EC487 Advanced Microeconomics, Part I: Lecture 9

EC487 Advanced Microeconomics, Part I: Lecture 9 Leonardo Felli 32L.LG.04 24 November 2017

Bargaining Games: Recall Two players, i {A, B} are trying to share a surplus. The size of the surplus is normalized to 1. Payoffs to the players in case of disagreement are normalized to 0. Denote (δ A, δ B ) the parties discount factors: 0 δ i 1, i {A, B}; Denote: x the share of the pie to party A; (1 x) the share of the pie to party B. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 2 / 65

T periods alternating offers bargaining Assume for simplicity that δ A = δ B = δ. Assume now that the game lasts for T periods where T is even. We compute the Subgame Perfect Equilibrium of the bargaining game. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 3 / 65

T periods alternating offers bargaining (cont d) Extensive form: Odd periods n T 1: Stage I : A makes an offer x A to B; Stage II : B observes the offer and can accept or reject it; If the offer is accepted then x = x A and the game terminates and the players payoffs are: Π A (x A, y) = δ n 1 x A Π B (x A, y) = δ n 1 (1 x A ) If the offer is rejected the game moves to Stage I of the following period. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 4 / 65

T periods alternating offers bargaining (cont d) Even periods τ < T : Stage I : B makes an offer x B to A; Stage II : A observes the offer and can accept or reject it; If the offer is accepted then x = x B and the game terminates and the players payoffs are: Π A (x B, y) = δ τ 1 x B Π B (x B, y) = δ τ 1 (1 x B ) If the offer is rejected the game moves to Stage I of the following period. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 5 / 65

T periods alternating offers bargaining (cont d) Period T : Stage I : B makes an offer x B to A; Stage II : A observes the offer and can accept or reject it; If the offer is accepted then x = x B and the game terminates and the players payoffs are: Π A (x B, y) = δ T 1 x B Π B (x B, y) = δ T 1 (1 x B ) If the offer is rejected the game ends and the players payoffs are: Π A (x B, n) = 0 Π B (x B, n) = 0. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 6 / 65

T periods alternating offers bargaining (cont d) Notice that: In the T -th period (even) B makes an offer and the size of the pie is δ T 1. Player B makes a take-it-or-leave-it offer: Π A = 0 Π B = δ T 1. In the (T 1)-th period (odd) A makes an offer and the share of the pie that will not be available any more next period is (δ T 2 δ T 1 ). Player A makes a take-it-or-leave-it offer on this share: Π A = 0 + (δ T 2 δ T 1 ) Π B = δ T 1 + 0. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 7 / 65

T periods alternating offers bargaining (cont d). In the second period (even) B makes an offer and the share of the pie that will not be available any more next period is (δ δ 2 ). Player B makes a take-it-or-leave-it offer on this share: Π A = 0 + (δ T 2 δ T 1 ) +... + 0 Π B = δ T 1 + 0 +... + (δ δ 2 ). In the first period (odd) A makes an offer and the share of the pie that will not be available any more next period is (1 δ). Player A makes a take-it-or-leave-it offer on this share: Π A = (δ T 2 δ T 1 ) +... + 0 + (1 δ) Π B = δ T 1 +... + (δ δ 2 ) + 0. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 8 / 65

T periods alternating offers bargaining (cont d) In other words, the SPE payoffs are: Π A = (1 δ) + (δ 2 δ 3 ) +... + (δ T 2 δ T 1 ) Π B = (δ δ 2 ) + (δ 3 δ 4 ) +... + (δ T 3 δ T 2 ) + δ T 1. We can re-write both payoffs as: Π A = (1 δ) (1 + δ 2 + δ 4 +... + δ T 2 ) Π B = (1 δ) δ (1 + δ 2 + δ 4 +... + δ T 4 ) + δ T 1. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 9 / 65

