Bilateral Market Power & Bargaining Johan Stennek 1 Dont need - PowerPoint PPT Presentation

T rounds Time Bidder π B π S Resp. T S 0 1 yes T-1 B 1- δ δ yes T-2 S δ - δ 2 1- δ + δ 2 yes T-3 B 1- δ + δ 2 - δ 3 δ - δ 2 + δ 3 yes T-4 S δ - δ 2 + δ 3 - δ 4 1- δ + δ 2 - δ 3 + δ 4 yes … … … … … 1 S yes δ - δ 2 + δ 3 - δ 4 + … - δ T-1 1- δ + δ 2 - δ 3 + δ 4 - … + δ T-1 49

T rounds Time Bidder π B π S Resp. T S 0 1 yes T-1 B 1- δ δ yes T-2 S δ - δ 2 1- δ + δ 2 yes T-3 B 1- δ + δ 2 - δ 3 δ - δ 2 + δ 3 yes T-4 S δ - δ 2 + δ 3 - δ 4 1- δ + δ 2 - δ 3 + δ 4 yes … … … … … 1 S yes δ - δ 2 + δ 3 - δ 4 + … - δ T-1 1- δ + δ 2 - δ 3 + δ 4 - … + δ T-1 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 50

T rounds Geometric series 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 S's share π S = 1 − δ + δ 2 − δ 3 + δ 4 − ... + δ T − 1 52

T rounds S's share π S = 1 − δ + δ 2 − δ 3 + δ 4 − ... + δ T − 1 Multiply δπ S = δ − δ 2 + δ 3 − δ 4 + δ 5 − ... + δ T 53

T rounds S's share π S = 1 − δ + δ 2 − δ 3 + δ 4 − ... + δ T − 1 Multiply δπ S = δ − δ 2 + δ 3 − δ 4 + δ 5 − ... + δ T Add π S + δπ S = 1 + δ T 54

T rounds S's share π S = 1 − δ + δ 2 − δ 3 + δ 4 − ... + δ T − 1 Multiply δπ S = δ − δ 2 + δ 3 − δ 4 + δ 5 − ... + δ T Add π S + δπ S = 1 + δ T Solve π S = 1 + δ T 1 + δ 55

T rounds Equilibrium shares with T periods ( ) 1 π S = 1 + δ 1 + δ T δ ( ) π B = 1 + δ 1 − δ T − 1 56

T rounds Equilibrium shares with T periods ( ) 1 π S = 1 + δ 1 + δ T δ ( ) π B = 1 + δ 1 − δ T − 1 S has advantage of making last bid 1 + δ T > 1 − δ T − 1 To confirm this, solve model where - B makes last bid - S makes first bid 57

T rounds Equilibrium shares with T periods ( ) 1 π S = 1 + δ 1 + δ T δ ( ) π B = 1 + δ 1 − δ T − 1 S has advantage of making last bid 1 + δ T > 1 − δ T − 1 Disappears if T very large 58

T rounds Equilibrium shares with T ≈ ∞ periods 1 π S = 1 + δ δ π B = 1 + δ S has advantage of making first bid δ 1 1 + δ > 1 + δ To confirm this, solve model where - B makes first bid 59

T rounds Equilibrium shares with T ≈ ∞ periods 1 π S = 1 + δ δ π B = 1 + δ S has advantage of making first bid δ 1 1 + δ > 1 + δ To confirm this, solve model where - B makes first bid 60

T rounds Equilibrium shares with T ≈ ∞ periods 1 π S = 1 + δ δ π B = 1 + δ S has advantage of making first bid δ 1 1 + δ > First bidder’s advantage disappears if δ ≈ 1 1 + δ 61

T rounds Equilibrium shares with T ≈ ∞ periods and very patient players ( δ ≈ 1) π S = 1 2 π B = 1 2 62

Difference in Patience Equilibrium shares with T ≈ ∞ periods and different discount factors π S = 1 − δ B 1 − δ S δ B π B = 1 − δ S δ B 1 − δ S δ B (Easy to show using same method as above) 63

