Appendix B Subgame Perfect Equilibria

Si; if 0 Si< ¹₂(1 qj + Tj);

0; if 0 S_i< ₂¹( 1 + + q_j T_j);

1 qj+2 Si + Tj

2+2 ; if maxf¹₂(1 q_j+ T_j);₂¹( 1 + + q_j T_j)g S_i 1 ¹₂S_j;

1 2

1

2qj; if 1 ¹₂Sj < Si:

6.1 The reduced best response correspondence

Table 4 now follows immediately.

Appendix B Subgame Perfect Equilibria

We de…ne the sets A_j(1); : : : ; A_j(4); B_j(1); : : : ; B_j(7); C_j(1); : : : ; C_j(4) as the sets of quantities qj satisfying the constraints as presented in Table 4. Notice that each of these sets is a subset of [0; 1=2]: Moreover, we de…ne Aj(k₁; : : : ; k_{`</sub>) = Aj(k<sub>1</sub>) [ [ A<sub>j</sub>(k<sub>`}); and similarly for sets B_j(k₁; : : : ; k_{`</sub>) and C<sub>j</sub>(k<sub>1</sub>; : : : ; k<sub>`}): In the proofs we will make use of Table 4. That table presents the reduced best response of …rm i to a …rst-period sales quantity of …rm j with the use of coe¢ cients 1; : : : ; 8: In the sequel we will need the reduced best response of …rm j to a …rst-period sales quantity of …rm i;

which follows from Table 4 by reversing the roles of …rm i and j: The corresponding coe¢ cients are denoted by ₁; : : : ; ₈:

Proposition B.1 If (q_i; q_j) is a Nash equilibrium of the reduced game and q_j 2 A_j(1; 2; 3) [ Bj(1; 2; 3; 4; 5; 6) [ Cj(1), then S_i q_i ¹₃; so q_i 2 Ci(1; 2; 3; 4):

Proof. For q_j 2 A^j(1; 3) [ B^j(1; 6) [ C^j(1) it follows immediately from Table 4 that S_i q_i ¹₃. For q_j 2 Aj(2) [ Bj(4);

Si q_i = 2S_i 1 + q_j +¹₂ 2 +

2 3 +¹₃

2 + = 1 3;

where the inequality follows from q_{j 2}: For q_j 2 Bj(2);

Si q_i < 3 4

1 4

1

2Sj < 1 3;

A_j (q_j < S_j ¹₂) q_i r_i

where the …rst inequality follows from 5 < q_j ¹₂ and the second one from 1 and S_j > q_j +¹₃ ¹₃: For q_j 2 Bj(3),

Si q_i = 2Si 1 + q_j + Tj

2(1 + )

(1 + + q

(1 + )(1 +¹₂ ))(1 2Tj) 2(1 + )

1 + + q

(1 + )(1 +¹₂ ) 6(1 + )

1 3;

where the …rst inequality follows from q_{j 6} (i.e. S_i S_i^c), the second from

1

3 < Tj 1

2 and the third one from 2 (0; 1]. For qj 2 B^j(5);

S_i q_i = 2Si 1 + q_j + Tj

2(1 + ) 1 2T_j 1

3;

where the …rst inequality follows from q_{j 7}, i.e. Si S^ll_i; and the second one from

1

3 < T_j ¹₂.

Proposition B.2 If (q_i; q_j) is a Nash equilibrium of the reduced game and q_j 2 A_j(4) [ Bj(7) [ Cj(4), then S_i q_i > ¹₃; so q_i 2 Ai(1; 2; 3; 4) [ Bi(1; 2; 3; 4; 5; 6; 7).

Proof. If q_j 2 A^j(4) [ B^j(7), then since q_j > 3; we have Si > ⁵₆ ¹₂q_j; and q_i = ¹₂ ¹₂q_j. Therefore, S_i q_i > ¹₃: If q_j 2 Cj(4), then S_j q_j ¹₃; S_i > 1 ¹₂S_j; and q_i = ¹₂ ¹₂q_j. This implies Si q_i > ¹₃.