T periods alternating offers bargaining (cont d) Recall that: 1 + a + a 2 + a 3 +... + a n = 1 an+1 1 a Let now a = δ 2 then we have: (1 + δ 2 + δ 4 +... + δ T 2 ) = 1 (δ2 ) ( T 2 2 +1) 1 δ 2 = 1 δt 1 δ 2 (1 + δ 2 + δ 4 +... + δ T 4 ) = 1 (δ2 ) ( T 4 2 +1) 1 δ 2 = 1 δt 2 1 δ 2 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 10 / 65

T periods alternating offers bargaining (cont d) The SPE payoffs are then: Π A = (1 δ) 1 δt 1 δ 2 Π B = (1 δ) δ 1 δt 2 1 δ 2 + δ T 1. Recall also that (1 δ 2 ) = (1 δ) (1 + δ). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 11 / 65

T periods alternating offers bargaining (cont d) When T is even agreement is reached in the first period and the SPE payoffs are: Π A = 1 δt 1 + δ Π B = 1 Π A = δ + δt 1 + δ Strategies for t T remaining periods are: If t is odd A offers share x A = 1 δt 1 + δ ; B accepts any share x (1 δt ) 1 + δ If t is even: B offers share x A = δ δt 1 + δ ; A accepts any share x (δ δt ) 1 + δ and rejects x > (1 δt ) 1 + δ and rejects x < (δ δt ) 1 + δ Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 12 / 65

T periods alternating offers bargaining (cont d) When T is odd agreement is reached in the first period and the SPE payoffs are: Π A = 1 + δt 1 + δ Π B = δ δt 1 + δ Strategies for t T remaining periods are: If t is odd: A offers share x A = 1 + δt 1 + δ ; B accepts any share x (1 + δt ) 1 + δ If t is even: B offers share x A = δ + δt 1 + δ ; A accepts any share x (δ + δt ) 1 + δ and rejects x > (1 + δt ) 1 + δ and rejects x < (δ + δt ) 1 + δ Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 13 / 65

Alternating offers bargaining horizon (Rubinstein 1982): Consider the general case where δ A and δ B might differ. Extensive form: Odd periods: Stage I : A makes an offer x A to B; Stage II : B observes the offer and can accept or reject it; If the offer is accepted then x = x A and the game terminates; If the offer is rejected the game moves to Stage I of the following period. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 14 / 65

Alternating offers bargaining horizon (cont d) Even periods: Stage I : B makes an offer x B to A; Stage II : A observes the offer and can accept or reject it; If the offer is accepted then x = x B and the game terminates; If the offer is rejected the game moves to Stage I of the following period. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 15 / 65

Alternating offers bargaining horizon (cont d) Payoffs: If parties agree on x in period n: Π A (σ A, σ B ) = δ n 1 A x, or if they do not agree: Π B (σ A, σ B ) = δ n 1 B (1 x), Π i (σ A, σ B ) = 0 i {A, B}. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 16 / 65

Alternating offers bargaining horizon (cont d) Nash equilibria: any share of the surplus x [0, 1]. Strategies: A offers share xa = x [0, 1] in the first period; B accepts any share x x and rejects any share x > x in the first period; player i offers share xi = x, for i {A, B} in any other period; player i accepts any offer x i, i {A, B} such that x A x or x B x and rejects all other offers in every other period. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 17 / 65

Alternating offers bargaining horizon (cont d) Subgame Perfect Equilibrium Outcome: Agreement is reached in the first period and the equilibrium payoffs are: Π A = 1 δ B 1 δ A δ B and Π B = δ B (1 δ A ) 1 δ A δ B Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 18 / 65

Alternating offers bargaining horizon (cont d) Strategies: A always offers share xa = 1 δ B 1 δ A δ B in odd periods; B accepts any share x (1 δ B) 1 δ A δ B and rejects any share x > (1 δ B) 1 δ A δ B in odd periods; B offers share xb = δ A (1 δ B ) 1 δ A δ B in even periods; A accepts any offer x δ A (1 δ B ) 1 δ A δ B and rejects any share x < δ A (1 δ B ) 1 δ A δ B in even periods. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 19 / 65

