1
Bilateral Market Power & Bargaining
Johan Stennek
Don’tneedtoknowappendixesinlecturenotes
BilateralMarketPowerExample:FoodRetailing
4
Food Retailing • Food retailers are huge
!
The!world’s!largest!food!retailers!in!2003!
Company! Food!Sales!(US$mn)!
Wal$Mart( 121(566(
Carrefour( 77(330(
Ahold( 72(414(
Tesco( 40(907(
Kroger( 39(320(
Rewe( 36(483(
Aldi( 36(189(
Ito$Yokado( 35(812(
Metro(Group(ITM( 34(700((
½ Swedish GDP
5
Food Retailing • Retail markets are highly concentrated
Tabell&1a.&Dagligvarukedjornas&andel&av&den&svenska&marknaden&Kedja& Butiker&
(antal)&Butiksyta&(kvm)&
Omsättning&(miljarder&kr)&
Axfood& 803&(24%)&
625&855&(18%)&
34,6&&&&&&&&&(18%)&
Bergendahls& 229&(7%)&
328&196&(10%)&
13,6&(7%)&
Coop& 730&(22%)&
983&255&(29%)&
41,4&(21%)&
ICA& 1&379&(41%)&
1&240&602&(36%)&
96,6&(50%)&
Lidl& 146&(4%)&
170&767&(5%)&
5,2&&&&&&&&&(3%)&
Netto& 105&(3%)&
70&603&(2%)&
3,0&(2%)&
&
6
Food Retailing
• Food manufacturers – Some are huge:
• Kraft Food, Nestle, Scan • Annual sales tenth of billions of Euros
– Some are tiny: • local cheese
7
Food Retailing • Mutual dependence
– Some brands = Must have • ICA “must” sell Coke • Otherwise families would shop at Coop
– Some retailers = Must channel • Coke “must” sell via ICA to be active in Sweden • Probably large share of Coke’s sales in Sweden
– Both would lose if ICA would not sell Coke
8
Food Retailing • Mutual dependence
– Manufacturers cannot dictate wholesale prices – Retailers cannot dictate wholesale prices
• Thus – They have to negotiate and agree
• In particular – Also retailers have market power
= buyer power
9
Food Retailing • Large retailers pay lower prices
(= more buyer power) Retailer( Market(Share(
(CC#Table#5:3,#p.#44)#
Price(
(CC#Table#5,#p.#435)#
Tesco# 24.6# 100.0#
Sainsbury# 20.7# 101.6#
Asda# 13.4# 102.3#
Somerfield# 8.5# 103.0#
Safeway# 12.5# 103.1#
Morrison# 4.3# 104.6#
Iceland# 0.1# 105.3#
Waitrose# 3.3# 109.4#
Booth# 0.1# 109.5#
Netto# 0.5# 110.1#
Budgens# 0.4# 111.1#
#
10
Food Retailing
• Questions – How analyze bargaining in intermediate goods
markets?
– Why do large buyers get better prices?
11
Other examples
• Labor markets – Vårdförbundet vs Landsting
• Relation-specific investments – Car manufacturers vs producers of parts
12
Bilateral Monopoly
13
BilateralMonopoly
• Bilateral monopoly
– One Seller: MC(q) = inverse supply if price taker
– One Buyer: MV(q) = inverse demand if price taker
q
€
MC(q)
MV(q)
14
Bilateral Monopoly Intuitive Analysis
• Efficient quantity – Complete information – Maximize the surplus
to be shared
q
€
MC(q)
MV(q) q*
S*
15
Bilateral Monopoly Intuitive Analysis
• Efficient quantity – Complete information – Maximize the surplus
to be shared
q
€
MC(q)
MV(q) q*
S*
Efficiency from the point of view of the two firms = Same quantity as a vertically integrated firm would choose
16
BilateralMonopolyIntuiHveAnalysis
• Problem
– But what price?
