In this part, we assume that the consumer-friendly upstream firm involves in a centralized bargaining with the downstream firms. At stage two of game, the upstream firm and downstream firms determine the terms of the two-part tariff contract by maximizing the following generalized Nash bargaining expression:
Max [ α C S + ∑ i = 1 2 ( q i ρ ω i ρ + F i ρ ) ] β * [ ∑ i = 1 2 π i ρ ] ( 1 − β ) , (19)
where α denotes the consumer-friendly upstream firm’s valuation over consumer surplus, ρ denotes the mode of downstream competition. And q i ρ and π i ρ denote the output and net profit of the final goods producers, β (resp. ( 1 − β ) ) shows the bargaining power of the upstream firm (resp. downstream firms). We restrict the range of β : β ∈ ( 0 , 1 ) . If β = 1 , ω i and F i are set by the upstream firm and if β = 0 , ω i and F i are set by the downstream firms.
Maximizing the above with respect to F i gives the following:
F i ρ = 1 2 β ∑ i = 1 2 [ ( 1 − q i ρ − γ q j ρ − ω i ρ ) q i ρ ] − ( 1 − β ) ∑ i = 1 2 q i ρ [ 2 ( 1 + α ) ω i ρ − α ( 1 − p j ρ ) ] . (20)
Maximizing (16) subject to (1)-(15) and (17) gives the equilibrium input prices of downstream firms under the different modes of downstream competition as:
ω 1 q q = ω 2 q q = α − γ + α γ ( − 2 + α ) ( 1 + γ )
(where both downstream firms choose quantity contracts),
ω 1 p p = ω 2 p p = α − γ − 2 + α
(where both downstream firms choose price contracts),
ω 1 p q = α + ( − 1 + α ) γ ( − 2 + α ) ( 1 + γ ) , ω 2 p q = α − γ − 2 + α
(where D 1 adopts price contract and D 2 adopts quantity contract),
ω 1 q p = α − γ − 2 + α , ω 2 q p = α + ( − 1 + α ) γ ( − 2 + α ) ( 1 + γ )
(where D 1 adopts quantity contract and D 2 adopts price contract).
At the stage one, each downstream firm simultaneously makes a choice whether to adopt quantity contract or price contract. We can get the equilibrium outcomes shown in
We can find that consumer surplus and social welfare will not change irrespective of the mode of downstream competition. As
π 1 c p q − π 1 c q q = π 1 c p p − π 1 c p q = π 2 c p q − π 2 c q q = π 2 c p p − π 2 c p q = γ 2 2 ( 1 + γ ) 2 ( − 2 + α ) 2 > 0 , for the
downstream firms, choosing price contract is the best strategy for the final goods producers, which is similar to Leonard F.S. Wang et al. (2016). When the downstream firms choose price contract, the profits of the upstream firm is:
Π c p p = 2 β − α ( 1 + β ) ( − 2 + α ) 2 ( 1 + γ ) . And in this case, the social welfare is
S W c p p = 3 − 2 α ( 2 − α ) 2 ( 1 + γ ) , the consumer surplus is C S c p p = 1 ( 2 − α ) 2 ( 1 + γ ) and
| Competition model | Profit of the downstream firms | Social welfare | Consumer surplus |
|---|---|---|---|
| pp | |||
| pq | |||
| qp |
the profit of downstream firm i is π i c p p = 1 − β 2 ( 2 − α ) ( 1 + γ ) .
Social welfare comprised with three major part―consumer surplus, upstream firm’s profits and downstream firms’ profits. So the effect of a change in α on social welfare can be decomposed into three different effects as follow:
d S W c p p d α = d C S c p p d α + + d Π c p p d α − + d π i c p p d α + 1. The first term on the right-hand side captures
the direct effect of a change in α on consumer surplus. We know that the final product price decline and output rises, can make the consumer surplus rise. So consumer-friendly upstream firm reduces the intermediate product prices to lower the price of the final product, increase the output of the final product, to increase the consumer surplus. The degree of the upstream firm’s concern over consumer surplus ( α ) rises leading to the downstream manufacturers’ output
increased ( d q i p p d α = 1 ( 2 − α ) 2 ( 1 + γ ) ), price fell ( d p i c p p d α = − 1 ( 2 − α ) 2 < 0 ). So the rise
of α is beneficial to the consumer surplus. The second and third terms on the right-hand side denotes the direct effect of a change in α on the profits of upstream and downstream firms. Consumer-friendly upstream firm reduce the intermediate product prices to rise the consumer surplus, which is damage to its
profits ( d Π c p p d α < 0 ) and beneficial to the profits of downstream firms
( d π i c p p d α > 0 ) . As d S W c p p d α = 2 ( − 1 + α ) ( 1 + γ ) ( − 2 + α ) 3 > 0 , the sum effect of a change in
α on consumer surplus and downstream is stronger than downstream firm.