We continue by solving for all Nash equilibria (q_i; q_j) of the reduced game where q_j 2 Aj(1): Next we consider Nash equilibria (q_i; q_j) with q_j 2 Aj(2): We restrict attention to the case with q_i 2 A= i(1); since using the symmetry of the …rms such equilibria follow already from the …rst case. We continue with q_j 2 A^j(3); and so on.

q_j 2 A^j(1)

It holds that

q_j < S_j 1

2; (3)

q_j < 1 1

2 2Si; (4)

q_i = ^R_i (q_j) = S_i: (5)

By Proposition B.1, q_i 2 Cⁱ(1; 2; 3; 4). This gives the following possibilities:

q_i 2 Cⁱ(1) : q_j = Sj; q_i 2 Ci(2) : q_j = 0;

q_i 2 Ci(3) : q_j = 1 S_i+ 2 S_j

2 + 2 ; (6)

q_i 2 Ci(4) : q_j = 1 2

1

2S_i: (7)

If q_i 2 Cⁱ(2), then q_i > ₅ implies Sj < ¹_{2 2}¹ +₂¹Si < ¹₂ by (5) and Lemma 4.1, so (3) leads to a contradiction.

Next, (3) and (6) imply Sj > 1 ¹₂Si, whereas q_i 2 Cⁱ(3) implies q_{i 8}; so Sj 1 ¹₂Si, a contradiction.

When q_i 2 Ci(4); then q_i S_i ¹₃ and S_j > ₈: These inequalities together with the inequalities (3) and (4) lead to the conclusion that (q_j; q_i) is a Nash equilibrium with q_j 2 Aj(1) if and only if q_j = ¹₂ ¹₂S_i; q_i = S_i; S_j > 1 ¹₂S_i; and S_i< ¹₃ ¹₃ . q_j 2 A^j(2)

It holds that

q_j < S_j 1

2; (8)

1 1

2 2S_i q_j 5 3

1

6 2S_i; (9)

q_i = ^R_i (q_j) = 1 q_j ¹₂ + Si

2 +

1 2:

By Proposition B.1, q_i 2 Ci(2; 3; 4).⁴ This gives the following possibilities:

q_i 2 Cⁱ(2) : q_i = 1 ¹₂ + Si

2 + ; q_j = 0; (10)

q_i 2 Cⁱ(3) : q_i = 1 + 2 ²+ Si+ 2 ²Si 2 Sj

3 + 5 + 2 ² ; (11)

q_j = 2 3 ²+ 8 S_j + 4 ²S_j+ 2 S_i

6 + 10 + 4 ² ;

q_i 2 Ci(4) : q_i = 1 + 2 S_i

3 + 2 ; q_j = 2 + 3 2 S_i

6 + 4 : (12)

Consider q_i 2 Cⁱ(2): Then q_i > ₅; so Sj < ^{2+3 +}_{8 +4}²2^{2 Si} < ¹₂, and (8) leads to a contradiction.

Consider q_i 2 Ci(3): It holds that 5 + 2 + ²+ 2 S_i

6 + 2 < S_j 1 1

2S_i; (13)

where the …rst inequality follows from (8) and (11), and the second inequality from Sj 8: By rewriting the expression in (13), it follows that Si < ¹₃ ¹₃ :

However, this is contradicted by Si

1 + 2 ² 2 S_j 3 + 4

1 3

1 3 ;

where the …rst inequality follows from (9) and (11), and the second inequality from Sj 8:

Consider q_i 2 Ci(4). It is implied by (9) and (12) that 1

3(1 ) S_i 1 3(2 1

3 ):

4Note that, by Proposition B.1, q_i 2 C= ⁱ(1). By Proposition B.1, if q_i 2 Cⁱ(1), then q_j 2 Cj(1; 2; 3; 4).