• Only restrictions
– Seller must cover his costs, C(q*)
– Buyer must not pay more than wtp,
V(q*)
=> Any split of S* = V(q*) – C(q*) seems reasonable
q
€
MC(q)
MV(q)q*
S*
17
BilateralMonopolyIntuiHveAnalysis
• Note – If someone demands “too much”
– The other side will reject and make a counter-offer
• Problem – Haggling could go on forever
– Gains from trade delayed
• Thus – Both sides have incentive to be reasonable
– Party with less aversion to delay has strategic advantage
18
BilateralMonopolyDefiniHons
• Definitions
– Efficient quantity: q*
– Walrasian price: pw
– Maximum bilateral surplus: S*
q
€
MC(q)
MV(q)q*
pw S*
19
BilateralMonopoly
• First important insight: – Contract must specify both price and
quantity, (p, q)
– Q: Why?
– Otherwise inefficiency (If p > pw then q < q*)
q
€
MC(q)
MV(q)q*
pw S*
ExtensiveFormBargainingUlHmatumbargaining
21
UlHmatumbargaining
• One round of negotiations – One party, say seller, gets to propose a contract (p, q)
– Other party, say buyer, can accept or reject
• Outcome – If (p, q) accepted, it is implemented
– Otherwise game ends without agreement
• Payoffs – Buyer: V(q) – p q if agreement, zero otherwise
– Seller: p q – C(q) if agreement, zero otherwise
• Perfect information – Backwards induction
Solvethisgamenow!
22
UlHmatumbargaining
• Time 2
– Buyer accepts proposed contract (p, q) iff V(q) – p q ≥ 0
• Time 1
– Seller: maxp,q p q – C(q) such that V(q) – p q ≥ 0
23
UlHmatumbargaining
max p,q p q – C(q)
st : V (q) – p q ≥ 0
Must set p such that: p ⋅q = V (q)
maxq V (q) – C(q)
Must set q such that: MV q( ) = MC(q)
Sellertakeswholesurplus
EfficientquanHty
24
UlHmatumbargaining
• SPE of ultimatum bargaining game
– Unique equilibrium
– There is agreement
– Efficient quantity
– Proposer takes the whole (maximal) surplus
25
UlHmatumbargaining
• Assume rest of lecture – Always efficient quantity
– Surplus = 1
– Player S gets share πS
– Player B gets share πB = 1 – πS
• Ultimatum game
– πS = 1
– πB = 0
Tworounds(T=2)
27
Tworounds(T=2)
• Alternating offers
• Players are impatient
– For B: €1 in period 2 is equally good as €δB in period 1
– Where δB, δS < 1 are discount factors
28
Tworounds(T=2)
• Period 1 – B proposes
– S accepts or rejects
• Period 2 (in case S rejected)
– S proposes
– B accepts or rejects
• Perfect information => Use BI
Solvethisgamenow!
π BT ,π S
T( )
π BT −1,π S
T −1( )
29
Tworounds
• Period T = 2 (S bids) (in case S rejected) – B accepts iff:
– S proposes:
• Period T-1 = 1 (B bids) – S accepts iff:
– B proposes:
• Note – S willing to reduce his share to get an early agreement
– Both players get part of surplus
– B’s share determined by S’s impatience. If S very patient πS≈1
π BT ≥ 0
π BT = 0 π S
T = 1
π ST −1 ≥ δSπ S
T = δS < 1
π BT −1 = 1− δS > 0 π S
T −1 = δS
Trounds
31
T rounds
• Model
– Large number of periods, T
– Buyer and seller take turns to make offer
– Common discount factor δ = δB = δS
– Subgame perfect equilibrium (ie start analysis in last period)
32
T rounds
Time Bidder πB πS Resp. T S ? ? ?
33
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
34
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B ? ? ?
35
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B rest δ yes
36
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B 1-δ δ yes
37
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B 1-δ δ yes T-2 S ? ? ?