Lemma 1. Social welfare is positive correlated with the degree of the upstream firm’s concern over consumer surplus under centralized bargaining.
In this part, we will investigate the situation that the decentralized bargaining between the consumer-friendly upstream firm and the downstream firms. At stage two of the game, the upstream firm and downstream firms adopt the terms of the two-part tariff contract by maximizing the following generalized Nash bargaining expression:
Max [ α ( C S 1 ρ − C S 2 ρ ) + ∑ i = 1 2 ( q i ρ ω i ρ + F i ρ ) − ω j ρ ( 1 − ω j ρ ) 2 + F j ρ ] β * [ π i ρ ] 1 − β , i , j = 1 , 2 , i ≠ j (21)
where C S 1 ρ = ( q 1 ρ ) 2 + ( q 2 ρ ) 2 + 2 γ q 1 ρ q 2 ρ 2 denotes the consumer surplus that two downstream firms in the downstream market. C S 2 ρ = ( 1 − ω j ρ ) 2 2 2 represents the consumer surplus and ω j ρ ( 1 − ω j ρ ) 2 + F j ρ is U’s profits when the upstream firm’s
negotiation with D i breakdown and D j acts as a monopolist in the downstream market, i.e., it produces the monopoly quantity.
Maximizing the above with respect to F i gives the following
F i ρ = ( − 1 + β ) [ α ( C S 1 ρ − C S 2 ρ ) + ∑ i = 1 2 ( q i ρ ω i ρ ) − ( 1 + ω j ρ ) 2 ( 1 − ω j ρ ) 2 ] + β ( 1 − q i ρ − γ q j ρ − ω i ρ ) q i ρ . (22)
In this case, we can get the equilibrium wholesale prices of downstream firms under the different modes of competition.
ω 1 q q = ω 2 q q = − γ 2 + α ( − 2 + γ ) ( 1 + γ ) 4 − 2 γ 2 + α ( − 2 + γ ) ( 1 + γ )
(where both downstream firms choose quantity contracts),
ω 1 p p = ω 2 p p = γ 2 − α ( 2 + γ ) 4 − α ( 2 + γ )
(where both downstream firms choose price contracts),
ω 1 p q = 4 γ 2 + α 2 ( 1 + γ ) ( − 4 + 3 γ 2 ) + α ( 8 + γ ( 12 − γ ( 2 + 5 γ ) ) ) ( 1 + γ ) ( − 4 ( − 2 + α ) 2 + ( 8 + α ( − 8 + 3 α ) ) γ 2 ) , ω 2 p q = 2 γ 2 ( − 2 + γ 2 ) + α 2 ( 1 + γ ) ( − 4 + 3 γ 2 ) − α ( 1 + γ ) ( − 8 + γ ( − 4 + γ ( 2 + 3 γ ) ) ) ( 1 + γ ) ( − 4 ( − 2 + α ) 2 + ( 8 + α ( − 8 + 3 α ) ) γ 2 ) ,
(where D 1 adopts price contract and D 2 adopts quantity contract),
ω 1 q p = 2 γ 2 ( − 2 + γ 2 ) + α 2 ( 1 + γ ) ( − 4 + 3 γ 2 ) − α ( 1 + γ ) ( − 8 + γ ( − 4 + γ ( 2 + 3 γ ) ) ) ( 1 + γ ) ( − 4 ( − 2 + α ) 2 + ( 8 + α ( − 8 + 3 α ) ) γ 2 ) , ω 2 q p = 4 γ 2 + α 2 ( 1 + γ ) ( − 4 + 3 γ 2 ) + α ( 8 + γ ( 12 − γ ( 2 + 5 γ ) ) ) ( 1 + γ ) ( − 4 ( − 2 + α ) 2 + ( 8 + α ( − 8 + 3 α ) ) γ 2 ) ,
(where D 1 adopts quantity contract and D 2 adopts price contract).