From (8) and (12) it follows that Sj > ⁵⁺⁵₆₊₄^{2 Si}. In conclusion, (q_j; q_i) is a Nash equilibrium with q_j 2 Aj(2) if and only if q_j = ²⁺³₆₊₄^{2 Si}; q_i = ¹ ₃₊₂^{+2 Si}; ¹₃(1 ) S_i ¹₃(2 ¹₃ ); and S_j > ⁵⁺⁵₆₊₄^{2 Si}.

q_j 2 A^j(3)

It holds that

q_j < S_j 1

2; (14)

5 3

1

6 2S_i< q_j 5

3 2S_i; (15)

q_i = ^R_i (q_j) = Si

1 3:

By Proposition B.1, q_i 2 Ci(2; 3; 4). This gives the following possibilities:

q_i 2 Ci(2) : q_j = 0;

q_i 2 Cⁱ(3) : q_j =

4 3

2

3 + 2 Sj Si

2 + 2 ;

q_i 2 Cⁱ(4) : q_j = 2 3

1 2Si:

Consider q_i 2 Ci(2): Since q_j = 0; the second inequality in (15) implies S_i ⁵₆: We have that

1

2 < Sj < Si 4 3 +²₃

2 ; (16)

where the …rst inequality follows from (14) and the second from q_i > ₅: By rewriting the expression (16), we …nd that S_i> ⁴₃ +¹₃ ; contradicting S_i 5=6:

Consider q_i 2 Cⁱ(3): By (14), it should hold that Sj > 11

6+1 6

1 2Si; which contradicts with S_{j 8}.

Consider q_i 2 Cⁱ(4). It holds that 2

3 1 9 < Si

2 3;

where both inequalities follow from (15). From (14), it follows that

Sj > 7 6

1 2Si:

The other constraints are redundant. In conclusion, q_j 2 Aj(3) if and only if q_i 2 Cⁱ(4); Sj > ⁷₆ ¹₂Si and ²₃ ¹₉ < Si 2

3.

q_j 2 A^j(4)

It holds that

q_j < S_j 1

2; (17)

q_j > 5 3 2S_i: We have

q_i = ^R_i (q_j) = 1 2

1 2q_j 1

2:

By Proposition B.1 and Proposition B.2, q_i 2 Aⁱ(4) [ Bⁱ(7).⁵ This gives the following possibilities:

q_i 2 Ai(4) : q_i = q_j = 1 3; q_i 2 Bⁱ(7) : q_i = q_j = 1

3;

Consider q_i 2 Ai(4) [ Bi(7). It follows from (17) that S_j > 5

6:

For q_i 2 Aⁱ(4), it follows from q_i < Si 1

2 that Si> ⁵₆. Next, if q_i 2 Bⁱ(7), it follows from

S_i 1

2 q_i < S_j 1 3 that

2 3 < Si

5 6:

The other constraints are redundant. In conclusion, q_j 2 Aj(4) if and only if q_i 2 Aⁱ(4) [ Bⁱ(7) and Sj > ⁵₆; Si> ²₃.

q_j 2 B^j(1) It holds that

Sj

1

2 q_j < Sj

1

3; (18)

q_j < 1 + S_j 2S_i

1 + : (19)

We have

q_i = ^R_i (q_j) = S_i 1 2:

5Note that Proposition B.1 excludes that qi 2 Aⁱ(1; 2; 3) [ Bⁱ(1; 2; 3; 4; 5; 6)and qj 2 A^j(4).

By Proposition B.1, q_i 2 Cⁱ(2; 3; 4). This gives the following possibilities:

q_i 2 Cⁱ(2) : q_j = 0;

q_i 2 Cⁱ(3) : q_j = 1 S_i+ 2 S_j

2 + 2 ;

q_i 2 Cⁱ(4) : q_j = 1 2

1 2Si: For q_i 2 Ci(2), it can be found that

S_j < 1

2 (S_i 1 + ) < 1

3 ( 1 + ) 0;

where the …rst inequality follows from q_i > ₅, the second one from (19) and the last one from 1.

For q_i 2 Ci(3); (18) implies ⁵₆ ¹₆ ¹₂S_i< S_j 1 ¹₂S_i. By (19), S_i < ¹₃ ¹₃ . The other constraints are redundant.

Next, q_i 2 Ci(4) implies S_j > 1 ¹₂S_i, whereas (18) implies S_j 1 ¹₂S_i, a contradiction.