38
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B 1-δ δ yes T-2 S δ(1-δ) rest yes
39
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B 1-δ δ yes T-2 S δ(1-δ) 1-δ(1-δ) yes
40
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B 1-δ δ yes T-2 S δ(1-δ) 1-δ(1-δ) yes
multiply
41
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B 1-δ δ yes T-2 S δ-δ2 1-δ+δ2 yes
42
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B 1-δ δ yes T-2 S δ-δ2 1-δ+δ2 yes T-3 B ? ? ?
43
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B 1-δ δ yes T-2 S δ-δ2 1-δ+δ2 yes T-3 B rest δ(1-δ+δ2) yes
44
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B 1-δ δ yes T-2 S δ-δ2 1-δ+δ2 yes T-3 B 1-δ(1-δ+δ2) δ(1-δ+δ2) yes
45
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B 1-δ δ yes T-2 S δ-δ2 1-δ+δ2 yes T-3 B 1-δ+δ2-δ3 δ-δ2+δ3 yes
46
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B 1-δ δ yes T-2 S δ-δ2 1-δ+δ2 yes T-3 B 1-δ+δ2-δ3 δ-δ2+δ3 yes T-4 S δ(1-δ+δ2-δ3) rest yes
47
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B 1-δ δ yes T-2 S δ-δ2 1-δ+δ2 yes T-3 B 1-δ+δ2-δ3 δ-δ2+δ3 yes T-4 S δ(1-δ+δ2-δ3) 1-δ(1-δ+δ2-δ3) yes
48
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B 1-δ δ yes T-2 S δ-δ2 1-δ+δ2 yes T-3 B 1-δ+δ2-δ3 δ-δ2+δ3 yes T-4 S δ-δ2+δ3-δ4 1-δ+δ2-δ3+δ4 yes
49
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B 1-δ δ yes T-2 S δ-δ2 1-δ+δ2 yes T-3 B 1-δ+δ2-δ3 δ-δ2+δ3 yes T-4 S δ-δ2+δ3-δ4 1-δ+δ2-δ3+δ4 yes … … … … … 1 S δ-δ2+δ3-δ4+…-δT-1 1-δ+δ2-δ3+δ4-…+δT-1 yes
50
T rounds
Time Bidder πB πS Resp. T S 0 1 yes
T-1 B 1-δ δ yes T-2 S δ-δ2 1-δ+δ2 yes T-3 B 1-δ+δ2-δ3 δ-δ2+δ3 yes T-4 S δ-δ2+δ3-δ4 1-δ+δ2-δ3+δ4 yes … … … … … 1 S δ-δ2+δ3-δ4+…-δT-1 1-δ+δ2-δ3+δ4-…+δT-1 yes
π B = δ − δ 2 + δ 3 − δ 4 + ...− δ T −1
π S = 1− δ + δ 2 − δ 3 + δ 4 − ...+ δ T −1
51
T rounds
Geometric seriesπ B = δ − δ 2 + δ 3 − δ 4 + ...− δ T −1
π S = 1− δ + δ 2 − δ 3 + δ 4 − ...+ δ T −1
52
T rounds
S's shareπ S = 1− δ + δ 2 − δ 3 + δ 4 − ...+ δ T −1
53
T rounds
S's shareπ S = 1− δ + δ 2 − δ 3 + δ 4 − ...+ δ T −1
Multiplyδπ S = δ − δ 2 + δ 3 − δ 4 + δ 5 − ...+ δ T
54
T rounds
S's shareπ S = 1− δ + δ 2 − δ 3 + δ 4 − ...+ δ T −1
Multiplyδπ S = δ − δ 2 + δ 3 − δ 4 + δ 5 − ...+ δ T
Addπ S + δπ S = 1+ δ T
55
T rounds
S's shareπ S = 1− δ + δ 2 − δ 3 + δ 4 − ...+ δ T −1
Multiplyδπ S = δ − δ 2 + δ 3 − δ 4 + δ 5 − ...+ δ T
Addπ S + δπ S = 1+ δ T
Solve
π S =1+ δ T
1+ δ
56
T rounds
Equilibrium shares with T periods
π S =1
1+ δ1+ δ T( )
π B =δ
1+ δ1− δ T −1( )
57
T rounds
Equilibrium shares with T periods
π S =1
1+ δ1+ δ T( )
π B =δ
1+ δ1− δ T −1( )
S has advantage of making last bid1+ δ T > 1− δ T −1
To confirm this, solve model where - B makes last bid - S makes first bid
58
T rounds