At the stage 1, each downstream firm simultaneously makes a choice whether to adopt quantity contract or price contract. We can get the equilibrium outcomes as follows (
As Min ( π 1 d q q − π 1 d p q ) ≥ 0 , Min ( π 1 d q p − π 1 d p p ) ≥ 0 , Min ( π 2 d p q − π 2 d p p ) ≥ 0 ,
Min ( π 2 d q q − π 2 d q p ) ≥ 0 , for the downstream firms, choosing quantity contract is the dominant strategy which is similar to Maria Alipranti (2014). When the downstream firms choices quantity contract, the profits of upstream firm is
Π d q q = ( − 2 + γ ) { 4 γ 3 − 4 β ( − 2 + γ ) ( − 2 + γ 2 ) + α ( − 2 + γ ) [ − 4 ( 1 + β ) − 8 γ + 3 ( − 1 + β ) γ 2 ] } 4 ( 4 − 2 γ 2 + α ( − 2 + γ ) ( 1 + γ ) ) 2 .
And in this case, the social welfare is
S W d q q = ( 2 − γ ) [ 6 + 2 α ( − 2 + γ ) ( 1 + γ ) − γ ( 1 + 3 γ ) ] ( 4 − 2 γ 2 + α ( − 2 + γ ) ( 1 + γ ) ) 2 , the consumer surplus is
C S d q q = ( 2 − γ ) 2 ( 1 + γ ) ( 4 − 2 γ 2 + α ( − 2 + γ ) ( 1 + γ ) ) 2 and the profit of downstream firm is
π i d q q = ( 1 − β ) ( − 2 + γ ) 2 [ 8 − 4 γ 2 + α ( − 4 + 3 γ 2 ) ] 8 ( 4 − 2 γ 2 + α ( − 2 + γ ) ( 1 + γ ) ) 2 .
What is the relationship between social welfare and the degree of the upstream firm’s concern over consumer surplus ( α )? Same as over the centralized bargaining, the effect of a change in α can be decomposed into three different
effects as follow: d S W d p p d α = d C S d p p d α + + d Π d p p d α − + d π i d p p d α + . The degree of upstream firm’s
concern about consumer surplus is positive correlated with consumer surplus and the net profit of the downstream firms, negatively correlated with the net profit of the upstream. But Social welfare and the attention of consumer surplus is inverted U-shaped relationship in this case. In other word, when the degree of
concern is relatively low ( 0 < α < 2 − γ − γ 2 2 + γ − γ 2 ), the sum effect of a change in α
on consumer surplus and downstream is stronger than upstream firm and in this case the social welfare increase as α rises. And when the degree of concern is
relatively high ( 2 − γ − γ 2 2 + γ − γ 2 < α < 1 ), the sum effect of a change in α on consumer
surplus and downstream is weaker than upstream firm, and in this case the social welfare falls as α rises. As shown in
Lemma 2. Social welfare and the degree of upstream firm’s concern about consumer surplus is inverted U-shaped relationship under decentralized bargaining.
Proof:
d S W d p p d α = d C S d p p d α + + d Π d p p d α − + d π i d p p d α + = 2 ( − 2 + γ ) 2 ( 1 + γ ) ( α ( − 2 + γ ) ( 1 + γ ) − ( − 1 + γ ) ( 2 + γ ) ) ( 4 − 2 γ 2 + α ( − 2 + γ ) ( 1 + γ ) ) 3 ,
when
0 < α < 2 − γ − γ 2 2 + γ − γ 2 , d S W d p p d α > 0 ,
when
2 − γ − γ 2 2 + γ − γ 2 < α < 1 , d S W d p p d α < 0 ,
and when
2 − γ − γ 2 2 + γ − γ 2 = α , d S W d p p d α = 0 .
According to the content of the above, we can get the following corollary:
Corollary 1. When the upstream firm is a consumer-friendly firm, the social welfare and the consumer surplus is greater than a profit maximization enterprise.
If the upstream firm is profit-maximizing, the endogenous choice of competition mode in the downstream market will not change. And the social welfare is
S W p c = 3 4 ( 1 + γ ) 2, the consumer surplus is C S p c = 1 4 ( 1 + γ ) under the centralized bargaining, the social welfare is S W p d = ( 2 − γ ) ( 6 − γ − 3 γ 2 ) 4 ( 2 − γ 2 ) 2 , the consumer surplus is C S p d = ( 2 − γ ) 2 ( 1 + γ ) 4 ( 2 − γ 2 ) 2 under the decentralized bargaining. We can
find that: S W c − S W p ≥ 0 and C S c − C S p ≥ 0 .