In conclusion, q_j 2 B^j(1) if and only if q_i 2 Cⁱ(3) and ⁵₆ ¹₆ ¹₂Si < Sj

1 ¹₂S_i; S_i < ¹₃ ¹₃ . q_j 2 B^j(2)

It holds that S_j 1

2 q_j < S_j 1

3; (20)

q_j > 1 + S_j+ 2 S_i

1 + : (21)

We have

q_i = ^R_i (q_j) = 0:

By Proposition B.1, q_i 2 Cⁱ(2; 3; 4). This gives the following possibilities:

q_i 2 Ci(2) : q_j = 0;

q_i 2 Ci(3) : q_j = 1 + 2 S_j + S_i

2 + 2 ;

q_i 2 Ci(4) : q_j = 1 2:

For q_i 2 Cⁱ(2), from q_i > ₅ it follows that Sj < ₂¹( 1 + Si) 0.

Consider q_i 2 Ci(3). Inequality (20) implies S_i < ₃¹( 1 + ) 0.

If q_i 2 Ci(4), it holds again that S_i < 1

4 ( 1 + 3 2 S_j) < 1

3 ( 1 + ) 0;

where the …rst inequality follows from (21) and the second one from Sj > ₈. In conclusion, q_j 2 B= j(2):

q_j 2 B^j(3)

By Proposition B.1, q_i 2 Ci(2; 3; 4). This gives the following possibilities:

q_i 2 Cⁱ(2) : q_i = 1 + 2 Si + Sj

We have

q_i = ^R_i (q_j) = 1 qj 1

2 + Si

2 + :

By Proposition B.1, q_i 2 Ci(2; 3; 4). This gives the following possibilities:

q_i 2 Ci(2) : q_i = 1 ¹₂ + S_i (25) respectively that S_j > ⁷⁺⁶

1

6+4 . The other constraints are redundant.

In conclusion, q_j 2 Bj(4) if and only if q_i 2 Ci(4) and S_i ¹₃(2 ¹₃ );

We have

q_i = ^R_i (q_j) = 1 q_j+ 2 S_i + S_j q_j

2 + 2 :

By Proposition B.1, q_i 2 Cⁱ(2; 3; 4). This gives the following possibilities:

q_i 2 Ci(2) : q_i = 1 + 2 S_i + S_j that Sj > ²₃. The other constraints are redundant.

If q_i 2 Cⁱ(4), it holds that Sj > 1 ¹₂Si. From (32) it follows that Sj 1 ¹₂Si,

By Proposition B.1, q_i 2 Cⁱ(2; 3; 4). This gives the following possibilities:

q_i 2 Ci(2) : q_j = 0; (36)

where the …rst inequality follows from (36) and the second one from (34). This contradicts with (33).

Consider q_i 2 Cⁱ(3):It holds that Sj 1 ¹₂Si. From (33) it follows that Sj >

1 ¹₂Si, a contradiction.

Next, if q_i 2 Ci(4), it follows from (33) that S_j ⁷₆ ¹₂S_i: By (34) and by (35) it is implied respectively that ¹₃(2 ¹₃ ) < Si 2

3 and Sj > ¹⁰⁺⁶ ₈₊₆^{7Si 3 Si}. The other constraints are redundant.

In conclusion, q_j 2 Bj(6) if and only if q_i 2 Ci(4) and ¹₃(2 ¹₃ ) < S_i

2

3;¹⁰⁺⁶ ₈₊₆^{7Si 3 Si} < Sj 7 6

1 2Si: q_j 2 B^j(7)

It holds that S_j 1

2 q_j < S_j 1

3; (37)

q_j > 5 3 2Si: We have

q_i = ^R_i (q_j) = 1 2

1 2q_j 1

2:

By Proposition B.1 and Proposition B.2, q_i 2 Bⁱ(7). This gives the following possibilities:

q_i 2 Bⁱ(7) : q_i = q_j = 1 3:

For q_i 2 Bj(7), it follows from S_i ¹₂ q_i < S_j ¹₂ that ²₃ < S_i ⁵₆. From (37) it follows that ²₃ < S_j ⁵₆. The rest of the constraints is redundant.