Equilibrium shares with T periods
π S =1
1+ δ1+ δ T( )
π B =δ
1+ δ1− δ T −1( )
S has advantage of making last bid1+ δ T > 1− δ T −1 Disappears if T very large
59
T rounds
Equilibrium shares with T ≈ ∞ periods
π S =1
1+ δ
π B =δ
1+ δ
S has advantage of making first bid1
1+ δ>
δ1+ δ
To confirm this, solve model where - B makes first bid
60
T rounds
Equilibrium shares with T ≈ ∞ periods
π S =1
1+ δ
π B =δ
1+ δ
S has advantage of making first bid1
1+ δ>
δ1+ δ
To confirm this, solve model where - B makes first bid
61
T rounds
Equilibrium shares with T ≈ ∞ periods
π S =1
1+ δ
π B =δ
1+ δ
S has advantage of making first bid1
1+ δ>
δ1+ δ
First bidder’s advantage disappears if δ ≈ 1
62
T rounds
Equilibrium shares with T ≈ ∞ periods and very patient players (δ ≈ 1)
π S =12
π B =12
63
Difference in Patience
Equilibrium shares with T ≈ ∞ periods and different discount factors
π S =1− δB
1− δSδB
π B =1− δS
1− δSδB
δB
(Easy to show using same method as above)
64
Difference in Patience
• Recall – ri = continous-time discount factor
– Δ = length of time period
• Then, as Δ è 0:
–
– Using l’Hopital’s rule
δ i = e−riΔ
π S =1− δB
1− δSδB
≈rB
rS + rB
65
Extensive form bargaining
• Conclusions
– Exists unique equilibrium (SPE)
– There is agreement
– Agreement is immediate
– Efficient agreement (here: quantity)
– Split of surplus (price) determined by relative patience • Right to propose in last round gives advantage (T < ∞)
• Right to propose in first round gives advantage (δ < 1)
66
Implications for Bilateral Monopoly
67
Implications for Bilateral Monopoly
• Equal splitting
ΠS = ΠB
p ⋅q − C q( ) = V q( ) − p ⋅q
2 ⋅ p ⋅q = V q( ) + C q( )
p = 12V q( )q
+C q( )q
⎡
⎣⎢
⎤
⎦⎥
68
Implications for Bilateral Monopoly
• Equal splitting
ΠS = ΠB
p ⋅q − C q( ) = V q( ) − p ⋅q
2 ⋅ p ⋅q = V q( ) + C q( )
p = 12V q( )q
+C q( )q
⎡
⎣⎢
⎤
⎦⎥
Retailer’s average revenues
69
Implications for Bilateral Monopoly
• Equal splitting
ΠS = ΠB
p ⋅q − C q( ) = V q( ) − p ⋅q
2 ⋅ p ⋅q = V q( ) + C q( )
p = 12V q( )q
+C q( )q
⎡
⎣⎢
⎤
⎦⎥
Manufacturer’s average costs
70
Implications for Bilateral Monopoly
• Equal splitting
ΠS = ΠB
p ⋅q − C q( ) = V q( ) − p ⋅q
2 ⋅ p ⋅q = V q( ) + C q( )
p = 12V q( )q
+C q( )q
⎡
⎣⎢
⎤
⎦⎥
The firms share the Retailer’s revenues
and the Manufacturer’s costs
equally
71
Nash Bargaining Solution -- A Reduced Form Model
72
Nash Bargaining Solution
• Extensive form bargaining model – Intuitive
– But tedious
• Nash bargaining solution – Less intuitive
– But easier to find the same outcome
73
Nash Bargaining Solution
• Three steps 1. Describe bargaining situation
2. Define Nash product
3. Maximize Nash product
74
Nash Bargaining Solution
• Step 1: Describe bargaining situation 1. Who are the two players?