Alternating offers bargaining horizon (cont d) Proof: (Shaked and Sutton 1984) Notice that the game is stationary: the continuation game starting at every odd period is the same, the continuation game starting in every even period also looks identical. Denote xi H the highest equilibrium share player A can get in a subgame starting in a period in which player i {A, B} makes an offer. Denote xi L the lowest equilibrium share player A can get in a subgame starting in a period in which player i {A, B} makes the offer. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 20 / 65

Alternating offers bargaining horizon (cont d) Then: x H B δ A x H A 1 x L A δ B ( ) 1 xb L moreover: x L B δ A x L A 1 x H A δ B ( ) 1 xb H Substituting we get: 1 x L A δ B 1 x H A δ B ( ) 1 δ A xa L ( ) 1 δ A xa H Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 21 / 65

Alternating offers bargaining horizon (cont d) and x L B δ A ( )] [1 δ B 1 xb L x H B δ A From these four inequalities we get: ( )] [1 δ B 1 xb H x L A 1 δ B 1 δ A δ B x H A and x L B δ A[1 δ B ] 1 δ A δ B x H B Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 22 / 65

Alternating offers bargaining horizon (cont d) These inequalities are therefore satisfied with equality: x A = 1 δ B 1 δ A δ B and x B = δ A[1 δ B ] 1 δ A δ B These are the offers in odd and even periods respectively. We also get: and x B = δ A x A 1 x A = δ B (1 x B ) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 23 / 65

Robustness of Rubinstein s SPE We here identify the sense in which the unique SPE equilibrium of the Rubinstein s game is a robust and meaningful solution. Let us consider the Rubinstein s equilibrium payoffs in the case: δ A = δ B = δ: Π A = 1 1 + δ, Π B = δ 1 + δ. (1) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 24 / 65

Robustness of Rubinstein s SPE (cont d) Recall also the equilibrium payoffs in the case, seen above, where the alternating offer bargaining game terminates in T (even) periods: Π A = 1 δt 1 + δ (2) Π B = δ + δt 1 + δ = (1 δt 2 ) 1 δ T δ 1 δt 1 + δ + δt 1 (3) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 25 / 65

Robustness of Rubinstein s SPE (cont d) The payoffs of the two games coincide in the limit T. Indeed 1 δ T lim T 1 + δ = 1 1 + δ δ + δ T lim T 1 + δ = δ 1 + δ Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 26 / 65

Robustness of Rubinstein s SPE (cont d) Recall Π A = 1 δt 1 + δ Π B = (δ ) 1 δt 2 1 δ T 1 δ T 1 + δ + δt 1 The last term of Π B, δ T 1 can be interpreted as the last mover advantage. Clearly the last mover advantage converges to zero as T increases. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 27 / 65

Robustness of Rubinstein s SPE (cont d) The coefficient of Π B δ (1 δt 2 ) 1 δ T can be interpreted as the first mover advantage, (second mover disadvantage). To understand this coefficient better we need to consider a version of Rubinstein s game that eliminates asymptotically such an advantage. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 28 / 65

Robustness of Rubinstein s SPE (cont d) Denote t the length of the time period that lapses between each two offers. The two discount factors are then: δ A = exp{ r A t} δ B = exp{ r B t} Taylor expansion implies that in a neighborhood of t = 0 we have that: δ A (1 r A t) δ B (1 r B t) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 29 / 65

Robustness of Rubinstein s SPE (cont d) Substituting we get that in a neighborhood of t = 0: Π A = r B t 1 (1 r A t)(1 r B t) = r B r A + r B r A r B t Π B = r A t (1 r B t) 1 (1 r A t)(1 r B t) = r A r A r B t r A + r B r A r B t Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 30 / 65

Robustness of Rubinstein s SPE (cont d) If we now consider the limit of the above payoffs for t 0 we get: Π A = r B r A + r B Π B = r A r A + r B Notice that if r A = r B = r we get: Π A = 1 2, Π B = 1 2 This is clearly an intuitive result in the case there are no first mover or last mover advantages. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 31 / 65

Bargaining Power Notice that each party s bargaining power is determined by: each party s discount factor δ i or interest rate r i, the extensive form of the bargaining game: first mover, last mover advantage etc... Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 32 / 65