Therefore, q_j 2 B^j(7) if q_i 2 Bⁱ(7) and ²₃ < Si 5

6;²₃ < Sj 5 6. q_j 2 C^j(1)

It holds that q_j Sj

1

3; (38)

q_j < 1 + S_j 2S_i

1 + : (39)

We have

q_i = ^R_i (q_j) = S_i 1 2:

By Proposition B.1 and Proposition B.2, q_i 2 Cⁱ(1; 2; 3). This gives the following possibilities:

q_i 2 Ci(1) : q_j = S_j; q_i 2 Ci(2) : q_j = 0;

q_i 2 Ci(3) : q_j = 1 Si+ 2 Sj

2 + 2 :

If q_i 2 Cⁱ(1), then, by qi < ₄, it holds that Sj < ¹₂ ¹₂ ¹₂Si. From (39), it follows that S_i< ¹₂ ¹₂ ¹₂S_j.

Consider q_i 2 Cⁱ(2): It holds that S_j < 1

2 (S_i 1 + ) < 1

3 ( 1 + ) 0;

where the …rst inequality follows from q_i > ₅ and the second one from (39).

For q_i 2 Ci(3), it follows from q_{i 4} that S_j ¹₂ ¹₂ ¹₂S_i. By (39), it is implied that Si < ¹₃ ¹₃ . From (38), it follows that Sj 5

6 1 6

1

2Si. The rest of the constraints is redundant.

In conclusion, q_j 2 Cj(1) if and only if q_i 2 Ci(1) and S_i < ¹₂ ¹₂ ¹₂S_j; S_j <

1 2

1 2

1

2Si or q_i 2 Cⁱ(3) and Si< ¹₃ ¹₃ ;¹₂ ¹₂ ¹₂Si Sj 5 6

1 6

1 2Si. q_j 2 C^j(2)

It holds that

q_j S_j 1

3;

q_j > 1 + S_j+ 2 S_i

1 + : (40)

We have

q_i = ^R_i (q_j) = 0:

By Proposition B.2, q_i 2 Ci(2; 3). This gives the following possibilities:

q_i 2 Cⁱ(2) : q_j = 0;

q_i 2 Cⁱ(3) : q_j = 1 + 2 S_j + S_i

2 + 2 :

For q_i 2 Cⁱ(2); it follows from (40) that Si< ₂¹( 1 + Sj) 0.

For q_i 2 Ci(3), it is implied by (40) that S_i< ₃¹( 1 + ) 0.

In conclusion, q_j 2 C= ^j(2) if q_i 2 Cⁱ(2; 3).

q_j 2 C^j(3)

It holds that

q_j S_j 1

3; (41)

1 + Sj 2Si

1 + q_j 1 + Sj+ 2 Si

1 + ; (42)

S_i 1 1

2S_j: (43)

We have

q_i = ^R_i (q_j) = 1 q_j+ 2 S_i + S_j q_j

1 + :

By Proposition B.2, q_i 2 Ci(3). This gives the following possibility:

q_i 2 Ci(3) : q_i = 1 + 3 S_i

3 + 3 ; q_j = 1 + 3 S_j 3 + 3 :

For q_i 2 Ci(3), it follows from q_i S_i ¹₃ and (41) respectively that S_i ²₃ and S_j ²₃.

Since q_i 2 Cⁱ(3) and by (41), Si 2

3 and Sj 2

3. Next, it follows from q_{i 4} that S_j ¹₃ ¹₃ . By (42), S_i ¹₃ ¹₃ . The remaining constraints are redundant.

In conclusion, q_j 2 Cj(3) if q_i 2 Ci(3) and ¹₃(1 ) S_i ²₃;¹₃(1 ) S_j ²₃. q_j 2 C^j(4)

This case does not need to be calculated here, since, by proposition B.2, it can only be combined with the situations A_i and B_i, and all these situations are already calculated.

In document Dynamic duopoly with intertemporal capacity constraints (Page 30-42)