2. What contracts can they agree upon?
3. What payoff would they get from every possible contract?
4. What payoff do they have before agreement?
5. What is their relative patience (= bargaining power)
75
Nash Bargaining Solution Example 1: Bilateral monopoly
• Step 1: Describe the bargaining situation – Players: Manufacturer and Retailer
– Contracts: (T, q)
– Payoffs: • Retailer:
• Manufacturer:
– Payoff if there is no agreement (while negotiating) • Retailer:
• Manufacturer:
– Same patience => same bargaining power
!π R = 0
!πM = 0
π R T ,q( ) = V q( ) − TπM T ,q( ) = T − C q( )
T = total price for q units.
76
Nash Bargaining Solution Example 1: Bilateral monopoly
• Step 2: Set up Nash product
N T ,q( ) = π R T ,q( ) − !π R⎡⎣ ⎤⎦ ⋅ πM T ,q( ) − !πM⎡⎣ ⎤⎦
Retailer’s profit from contract Manufacturer’s profit from contract
77
Nash Bargaining Solution Example 1: Bilateral monopoly
• Step 2: Set up Nash product
N T ,q( ) = π R T ,q( ) − !π R⎡⎣ ⎤⎦ ⋅ πM T ,q( ) − !πM⎡⎣ ⎤⎦
Retailer’s extra profit from contract Manufacturer’s extra profit from contract
78
Nash Bargaining Solution Example 1: Bilateral monopoly
• Step 2: Set up Nash product
N T ,q( ) = π R T ,q( ) − !π R⎡⎣ ⎤⎦ ⋅ πM T ,q( ) − !πM⎡⎣ ⎤⎦
Nash product - Product of payoff increases
Depends on contract
79
Nash Bargaining Solution Example 1: Bilateral monopoly
• Step 2: Set up Nash product
N T ,q( ) = π R T ,q( ) − !π R⎡⎣ ⎤⎦ ⋅ πM T ,q( ) − !πM⎡⎣ ⎤⎦
Claim: The contract (T, q) maximizing N is the same contract that the parties would agree upon in an extensive form bargaining game!
80
Nash Bargaining Solution Example 1: Bilateral monopoly
• Step 2: Set up Nash product
N T ,q( ) = π R T ,q( ) − !π R⎡⎣ ⎤⎦ ⋅ πM T ,q( ) − !πM⎡⎣ ⎤⎦
N T ,q( ) = V q( ) − T⎡⎣ ⎤⎦ ⋅ T − C q( )⎡⎣ ⎤⎦
81
Nash Bargaining Solution Example 1: Bilateral monopoly
• Maximize Nash product
N T ,q( ) = V q( ) − T⎡⎣ ⎤⎦ ⋅ T − C q( )⎡⎣ ⎤⎦
∂N∂T
= − T − C q( )⎡⎣ ⎤⎦ + V q( ) − T⎡⎣ ⎤⎦ = 0 ⇒ T = 12 V q( ) + C q( )⎡⎣ ⎤⎦
⇒ p = 12V q( )q
+C q( )q
⎡
⎣⎢
⎤
⎦⎥
Equal profits = Equal split of surplus
82
Nash Bargaining Solution Example 1: Bilateral monopoly
• Maximize Nash product
N T ,q( ) = V q( ) − T⎡⎣ ⎤⎦ ⋅ T − C q( )⎡⎣ ⎤⎦
∂N∂T
= − T − C q( )⎡⎣ ⎤⎦ + V q( ) − T⎡⎣ ⎤⎦ = 0 ⇒ T = 12 V q( ) + C q( )⎡⎣ ⎤⎦
⇒ p = 12V q( )q
+C q( )q
⎡
⎣⎢
⎤
⎦⎥
83
Nash Bargaining Solution Example 1: Bilateral monopoly
• Maximize Nash product
N T ,q( ) = V q( ) − T⎡⎣ ⎤⎦ ⋅ T − C q( )⎡⎣ ⎤⎦
∂N∂T
= − T − C q( )⎡⎣ ⎤⎦ + V q( ) − T⎡⎣ ⎤⎦ = 0 ⇒ T = 12 V q( ) + C q( )⎡⎣ ⎤⎦
⇒ p = 12V q( )q
+C q( )q
⎡
⎣⎢
⎤
⎦⎥
Convert to price per unit.