Difference in Patience • Recall δ i = e − r i Δ – r i = continous-time discount factor – Δ = length of time period • Then, as Δ è 0: π S = 1 − δ B r B ≈ – 1 − δ S δ B S + r B r – Using l’Hopital’s rule 64

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) 65

Implications for Bilateral Monopoly 66

Implications for Bilateral Monopoly • Equal splitting Π S = Π B ( ) = V q ( ) − p ⋅ q p ⋅ q − C q ( ) + C q ( ) 2 ⋅ p ⋅ q = V q ( ) ( ) ⎡ ⎤ p = 1 V q + C q ⎢ ⎥ ⎣ ⎦ 2 q q 67

Implications for Bilateral Monopoly • Equal splitting Π S = Π B ( ) = V q ( ) − p ⋅ q p ⋅ q − C q Retailer’s average revenues ( ) + C q ( ) 2 ⋅ p ⋅ q = V q ( ) ( ) ⎡ ⎤ p = 1 V q + C q ⎢ ⎥ ⎣ ⎦ 2 q q 68

Implications for Bilateral Monopoly • Equal splitting Π S = Π B ( ) = V q ( ) − p ⋅ q p ⋅ q − C q Manufacturer’s average costs ( ) + C q ( ) 2 ⋅ p ⋅ q = V q ( ) ( ) ⎡ ⎤ p = 1 V q + C q ⎢ ⎥ ⎣ ⎦ 2 q q 69

Implications for Bilateral Monopoly • Equal splitting Π S = Π B The firms share ( ) = V q ( ) − p ⋅ q the Retailer’s revenues p ⋅ q − C q and the Manufacturer’s costs ( ) + C q ( ) 2 ⋅ p ⋅ q = V q equally ( ) ( ) ⎡ ⎤ p = 1 V q + C q ⎢ ⎥ ⎣ ⎦ 2 q q 70

Nash Bargaining Solution -- A Reduced Form Model 71

Nash Bargaining Solution • Extensive form bargaining model – Intuitive – But tedious • Nash bargaining solution – Less intuitive – But easier to find the same outcome 72

Nash Bargaining Solution • Three steps 1. Describe bargaining situation 2. Define Nash product 3. Maximize Nash product 73

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) 74

Nash Bargaining Solution Example 1: Bilateral monopoly • Step 1: Describe the bargaining situation – Players: Manufacturer and Retailer – Contracts: (T, q) T = total price for q units. – Payoffs: ( ) = V q ( ) − T π R T , q • Retailer: ( ) = T − C q ( ) π M T , q • Manufacturer: – Payoff if there is no agreement (while negotiating) π R = 0 ! • Retailer: π M = 0 ! • Manufacturer: – Same patience => same bargaining power 75

Nash Bargaining Solution Example 1: Bilateral monopoly • Step 2: Set up Nash product ( ) = π R T , q ( ) − ! ( ) − ! ⎡ π R ⎤ ⎦ ⋅ π M T , q ⎡ π M ⎤ N T , q ⎣ ⎣ ⎦ Retailer’s profit from contract Manufacturer’s profit from contract 76

Nash Bargaining Solution Example 1: Bilateral monopoly • Step 2: Set up Nash product ( ) = π R T , q ( ) − ! ( ) − ! ⎡ π R ⎤ ⎦ ⋅ π M T , q ⎡ π M ⎤ N T , q ⎣ ⎣ ⎦ Retailer’s extra profit from contract Manufacturer’s extra profit from contract 77

Nash Bargaining Solution Example 1: Bilateral monopoly • Step 2: Set up Nash product ( ) = π R T , q ( ) − ! ( ) − ! ⎡ π R ⎤ ⎦ ⋅ π M T , q ⎡ π M ⎤ N T , q ⎣ ⎣ ⎦ Nash product - Product of payoff increases Depends on contract 78

