c 2 c 2
O
y c 2 c
HF G
J
S S i
T T i
h j h j
j [
⁄ T, j ±i
j [T S j [
⁄ T, j ±i
2 2
2 2
j
¯ 5 c 2 c
2 c 2 c 1 2
O
y [c 2 c 2 c 2 c
]
S S i
T T i
S S i
T T i
h j h j
h j h j
j [ ⁄ T, j ±i
j i
i
¯ ¯
¯ 1 2
O
y [y 1 c 2 c ] 1 2y c 2 C .
S S i
S T
h j j [T S
Clearly,
i
¯y 1 c 2 c c 2 c
2 c 2 c ;
S S i
S S i
T T i
h j h j
h j
and c 2 c 5 c 2 c
1 c 2 c c 2 c
1 c 2 c
S T
S S i
S i T
S S i
T i T
h j h j
h j h j
5 c 2 c 2 c 2 c
.
S S i
T T i
h j h j
It follows from the above two expressions and Part ii of Claim 3 that fS, T, i c 1 c
[c 2 c 2 c 2 c
]
T T i
S S i
T T i
h j h j
h j j
j
¯ ¯
1 2
O
y [c 2 c 2 c 2 c
] 1 2
O
y [c 2 c
S S i
T T i
S S i
h j h j
h j j [
⁄ T, j ±i
j [T S i
¯ 2 c 2 c
] 1 2y [c 2 c 2 c 2 c
]
T T i
S S i
T T i
h j h j
h j j
¯ 5 [c 1 c
1 2
O
y ][c 2 c 2 c 2 c
].
T T i
S S i
T T i
h j h j
h j j [
⁄ S
This leads to fS, T, i
j
]]]]]]]] ¯
FS, T, i 5 c 1 c
1 2
O
y
Y
2
F G
T T i
h j
2[c 2 c 2 c 2 c
]
S S i
T T i
j [ ⁄ S
h j h j
j
¯ 5 c 1 c
2 1
O
y .
T T i
h j j [
⁄ S
Because the number of elements in V is finite, we have
202 J
. Zhao Mathematical Social Sciences 37 1999 189 –204
j
¯ v 5 Min FS, T, i
uS, T, i [ V Min c 1 c 2 1
O
y uS, T, i [ V
h j
H J
T T i
h j j [
⁄ S
. 0. Q.E.D The above results and claims lead directly to a proof for the main theorem.
Proof of Theorem 3. Let S, T, i [V, then c 2c
.c 2c . Substituting
S S
hi j T
T hi j
2
1
2 j
¯ ˜
] ¯
v S 5 X 5
a 2
O
y 2 c
S D
S S
4
j [ ⁄ S
˜ ˜
˜ ˜
into the expressions for vS i -vS -vT i -vT , we have
2 j
˜ ˜
˜ ˜
¯ 4[vS i 2 vS 2 vT i 2 vT ] 5
a 2
O
y 2 c
HS D
S i h j
j [ ⁄ S, j ±i
2 2
2 j
j j
¯ ¯
¯ 2 a 2
O
y 2 c 2
a 2
O
y 2 c 2 a 2
O
y 2 c
S D J HS
D S D J
S T i
T h j
j [ ⁄ S
j [ ⁄ T, j ±i
j [ ⁄ T
5 2a[c 2 c 2 c 2 c
] 2 fS, T, i
h j
S S i
T T i
h j h j
5 2[c 2 c 2 c 2 c
] a 2 FS, T, i , h
j
S S i
T T i
h j h j
where fS, T, i and FS, T, i are given in 17 and 18. It follows from c 2 c
. c 2 c
S S i
T T i
h j h j
and the above expression that ˜
˜ ˜
˜ [vS i 2 vS 2 vT i 2 vT ] 0
⇔ a FS, T, i
⇔ a v
5 Min FS, T, i uS, T, i [ V .
h j
Thus the TU game 14 is convex if and only if ‘‘a v.’’ Q.E.D
5. Extensions and concluding remarks