84
Nash Bargaining Solution Example 1: Bilateral monopoly
• Maximize Nash product
N T ,q( ) = V q( ) − T⎡⎣ ⎤⎦ ⋅ T − C q( )⎡⎣ ⎤⎦
∂N∂T
= − T − C q( )⎡⎣ ⎤⎦ + V q( ) − T⎡⎣ ⎤⎦ = 0
∂N∂q
= V ' q( ) ⋅ T − C q( )⎡⎣ ⎤⎦ − C ' q( ) ⋅ V q( ) − T⎡⎣ ⎤⎦ = 0 ⇒ V ' q( ) = C ' q( )
85
Nash Bargaining Solution Example 1: Bilateral monopoly
• Maximize Nash product
N T ,q( ) = V q( ) − T⎡⎣ ⎤⎦ ⋅ T − C q( )⎡⎣ ⎤⎦
∂N∂T
= − T − C q( )⎡⎣ ⎤⎦ + V q( ) − T⎡⎣ ⎤⎦ = 0
∂N∂q
= V ' q( ) ⋅ T − C q( )⎡⎣ ⎤⎦ − C ' q( ) ⋅ V q( ) − T⎡⎣ ⎤⎦ = 0 ⇒ V ' q( ) = C ' q( )
Efficiency
86
Nash Bargaining Solution Example 1: Bilateral monopoly
• Conclusion – Maximizing Nash product is easy way to find equilibrium
– Efficient quantity
– Price splits surplus equally
87
Nash Bargaining Solution
• With different bargaining power
N T ,q( ) = π R T ,q( )− !π R⎡⎣ ⎤⎦β⋅ πM T ,q( )− !πM⎡⎣ ⎤⎦
1−β
Exponents determined by relative patience
88
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
Manufacturer
Retailer 1 Retailer 2
Consumers Consumers
89
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
Manufacturer
Retailer 1 Retailer 2
Consumers Consumers
π1 T1,q1( ) = V q1( ) − T1 π 2 T2 ,q2( ) = V q2( ) − T2
Retailers independent
90
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
Manufacturer
Retailer 1 Retailer 2
Consumers Consumers
π1 T1,q1( ) = V q1( ) − T1 π 2 T2 ,q2( ) = V q2( ) − T2
πM T1,T2 ,q1,q2( ) = T1 + T2 − C q1 + q2( )
91
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
• Step 1: Describe one of the bargaining situations – Contracts: (T1, q1)
92
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
• Step 1: Describe one of the bargaining situations – Contracts: (T1, q1)
– Payoffs: • Retailer:
• Manufacturer:
π1 T1,q1( ) = V q1( ) − T1πM T1,T2 ,q1,q2( ) = T1 + T2 − C q1 + q2( )
93
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
• Step 1: Describe one of the bargaining situations – Contracts: (T1, q1)
– Payoffs: • Retailer:
• Manufacturer:
– Payoff if no agreement • Retailer:
• Manufacturer: !π1 = 0
!πM = T2 − C q2( )
π1 T1,q1( ) = V q1( ) − T1πM T1,T2 ,q1,q2( ) = T1 + T2 − C q1 + q2( )
Manufacturer only supplies other retailer
94
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
• Step 1: Describe one of the bargaining situations – Contracts: (T1, q1)
– Payoffs: • Retailer:
• Manufacturer:
– Payoff while negotiating • Retailer:
• Manufacturer:
– Same patience => same bargaining power
!π1 = 0
!πM = T2 − C q2( )