Coase Theorem What could parties achieve in an economic environment in which they can costlessly negotiate a contractual agreement? Theorem (Coase Theorem: Coase (1960)) In an economy where ownership rights are well defined and transacting is costless gains from trade will be exploited (a contract will be agreed upon) and efficiency achieved whatever the distribution of entitlements. That is rational agents negotiate agreements that are individually rational and Pareto efficient. An agreement is individually rational if each contracting party is not worse off by deciding to sign the contract rather then choosing not to sign it. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 33 / 65

Freedom of Contract This is the reflection of a basic principle of a well functioning legal system known as: freedom of contract. This is equivalent to assume that the action space of the contracting parties always contains the option not to sign the contract. A contract is Pareto efficient if there does not exist an other feasible contract that makes at least one of the contracting party strictly better off without making any other contracting party worse off. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 34 / 65

A Model of Production Externality Consider the following simple model of a production externality. Consider two parties, labelled A and B. Party A generates revenue R A (e A ) (strictly concave) by choosing the input e A at a linear cost c e A (c > 0). A s payoff function is then: Π A (e A ) = R A (e A ) c e A Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 35 / 65

A Model of Production Externality (cont d) Party B generates revenue R B (e B ) (strictly concave) by choosing the input e B at the linear cost c e B (c > 0). Party B also suffers from an externality γ e A (γ > 0) imposed by A on B. B s payoff function is then: Π B (e B ) γ e A = R B (e B ) c e B γ e A Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 36 / 65

Social Efficient Outcome Consider first the social efficient amounts of input e A and e B. These solve the Central Planner s problem: max e A,e B Π A (e A ) + Π B (e B ) γ e A In other words (ea, e B ) are such that: R A (e A ) = c + γ R B (e B ) = c Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 37 / 65

No Agreement Outcome Assume now that parties choose the amounts of input e A and e B simultaneously and independently. Party A s problem: max e A Π A (e A ) Party B s problem: max e B Π B (e B ) γ e A In equilibrium the inputs chosen (ê A, ê B ) are: R A (ê A) = c, R B (ê B) = c Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 38 / 65

Gains form Trade Comparing (ê A, ê B ) and (ea, e B ) we obtain using concavity of R A ( ): eb = ê B, ea < ê A In other words: [Π A (e A ) + Π B(e B ) γ e A ] [Π A(ê A ) + Π B (ê B ) γ ê A ] = = [Π A (e A ) Π A(ê A )] + γ (ê A e A ) > 0 The joint surplus is reduced by the inefficiency generated by the externality. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 39 / 65

Gains form Trade (cont d) Assume now that the two contracting parties get together and agree on a contract before the amounts of input are chosen. A reduction of input e A from ê A to e A generates: a decrease in the net revenues from A s technology: Π A (e A) < Π A (ê A ) reduction in the negative externality γ e A < γ ê A and the former effect is more than compensated by the latter one γ (ê A e A ) > [Π A(ê A ) Π A (e A )] Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 40 / 65

Negotiation and Ownership Rights This may create room for negotiation. For simplicity normalize to 1 the total size of the surplus that is available to share between the two contracting parties (parties negotiate on which percentage of the surplus accrues to each one). To establish a well defined negotiation ownership rights need to be specified. Entitlements/ownership rights define the outside option of each party to the contract. In other words they define the payoff each party is entitled to without need for the other party to agree. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 41 / 65

Bargaining Denote w A and w B the entitlements of party A, respectively B where: w A + w B < 1. In general, the Coase Theorem is stated without a specific reference to the extensive form of the costless negotiation between the two parties. In what follows we will show the result for two examples of a bargaining game with outside options. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 42 / 65

Bargaining (cont d) Once again denote: δ the parties common discount factor, x the share of the pie to party A, (1 x) the share of the pie to party B. w A and w B each party s outside option. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 43 / 65

Take-it-or-leave-it offer by Party A Extensive form: A makes an offer x [0, 1] to B; B observes the offer x and decides whether to accept or reject it. If the offer is accepted the game ends and the players payoffs are: P A = x [Π A (e A ) + Π B(e B ) γ e A ], P B = (1 x)[π A (e A ) + Π B(e B ) γ e A ] If the offer is rejected the game ends and the players payoffs are: P A = w A [Π A (e A ) + Π B(e B ) γ e A ], P B = w B [Π A (e A ) + Π B(e B ) γ e A ] Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 44 / 65

Take-it-or-leave-it offer by Party A (cont d) Subgame Perfect Equilibria Outcome: Shares: x = 1 w B (1 x) = w B SPE Strategies: A offers share 1 x = wb ; B accepts any share 1 x w B. Proof: backward induction Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 45 / 65

Take-it-or-leave-it offer by Party A (cont d) The Payoffs associated with this equilibrium agreement are then: P A = (1 w B ) [Π A (e A ) + Π B(e B ) γ e A ], Clearly, efficiency applies: P B = w B [Π A (e A ) + Π B(e B ) γ e A ] P A + P B = [Π A (e A ) + Π B(e B ) γ e A ] In other words input choices are efficient (e A, e B ). The ownership rights/entitlements of player B determine the shares of the two parties. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 46 / 65

Two Periods Alternating Offers Period 1: Stage I : A makes an offer x A to B, Stage II : B observes the offer and has three alternatives: he can accept the offer, then x = x A and the game terminates; he can reject the offer and take his outside option w B and the game terminates; he can reject the offer and not take his outside option, then the game moves to Stage I of the following period. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 47 / 65

Two Periods Alternating Offers (cont d) Period 2: Stage I : B makes an offer x B to A, Stage II : A observes the offer and has two alternative choices: he can accept the offer, then x = x B and the game terminates; he can reject the offer and take his outside option w A and the game terminates. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 48 / 65

Two Periods Alternating Offers, Equilibrium Subgame Perfect Equilibrium Outcome: Agreement is reached in the first period with payoffs: (1 max{w B, δ(1 w A )}, max{w B, δ(1 w A )}) SPE Strategies: A offers share 1 xa = max{w B, δ(1 w A )} in period 1; B accepts any share 1 x max{w B, δ(1 w A )} in period 1; B rejects any share 1 x < max{w B, δ(1 w A )} in period 1; B offers share x B = w A in the period 2; A accepts any share x w A in the period 2. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 49 / 65

Efficiency and Ownership Rights Notice that an efficient agreement is reached in all cases independently of the size of the entitlements (w A, w B ). Clearly in all cases the result above implies that we would get the efficient outcome: (e A, e B ). However, the share that accrues to each party depends on the entitlements w A and w B. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 50 / 65

Comments If each party is entitled to the choice of his input, then: w A = Π A (ê A ) Π A (e A ) + Π B(e B ) γ e A w B = Π B (ê B ) γ ê A Π A (e A ) + Π B(e B ) γ e A It is important to recall that the parties need to agree to the contract before choosing the investments (e A, e B ). The Coasian contract specifies: their choice of investments (ea, e B ) and the transfers that the parties have to make to each other. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 51 / 65

Costly Bargaining Recall that the Coase Theorem explicitly stated that bargaining was costless. Let us now instead consider bargaining when there exists an opportunity cost of time: spending the period bargaining has a positive cost. Assume that this costs are (c A, c B ). If either party decides not to pay the cost than negotiation breaks down since either party does not spend the time at the negotiation table. Assume w A = w B = 0 and discount factors are δ A and δ B. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 52 / 65

Costly Bargaining (cont d) Consider first the take-it-or-leave-it offer: Stage 0 both parties decide, simultaneously and independently, whether to pay (c A, c B ); if either or both parties do not pay the game ends; Stage I if both parties pay, A makes an offer x A to B, Stage II B observes the offer and can accept or reject it; if the offer is accepted then x = x A and the game terminates; if the offer is rejected the game ends. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 53 / 65

Costly Bargaining (cont d) Payoffs: if parties agree on x: Π A = x c A Π B = (1 x c B ), if they do not agree and both pay the costs : Π A = c A, Π B = c B, if only one party, i {A, B} pays the cost: Π i = c i, Π i = 0. if neither party pays the cost: Π A = 0, Π B = 0. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 54 / 65

Costly Bargaining (cont d) We solve for the SPE using backward induction. Consider Stages I and II : The SPE of these stages is A offers xa = 1 and B accepts. Payoffs associated with the SPE of this subgame: (1, 0). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 55 / 65

Costly Bargaining (cont d) Consider now Stage 0. The transformed game is described by the following normal form: pay c B do not pay c B pay c A 1 c A, c B c A, 0 do not pay c A 0, c B 0, 0 The unique NE of this game is: (do not pay c A, do not pay c B ). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 56 / 65

Costly Bargaining (cont d) Therefore the unique SPE of the entire game is: Strategies: A does not pay ca ; A offers xa = 1; B does not pay cb ; B accepts any offer x A 1. The outcome is then a payoff of (0, 0). Notice that this result applies for every (even very small) costs c A 0 and c B > 0. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 57 / 65

Costly Bargaining (cont d) Notice that we can choose c A + c B < 1 which means that is efficient to reach an agreement, but agreement is not an equilibrium of the game. In other words, this is a failure of the Coase Theorem. The question is then whether it is the take-it-or-leave-it nature of the game that generates this very inefficient outcome. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 58 / 65

Two-Periods bargaining with Transaction Costs First period: Stage 0 both parties decide, simultaneously and independently, whether to pay (c A, c B ); if either or both parties do not pay the game ends; Stage I if both parties pay, A makes an offer x A to B, Stage II B observes the offer and can accept or reject it; if the offer is accepted then x = x A and the game terminates; if the offer is rejected the game moves to Stage I of the following period. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 59 / 65

Two-Periods bargaining with Transaction Costs (cont d) Second period: Stage I if both parties pay, B makes an offer x B to A, Stage II A observes the offer and can accept or reject it; if the offer is accepted then x = x B and the game terminates; if the offer is rejected the game ends. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 60 / 65

Two-Periods bargaining with Transaction Costs (cont d) Payoffs: if parties agree on x in period n: Π A = δ n 1 A x c A Π B = δ n 1 B (1 x) c B, if they do not agree and both pay the costs: Π A = c A, Π B = c B, if only one party, i {A, B} pays the cost: Π i = c i, Π i = 0. If neither party pays the cost: Π A = 0, Π B = 0. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 61 / 65

Two-Periods bargaining with Transaction Costs (cont d) We solve for the SPE using backward induction. Consider the second period and Stages I and II of the first period: This is a familiar subgame and the SPE of these stages is: A offers x A = 1 δ B in period 1; B accepts any offer x 1 δ B and rejects any offer x > 1 δ B in period 1; B offers xb = 0 in period 2; A accepts any offer x 0 in period 2. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 62 / 65

Two-Periods bargaining with Transaction Costs (cont d) Payoffs associated with the SPE of this subgame are: (1 δ B, δ B ). Consider now Stage 0 of period 1. The transformed game is described by the following normal form: pay c B do not pay c B pay c A 1 δ B c A, δ B c B c A, 0 do not pay c A 0, c B 0, 0 The Nash equilibrium of this game will depend on whether (1 δ B ) c A and δ B c B. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 63 / 65

Two-Periods bargaining with Transaction Costs (cont d) In particular the unique SPE if δ B > 1 c A or δ B < c B is such that: No player i {A, B} pays his cost ci ; A offers xa = 1 δ B in period 1; B accepts any offer x 1 δ B and rejects any offer x > 1 δ B in period 1; B offers xb = 0; A accepts any offer x 0. The outcome is a payoff for both players of (0, 0). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 64 / 65

Two-Periods bargaining with Transaction Costs (cont d) Notice that the equilibrium we just described may be inefficient. Indeed, we just envisage a situation where the strong version of the Coase Theorem fails. This corresponds to the case: c A + c B < 1 δ B > 1 c A or δ B < c B The inefficiency does not depend on the take-it-or-leave-it nature of the game. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 24 November 2017 65 / 65