Nash Bargaining Solution Example 1: Bilateral monopoly • Step 2: Set up Nash product ( ) = π R T , q ( ) − ! ( ) − ! ⎡ π R ⎤ ⎦ ⋅ π M T , q ⎡ π M ⎤ N T , q ⎣ ⎣ ⎦ Claim: The contract (T, q) maximizing N is the same contract that the parties would agree upon in an extensive form bargaining game! 79

Nash Bargaining Solution Example 1: B ilateral monopoly • Step 2: Set up Nash product ( ) = π R T , q ( ) − ! ( ) − ! π R ⎦ ⋅ π M T , q π M ⎡ ⎤ ⎡ ⎤ N T , q ⎣ ⎣ ⎦ ( ) = V q ( ) − T ( ) ⎦ ⋅ T − C q ⎡ ⎤ ⎡ ⎤ N T , q ⎣ ⎣ ⎦ 80

Nash Bargaining Solution Example 1: Bilateral monopoly • Maximize Nash product ( ) = V q ( ) − T ( ) ⎡ ⎤ ⎦ ⋅ T − C q ⎡ ⎤ N T , q ⎣ ⎣ ⎦ ∂ N ( ) ( ) − T ( ) + C q ( ) ∂ T = − T − C q ⎦ + V q ⎦ = 0 ⇒ T = 1 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 2 V q ⎣ ⎣ ⎣ ⎦ ( ) ( ) ⎡ ⎤ V q + C q ⇒ p = 1 ⎢ ⎥ 2 ⎣ ⎦ q q Equal profits = Equal split of surplus 81

Nash Bargaining Solution Example 1: Bilateral monopoly • Maximize Nash product ( ) = V q ( ) − T ( ) ⎡ ⎤ ⎦ ⋅ T − C q ⎡ ⎤ N T , q ⎣ ⎣ ⎦ ∂ N ( ) ( ) − T ( ) + C q ( ) ∂ T = − T − C q ⎦ + V q ⎦ = 0 ⇒ T = 1 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 2 V q ⎣ ⎣ ⎣ ⎦ ( ) ( ) ⎡ ⎤ V q + C q ⇒ p = 1 ⎢ ⎥ 2 ⎣ ⎦ q q 82

Nash Bargaining Solution Example 1: Bilateral monopoly • Maximize Nash product ( ) = V q ( ) − T ( ) ⎡ ⎤ ⎦ ⋅ T − C q ⎡ ⎤ N T , q ⎣ ⎣ ⎦ ∂ N ( ) ( ) − T ( ) + C q ( ) ∂ T = − T − C q ⎦ + V q ⎦ = 0 ⇒ T = 1 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 2 V q ⎣ ⎣ ⎣ ⎦ ( ) ( ) ⎡ ⎤ V q + C q ⇒ p = 1 ⎢ ⎥ 2 ⎣ ⎦ q q Convert to price per unit. 83

Nash Bargaining Solution Example 1: Bilateral monopoly • Maximize Nash product ( ) = V q ( ) − T ( ) ⎡ ⎤ ⎦ ⋅ T − C q ⎡ ⎤ N T , q ⎣ ⎣ ⎦ ∂ N ( ) ( ) − T ∂ T = − T − C q ⎦ + V q ⎦ = 0 ⎡ ⎤ ⎡ ⎤ ⎣ ⎣ ∂ N ( ) ⋅ T − C q ( ) ( ) ⋅ V q ( ) − T ( ) = C ' q ( ) ∂ q = V ' q ⎦ − C ' q ⎦ = 0 ⇒ ⎡ ⎤ ⎡ ⎤ V ' q ⎣ ⎣ 84

Nash Bargaining Solution Example 1: Bilateral monopoly • Maximize Nash product ( ) = V q ( ) − T ( ) ⎡ ⎤ ⎦ ⋅ T − C q ⎡ ⎤ N T , q ⎣ ⎣ ⎦ ∂ N ( ) ( ) − T ∂ T = − T − C q ⎦ + V q ⎦ = 0 ⎡ ⎤ ⎡ ⎤ ⎣ ⎣ ∂ N ( ) ⋅ T − C q ( ) ( ) ⋅ V q ( ) − T ( ) = C ' q ( ) ∂ q = V ' q ⎦ − C ' q ⎦ = 0 ⇒ ⎡ ⎤ ⎡ ⎤ V ' q ⎣ ⎣ Efficiency 85

Nash Bargaining Solution Example 1: Bilateral monopoly • Conclusion – Maximizing Nash product is easy way to find equilibrium – Efficient quantity – Price splits surplus equally 86

Nash Bargaining Solution • With different bargaining power β ⋅ π M T , q ( ) = π R T , q ( ) − ! ( ) − ! 1 − β ⎡ π R ⎤ ⎡ π M ⎤ N T , q ⎣ ⎦ ⎣ ⎦ Exponents determined by relative patience 87

Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers Manufacturer Retailer 1 Retailer 2 Consumers Consumers 88

Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers Retailers independent Manufacturer ( ) = V q 1 ( ) − T 1 ( ) = V q 2 ( ) − T 2 π 1 T 1 , q 1 π 2 T 2 , q 2 Retailer 1 Retailer 2 Consumers Consumers 89

Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers ( ) = T 1 + T 2 − C q 1 + q 2 ( ) π M T 1 , T 2 , q 1 , q 2 Manufacturer ( ) = V q 1 ( ) − T 1 ( ) = V q 2 ( ) − T 2 π 1 T 1 , q 1 π 2 T 2 , q 2 Retailer 1 Retailer 2 Consumers Consumers 90

Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers • Step 1: Describe one of the bargaining situations – Contracts: (T 1 , q 1 ) 91

Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers • Step 1: Describe one of the bargaining situations – Contracts: (T 1 , q 1 ) – Payoffs: ( ) = V q 1 ( ) − T 1 π 1 T 1 , q 1 • Retailer: ( ) = T 1 + T 2 − C q 1 + q 2 ( ) π M T 1 , T 2 , q 1 , q 2 • Manufacturer: 92

Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers • Step 1: Describe one of the bargaining situations – Contracts: (T 1 , q 1 ) – Payoffs: ( ) = V q 1 ( ) − T 1 π 1 T 1 , q 1 • Retailer: ( ) = T 1 + T 2 − C q 1 + q 2 ( ) π M T 1 , T 2 , q 1 , q 2 • Manufacturer: – Payoff if no agreement Manufacturer only supplies π 1 = 0 ! • Retailer: other retailer ( ) π M = T 2 − C q 2 ! • Manufacturer: 93

Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers • Step 1: Describe one of the bargaining situations – Contracts: (T 1 , q 1 ) – Payoffs: ( ) = V q 1 ( ) − T 1 π 1 T 1 , q 1 • Retailer: ( ) = T 1 + T 2 − C q 1 + q 2 ( ) π M T 1 , T 2 , q 1 , q 2 • Manufacturer: – Payoff while negotiating Manufacturer only supplies π 1 = 0 ! • Retailer: other retailer ( ) π M = T 2 − C q 2 ! • Manufacturer: – Same patience => same bargaining power 94

Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers • Step 2: Set up Nash product ( ) = π R T 1 , q 1 ( ) − ! ( ) − ! ⎡ π R ⎤ ⎦ ⋅ π M T 1 , T 2 , q 1 , q 2 ⎡ π M ⎤ N T 1 , q 1 ⎣ ⎣ ⎦ { } − T 2 − C q 2 { } ( ) = V q 1 ( ) − T 1 ( ) ( ) ⎡ ⎤ ⎡ ⎤ ⎦ ⋅ T 1 + T 2 − C q 1 + q 2 N T 1 , q 1 ⎣ ⎣ ⎦ { } ( ) = V q 1 ( ) − T 1 ( ) − C q 2 ( ) ⎡ ⎤ ⎡ ⎤ ⎦ ⋅ T 1 − C q 1 + q 2 N T 1 , q 1 ⎣ ⎣ ⎦ Incremental cost 95

Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers • Step 2: Set up Nash product ( ) = π R T 1 , q 1 ( ) − ! ( ) − ! ⎡ π R ⎤ ⎦ ⋅ π M T 1 , T 2 , q 1 , q 2 ⎡ π M ⎤ N T 1 , q 1 ⎣ ⎣ ⎦ { } − T 2 − C q 2 { } ( ) = V q 1 ( ) − T 1 ( ) ( ) ⎡ ⎤ ⎡ ⎤ ⎦ ⋅ T 1 + T 2 − C q 1 + q 2 N T 1 , q 1 ⎣ ⎣ ⎦ { } ( ) = V q 1 ( ) − T 1 ( ) − C q 2 ( ) ⎡ ⎤ ⎡ ⎤ ⎦ ⋅ T 1 − C q 1 + q 2 N T 1 , q 1 ⎣ ⎣ ⎦ Incremental cost 96

Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers • Step 2: Set up Nash product ( ) = π R T 1 , q 1 ( ) − ! ( ) − ! ⎡ π R ⎤ ⎦ ⋅ π M T 1 , T 2 , q 1 , q 2 ⎡ π M ⎤ N T 1 , q 1 ⎣ ⎣ ⎦ { } − T 2 − C q 2 { } ( ) = V q 1 ( ) − T 1 ( ) ( ) ⎡ ⎤ ⎡ ⎤ ⎦ ⋅ T 1 + T 2 − C q 1 + q 2 N T 1 , q 1 ⎣ ⎣ ⎦ { } ( ) = V q 1 ( ) − T 1 ( ) − C q 2 ( ) ⎡ ⎤ ⎡ ⎤ ⎦ ⋅ T 1 − C q 1 + q 2 N T 1 , q 1 ⎣ ⎣ ⎦ Incremental cost 97

Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers • Step 2: Set up Nash product ( ) = π R T 1 , q 1 ( ) − ! ( ) − ! ⎡ π R ⎤ ⎦ ⋅ π M T 1 , T 2 , q 1 , q 2 ⎡ π M ⎤ N T 1 , q 1 ⎣ ⎣ ⎦ { } − T 2 − C q 2 { } ( ) = V q 1 ( ) − T 1 ( ) ( ) ⎡ ⎤ ⎡ ⎤ ⎦ ⋅ T 1 + T 2 − C q 1 + q 2 N T 1 , q 1 ⎣ ⎣ ⎦ { } ( ) = V q 1 ( ) − T 1 ( ) − C q 2 ( ) ⎡ ⎤ ⎡ ⎤ ⎦ ⋅ T 1 − C q 1 + q 2 N T 1 , q 1 ⎣ ⎣ ⎦ Incremental cost 98

Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers • Maximize Nash product { } ( ) = V q 1 ( ) − T 1 ( ) − C q 2 ( ) ⎡ ⎤ ⎡ ⎦ ⋅ T 1 − C q 1 + q 2 ⎤ N T 1 , q 1 ⎣ ⎣ ⎦ ∂ N { } ( ) − C q 2 ( ) ( ) − T 1 ⎡ ⎤ ⎡ ⎤ = T 1 − C q 1 + q 2 ⎦ − V q 1 ⎦ = 0 ⎣ ⎣ ∂ T 1 { } ( ) + C q 1 + q 2 ( ) − C q 2 ( ) ⎡ ⎤ ⇒ T 1 = 1 2 V q 1 ⎣ ⎦ ( ) ( ) − C q 2 ( ) ⎡ ⎤ + C q 1 + q 2 V q 1 ⇒ p 1 = 1 ⎢ ⎥ 2 ⎣ q 1 q 1 ⎦ Average incremental cost 99

Nash Bargaining Solution Example 2: Bilateral oligopoly with two retailers € 10 Example MC(q) 9 - V(q) = 6 q 8 - C(q) = ½ q 2 7 MV(q) 6 5 ( ) − C q 2 ( ) 2 6 + C q 1 + q 2 ⎡ ⎤ p 1 = 1 ⎢ ⎥ 4 ⎣ q 1 ⎦ 3 2 Average incremental cost 1 0 0 1 2 3 4 5 6 7 8 9 10 Quantity 100