π1 T1,q1( ) = V q1( ) − T1πM T1,T2 ,q1,q2( ) = T1 + T2 − C q1 + q2( )
Manufacturer only supplies other retailer
95
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
• Step 2: Set up Nash product
N T1,q1( ) = π R T1,q1( ) − !π R⎡⎣ ⎤⎦ ⋅ πM T1,T2 ,q1,q2( ) − !πM⎡⎣ ⎤⎦
N T1,q1( ) = V q1( ) − T1⎡⎣ ⎤⎦ ⋅ T1 + T2 − C q1 + q2( ){ } − T2 − C q2( ){ }⎡⎣ ⎤⎦
N T1,q1( ) = V q1( ) − T1⎡⎣ ⎤⎦ ⋅ T1 − C q1 + q2( ) − C q2( ){ }⎡⎣ ⎤⎦
Incremental cost
96
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
• Step 2: Set up Nash product
N T1,q1( ) = π R T1,q1( ) − !π R⎡⎣ ⎤⎦ ⋅ πM T1,T2 ,q1,q2( ) − !πM⎡⎣ ⎤⎦
N T1,q1( ) = V q1( ) − T1⎡⎣ ⎤⎦ ⋅ T1 + T2 − C q1 + q2( ){ } − T2 − C q2( ){ }⎡⎣ ⎤⎦
N T1,q1( ) = V q1( ) − T1⎡⎣ ⎤⎦ ⋅ T1 − C q1 + q2( ) − C q2( ){ }⎡⎣ ⎤⎦
Incremental cost
97
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
• Step 2: Set up Nash product
N T1,q1( ) = π R T1,q1( ) − !π R⎡⎣ ⎤⎦ ⋅ πM T1,T2 ,q1,q2( ) − !πM⎡⎣ ⎤⎦
N T1,q1( ) = V q1( ) − T1⎡⎣ ⎤⎦ ⋅ T1 + T2 − C q1 + q2( ){ } − T2 − C q2( ){ }⎡⎣ ⎤⎦
N T1,q1( ) = V q1( ) − T1⎡⎣ ⎤⎦ ⋅ T1 − C q1 + q2( ) − C q2( ){ }⎡⎣ ⎤⎦
Incremental cost
98
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
• Step 2: Set up Nash product
N T1,q1( ) = π R T1,q1( ) − !π R⎡⎣ ⎤⎦ ⋅ πM T1,T2 ,q1,q2( ) − !πM⎡⎣ ⎤⎦
N T1,q1( ) = V q1( ) − T1⎡⎣ ⎤⎦ ⋅ T1 + T2 − C q1 + q2( ){ } − T2 − C q2( ){ }⎡⎣ ⎤⎦
N T1,q1( ) = V q1( ) − T1⎡⎣ ⎤⎦ ⋅ T1 − C q1 + q2( ) − C q2( ){ }⎡⎣ ⎤⎦
Incremental cost
99
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
• Maximize Nash product
N T1,q1( ) = V q1( ) − T1⎡⎣ ⎤⎦ ⋅ T1 − C q1 + q2( ) − C q2( ){ }⎡⎣ ⎤⎦
∂N∂T1
= T1 − C q1 + q2( ) − C q2( ){ }⎡⎣ ⎤⎦ − V q1( ) − T1⎡⎣ ⎤⎦ = 0
⇒ T1 = 12 V q1( ) + C q1 + q2( ) − C q2( ){ }⎡⎣ ⎤⎦
⇒ p1 = 12
V q1( )q1
+C q1 + q2( ) − C q2( )
q1
⎡
⎣⎢
⎤
⎦⎥
Average incremental cost
100
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
p1 = 12 6 +
C q1 + q2( )−C q2( )q1
⎡
⎣⎢
⎤
⎦⎥
Example - V(q) = 6 q - C(q) = ½ q2
0 1 2 3 4 5 6 7 8 9 10
10
0
1
2
3
4
5
6
7
8
9
Quantity
€
MC(q)
MV(q)
Average incremental cost
101
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
0 1 2 3 4 5 6 7 8 9 10
10
0
1
2
3
4
5
6
7
8
9
Quantity
€
MC(q)
MV(q)
Incremental Cost
Units sold to other firm
Retailer 1buys 4 units
Example - Retailer 1 buys 4 units - Retailer 2 buys 2 units
p1 = 12 6 +
C q1 + q2( )−C q2( )q1
⎡
⎣⎢
⎤
⎦⎥ = 1
2 6 +164
⎡⎣⎢
⎤⎦⎥= 5
102
• Compare prices when retailer 1 buys more than 2
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
• Average incremental cost is lower for supplying 1!