S W ↑ = C S ↑ + N Π ↓ 3. (23)
Equation (23) means that consumer surplus rise and total profits of downstream and upstream firms declined, but the social welfare will rise when the upstream firm is consumer-friendly. If the upstream firm is a consumer-friendly firm, the upstream firm not only pay attention to its own profit, but also con-
cerned about the consumer surplus in the bargaining process. As d p d α < 0 and d q d α > 0 , denote the degree of upstream firm’s concern about consumer surplus
is significantly positive correlation with quantity and negative correlation with price. The consumer surplus rise when the upstream firm is consumer-friendly. And in this case, the increase in consumer surplus is greater than the decrease in total profits of firms, resulting in the rise of social welfare.
According to lemma 1 and lemma 2, social welfare is positive correlated with the degree of upstream firm’s concern about consumer surplus under centralized bargaining, and social welfare and the degree of upstream firm’s concern about consumer surplus is inverted U-shaped relationship under decentralized bargaining. In which case social welfare is higher?
According to the above analysis, we can get that when the consumer-friendly upstream firm is involved in a centralized bargaining with the downstream firms,
the social welfare is S W c = 3 − 2 α ( − 2 + α ) 2 ( 1 + γ ) . And the social welfare is
S W d = ( 2 − γ ) ( 6 + 2 α ( − 2 + γ ) ( 1 + γ ) − γ ( 1 + 3 γ ) ) ( 4 − 2 γ 2 + α ( − 2 + γ ) ( 1 + γ ) ) 2 under decentralized bargain-
ing.
Proposition 1.
When
− 6 + γ ( − 1 + 3 γ ) + 4 + γ ( 4 + ( − 1 + γ ) 2 γ ) 2 ( − 2 + γ ) ( 1 + γ ) < α ≤ 1 , S W c > S W d ,
when
0 ≤ α < − 6 + γ ( − 1 + 3 γ ) + 4 + γ ( 4 + ( − 1 + γ ) 2 γ ) 2 ( − 2 + γ ) ( 1 + γ ) , S W c < S W d ,
when
α = − 6 + γ ( − 1 + 3 γ ) + 4 + γ ( 4 + ( − 1 + γ ) 2 γ ) 2 ( − 2 + γ ) ( 1 + γ ) , S W c = S W d .
The intuition for proposition 1 is straightforward. As shown in
As shown in
( 0 < α < 2 − γ − γ 2 2 + γ − γ 2 area I), the social welfare increases with α under the two
bargaining modes. And in this situation the social welfare under the decentralized bargaining is greater than under the centralized bargaining. When
2 − γ − γ 2 2 + γ − γ 2 < α < − 6 + γ ( − 1 + 3 γ ) + 4 + γ ( 4 + ( − 1 + γ ) 2 γ ) 2 ( − 2 + γ ) ( 1 + γ ) (area II), although the
social welfare is negatively correlated with α under decentralized bargaining and
positively correlated under centralized bargaining, the social welfare under the decentralized bargaining is also greater than under the centralized bargaining. But when the degree of concern is relatively high
( − 6 + γ ( − 1 + 3 γ ) + 4 + γ ( 4 + ( − 1 + γ ) 2 γ ) 2 ( − 2 + γ ) ( 1 + γ ) < α < 1 , area III), the social welfare
under the decentralized bargaining is less than under the centralized bargaining. In this situation, reduction in social welfare is sufficiently high under decentralized bargaining and added value of social welfare is sufficiently high under centralized bargaining.
The consumer surplus is C S c = 1 ( 1 + γ ) ( 2 − α ) 2 , when that the upstream firm
involves in a centralized bargaining with the downstream firms. And the con-
sumer surplus is C S d = ( 2 − γ ) 2 ( 1 + γ ) ( 4 + γ 2 ( − 2 + α ) − 2 λ − γ α ) 2 under decentralized bar-
gaining mode. It can be checked that the consumer surplus increase as α rises under two bargaining modes. In which case the consumer surplus is higher?
Proposition 2. Decentralized bargaining generates higher consumer surplus as compared to centralized bargaining, irrespective of both the product differentiation degree and the upstream firm’s valuation over consumer surplus.
Proof:
Δ = C S c − C S d = 4 γ [ − 4 − γ + α ( 2 + γ ) ] ( − 2 + α ) 2 ( 1 + γ ) [ − 4 + α ( 2 + γ ) ] 2 < 0 .
Centralized bargaining is that downstream firms together to bargain with the upstream firm and decentralized bargaining is that downstream firms respectively bargains with the upstream firm. So when the downstream firms get together, more conducive to protect their own profits, what is mean that more convenient to occupy the consumer surplus. So the consumer surplus is less under the centralized bargaining.