103
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
0 1 2 3 4 5 6 7 8 9 10
10
0
1
2
3
4
5
6
7
8
9
Quantity
€
MC(q)
MV(q)
Incremental Cost
Units sold to other firm
Retailer 2buys 2 units
0 1 2 3 4 5 6 7 8 9 10
10
0
1
2
3
4
5
6
7
8
9
Quantity
€
MC(q)
MV(q)
Incremental Cost
Units sold to other firm
Retailer 1buys 4 units
• Retailer 1 will pay a lower price per unit
104
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
0 1 2 3 4 5 6 7 8 9 10
10
0
1
2
3
4
5
6
7
8
9
Quantity
€
MC(q)
MV(q)
Incremental Cost
Units sold to other firm
Retailer 2buys 2 units
0 1 2 3 4 5 6 7 8 9 10
10
0
1
2
3
4
5
6
7
8
9
Quantity
€
MC(q)
MV(q)
Incremental Cost
Units sold to other firm
Retailer 1buys 4 units
105
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
0 1 2 3 4 5 6 7 8 9 10
10
0
1
2
3
4
5
6
7
8
9
Quantity
€
MC(q)
MV(q)
Incremental Cost
Units sold to other firm
Retailer 2buys 2 units
0 1 2 3 4 5 6 7 8 9 10
10
0
1
2
3
4
5
6
7
8
9
Quantity
€
MC(q)
MV(q)
Incremental Cost
Units sold to other firm
Retailer 1buys 4 units
p1 = 12 6 +
C q1 + q2( )−C q2( )q1
⎡
⎣⎢
⎤
⎦⎥ = 1
2 6 +12 ⋅36 − 1
2 ⋅44
⎡⎣⎢
⎤⎦⎥= 1
2 6 +164
⎡⎣⎢
⎤⎦⎥= 5 p2 = 1
2 6 +C q1 + q2( )−C q1( )
q1
⎡
⎣⎢
⎤
⎦⎥ = 1
2 6 +12 ⋅36 − 1
2 ⋅162
⎡⎣⎢
⎤⎦⎥= 1
2 6 +102
⎡⎣⎢
⎤⎦⎥= 5.5
106
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
• Maximize Nash product
N T1,q1( ) = V q1( ) − T1⎡⎣ ⎤⎦ ⋅ T1 − C q1 + q2( ) − C q2( ){ }⎡⎣ ⎤⎦
∂N∂q1
= V ' q1( ) ⋅ T1 − C q1 + q2( ) − C q2( ){ }⎡⎣ ⎤⎦ + C ' q1 + q2( ) ⋅ V q1( ) − T1⎡⎣ ⎤⎦ = 0
⇒ V ' q1( ) = C ' q1 + q2( )
Bilateral Efficiency
107
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
• Maximize Nash product
V ' q1( ) = C ' q1 + q2( )
V ' q2( ) = C ' q1 + q2( )
⎫
⎬⎪⎪
⎭⎪⎪
⇒V ' q1( ) = V ' q2( )
Bilateral efficiency 1
Bilateral efficiency 2
108
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers
• Maximize Nash product
V ' q1( ) = C ' q1 + q2( )
V ' q2( ) = C ' q1 + q2( )
⎫
⎬⎪⎪
⎭⎪⎪
⇒V ' q1( ) = V ' q2( )
Overall market efficiency - Efficient distribution of goods
109
• Conclusions – Prices of homogenous goods may differ
• Quantity discounts if producer has increasing MC
– Market with bilateral market power may be efficient
Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers