Leonardo Felli 9 January, 2002 Topics in Contract Theory Lecture 3 Consider now a different cause for the failure of the Coase Theorem: the presence of transaction costs. Of course for this to be an interesting argument we need to abstract for other source of failure of the Coase Theorem such as asymmetric information. Strong version: (Nicholson 1989) the Coase theorem guarantees efficiency: regardless of the way in which property rights are assigned, and whenever the mutual gains from trade exceed the necessary transaction costs. 1
Topics in Contract Theory 2 This is not necessarily the case. The reason is the strategic role of transaction costs. Key factor: some transaction costs have to be paid ex-ante, before the negotiation starts. These ex-ante transaction costs identify an inefficiency usually known as a hold-up problem. The hold-up problem yield an outcome that is constrained inefficient.
Topics in Contract Theory 3 Numerical Example Potential Surplus = 100, Ex-ante cost to each negotiating party = 20, Distribution of bargaining power = (10%, 90%), Ex-ante Payoff to party A = (10% 100 20) = 10, Ex-ante Payoff to party B = (90% 100 20) = 70, Social surplus = 60. Coasian negotiation opportunity is left unexploited.
Topics in Contract Theory 4 Natural question: whether it is possible to find a Coasian solution to this inefficiency. In other words we are asking whether the parties can agree ex-ante on a transfer contingent on each party entering a future negotiation. We are going to show that under plausible conditions a Coasian solution of this form may not be available. The reason is that any new negotiation may itself be associated with (possibly small) ex-ante transaction costs.
Topics in Contract Theory 5 Numerical Example Party B makes a transfer to party A contingent on the cost of 20 being paid by A. Assume that B makes a take-it-or-leave-it offer to A, Ex-ante costs to each party associated with this agreement contingent on future negotiation = 2, A accepts the agreement if B accepts it and: 10% 100 20 + x 0, or x 10. There always exists an equilibrium in which x = 10, Ex-ante payoff to party A = 10% 100 20+10 2 = 2, No negotiation (contingent or not) will take place: a Coasian solution is not available.
Topics in Contract Theory 6 Ex-ante transaction costs: time to arrange a meeting, time and effort to conceive and agree upon a suitable negotiation language, time and effort to collect information about the legal environment in which the agreement is enforced, time to collect and analyze background information for the negotiation, time and effort to think about the negotiation at hand. These costs can be monetized through the hiring of an outside party (lawyer). Provided that the lawyer needs to be paid independently of the success of the negotiation, then monetizing the costs increases the magnitude of the inefficiency.
Topics in Contract Theory 7 Consider the following simple coasian negotiation : two agents, i {A, B}; share a surplus, size of the surplus normalized to one, parties payoffs in case of disagreement to zero. Ex-ante costs: (c A, c B ). Assume that (c A, c B ) are: complements: each party i has to pay c i for the negotiation to be feasible; affordable: party i s endowment covers c i ; efficient: c A + c B < 1.
Topics in Contract Theory 8 Timing: Simult. Decisions on (c A, c B ) Contract Negotiated Contract enforced t = 0 t = 1 t = 2 Let λ be the given bargaining power of A. The division of surplus at t = 1 is then (λ, 1 λ). Result 1. for any given λ there exists a pair (c A, c B ) of affordable and efficient ex-ante costs such that the unique SPE is (not pay c A, not pay c B )
Topics in Contract Theory 9 Result 2. for any pair (c A, c B ) of affordable and efficient ex-ante costs there exists a value of λ such that the unique SPE is (not pay c A, not pay c B ) The reduced form of the two stage game is: pay c B not pay c B pay c A λ c A, 1 λ c B c A, 0 not pay c A 0, c B 0, 0 A pays c A iff λ c A and B pays c B, A does not pay c A if B does not pay c B, B pays c B iff 1 λ c B and A pays c A, B does not pay c B if A does not pay c A. Therefore the result holds when λ < c A or when 1 λ < c B.
Topics in Contract Theory 10 Assume that (c A, c B ) are: substitutes: either party has to pay c i ; affordable: party i s endowment covers c i ; efficient: min{c A, c B } < 1. Result 3. Both results 1 and 2 (for one type of inefficiency) hold. The reduced form game is now: pay c B not pay c B pay c A λ c A, 1 λ c B λ c A, 1 λ not pay c A λ, 1 λ c B 0, 0 A pays c A iff λ c A and B does not pay c B, A does not pay c A if B pays c B, B pays c B iff 1 λ c B and A does not pay c A, B does not pay c B if A pays c A.
Topics in Contract Theory 11 We then have two types of inefficiencies: an inefficiency that leads to a unique SPE with no agreement: λ < c A, and (1 λ) < c B an inefficiency that leads to an agreement obtained paying too high a cost: if c A < c B, λ < c A, and (1 λ) > c B Results 1 and 2 also generalize to the case in which (c A, c B ) are substitutes and they are strategic complements.
Topics in Contract Theory 12 Consider the following extensive form: A and B decide, simultaneously and independently, whether to pay (c A, c B ); if both pay then the parties share the surplus according to the given λ and payoffs are: (λ c A, 1 λ c B ) If only one party i pays c i then he makes a takeit-or-leave-it offer l to i; this offer does not have to satisfy the (IR) constraint of party i: l [ ɛ, 1 + ξ], ɛ > 0, ξ > 0. Party i now can pay c i and see l or not pay c i. If he does not pay he must accept/reject blind. If neither party pays the no-agreement outcome is realized.
Topics in Contract Theory 13 Assume now that λ < c A. Result 4. The unique SPE of the model has neither party paying the ex-ante costs at t = 0 and hence the equilibrium outcome obtains. Proof: (key step) There exists no equilibrium in which only one party, say B, pays c B at t = 0. Assume by contradiction that this equilibrium exists. Then A can either pay c A or not pay. Suppose A does not pay c A and A accepts the offer. The optimal offer for B is then l = ɛ. A is strictly better off by rejecting. A rejects the offer. Then B is strictly better off by not paying c B. Suppose A pays c A. Whether he accepts or not he is better off by not paying and rejecting.
Topics in Contract Theory 14 The Impossibility of a Coasian Solution Is there a Coasian solution to the hold-up problem we have identified? Consider the following simple contingent agreement. A transfer σ B 0 (σ A 0) payable contingent on whether the other party decides to pay c A, (c B ). Key assumption: this new negotiation is associated with a fresh set of ex-ante costs (c 1 A, c1 B ); the two sets of ex-ante costs are assumed to be complements, affordable and efficient.
Topics in Contract Theory 15 Assume: λ < c A Simult. Decision Simult. Decision on (c 1 A, c1 B ) on (c A, c B ) A Accepts/Rejects B makes offer to A Tranfers Negotiation Contract enforced A/B does not pay 2 1 0 1 2 Result 5. There always exists a SPE of this game in which both agents pay neither the second tier, (c 1 A, c1 B ), nor the first tier, (c A, c B ), of ex-ante costs.
Topics in Contract Theory 16 Proof: At each stage the two agents decide simultaneously and independently whether to pay their ex-ante costs. An agreement is achieved only if both agents pay (c 1 A, c1 B ) and (c A, c B ). Either agent will never pay if he expects the other not to pay his ex-ante cost. Result 6. There always exists a SPE of this game in which both agents pay the second tier, (c 1 A, c1 B ), and the first tier, (c A, c B ), of ex-ante costs and an agreement is successfully negotiated. Proof: Assume that both parties have paid the exante costs (c 1 A, c1 B ) at t = 2 and party A has accepted the transfer σ B 0.
Topics in Contract Theory 17 The parties continuation game is then: pay c B not pay c B pay c A λ c A + σ B, 1 λ c B σ B c A, 0 not pay c A 0, c B 0, 0 It follows: A pays c A if B pays c B and λ + σ B > c A and B pays c B if A pays c A and 1 λ σ B > c B. Therefore if 1 λ c B > σ B > c A λ the subgame has two Pareto-ranked equilibria: one in which an agreement is successfully negotiated, an other one in which an agreement does not arise.
Topics in Contract Theory 18 Therefore there always exist a SPE of the model such that at t = 0 if λ + σ B c A + c 1 A the (constrained) efficient equilibrium is played. if λ + σ B < c A + c 1 A the no-agreement equilibrium is played. In this equilibrium necessarily σ B = c A + c 1 A λ Therefore the agreement is successfully negotiated. Observation: All equilibria of the model are constrained inefficient. Observation: the equilibrium described in Result 6 is not renegotiation-proof.
Topics in Contract Theory 19 Definition. a SPE of the model is renegotiationproof (RP) if and only if the equilibria played in every proper subgame are not strictly Pareto-dominated by any other equilibrium of the same subgame (Benoît and Krishna, 1993). Result 7. The unique RP SPE of the game involves both agents paying neither the second (c 1 A, c1 B ) nor the first (c A, c B ) tier of ex-ante costs. Proof: (intuition) In any RP equilibrium the costs (c 1 A, c1 B ) (if paid) are strategically sunk. Therefore in equilibrium σ B does not pay c 1 A. = c A λ. Hence, A Notice that if the parties pay the ex-ante costs sequentially, rather than simultaneously the SPE of the model is unique and satisfies Result 7.
Topics in Contract Theory 20 Continuous Costs Ex-ante costs are continuous (Holmström 1982). The more detailed the agreement is, the higher the surplus. The surplus is a monotonic and concave function of costs: x(c A, c B ) where 2 x(c A, c B ) c A c B > 0. Payoff to party A is: λ x(c A, c B ) c A while the payoff to B is: (1 λ)x(c A, c B ) c B.
Topics in Contract Theory 21 Result 8. for every λ and every (c A, c B ) affordable and efficient every equilibrium is such that c A < c E A c B < c E B. Proof: The equilibrium costs (c A, c B ) are such that: max c A λ x(c A, c B ) c A max c B (1 λ)x(c A, c B ) c B. The first order conditions of both these problems are: x(c A, c B ) c A = 1 λ and x(c A, c B ) c B = 1 1 λ The (constrained) efficient level of costs (c E A, ce B ) are such that: max c A,c B x(c A, c B ) c A c B
Topics in Contract Theory 22 The first order conditions of this problem are: x(c E A, ce B ) c A = 1 and x(ce A, ce B ) c B = 1 Concavity of x(, ) and the fact that: 2 x(c A, c B ) c A c B > 0. imply c A < c E A c B < c E B. One key different between this result and the one we found for discrete costs is that this result holds for every λ (0, 1).
Topics in Contract Theory 23 Alternating Offer Bargaining Infinite horizon, alternating offers bargaining game with discounting: two agents, i {A, B}; share a surplus, size of the surplus normalized to one, payoffs in case of disagreement are zero. We 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, (c A, c B ) the costs that need to be paid by the parties to negotiate in every one period.
Topics in Contract Theory 24 odd periods: Extensive form: Stage I both parties decide, simultaneously and independently, whether to pay (c A, c B ); if either or both parties do not pay the game moves to Stage I of the following period; Stage II if both parties pay, A makes an offer x A to B, Stage III 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.
Topics in Contract Theory 25 even periods: Stage I both parties decide, simultaneously and independently, whether to pay (c A, c B ); if either or both parties do not pay the game moves to Stage I of the following period; Stage II if both parties pay, B makes an offer x B to A, Stage III 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.
Topics in Contract Theory 26 Payoffs: If parties agree on x in period n + 1: Π A (σ A, σ B ) = δ n x C A (σ A, σ B ), Π B (σ A, σ B ) = δ n (1 x) C B (σ A, σ B ), or if they do not agree: Π i (σ A, σ B ) = C i (σ A, σ B ). Result 9. Whatever the values of δ i and c i for i {A, B}, there exists an SPE of the game in which neither player pays his participation cost in any period, and therefore an agreement is never reached. Proof: By construction: in Stage I parties do not pay their costs; in Stage II party i demands the entire surplus (x A = 1, x B = 0); in Stage III party i accepts any offer x [0, 1].
Topics in Contract Theory 27 Result 10. The game has an SPE in which an agreement is reached in finite time if and only if δ i and c i for i {A, B} satisfy δ A (1 c A c B ) c A and δ B (1 c A c B ) c B First type of inefficiency: c B 1 δ B 1 + δ B SPE δ A 1 + δ A 1 c A
Topics in Contract Theory 28 Proof: only if: Notice that i {A, B}: c A x i 1 c B moreover: x H B δ ( ) A x H A c A 1 x L A δ ( ) B 1 x L B c B by substitution we get: δ A (1 c A c B ) c A δ B (1 c A c B ) c B if: we construct a SPE where x A = 1 c B, x B = c A
Topics in Contract Theory 29 in every period players pay their costs; odd periods: the offer is 1 c B and B accepts any x 1 c B ; even periods: the offer is c A and A accepts any x c A ; if either player does not pay the cost then we switch to (0, 0); if either player rejects an offer he is supposed to accept we switch to (0, 0). Notice that A cannot gain by accepting an offer x < c A since, by waiting until the next period, he gets δ A (1 c B c A ) c A The same is true for B from (1).
Assume that: Topics in Contract Theory 30 δ A (1 c A c B ) c A δ B (1 c A c B ) c B Result 11. There exists an SPE of the A subgames in which x A is agreed immediately, if an only if x A [1 δ B (1 c A c B ), 1 c B ] (1) There also exists an SPE of the B subgames in which x B is agreed immediately, if and only if x B [c A, δ A (1 c A c B )] (2) However there also exists a whole huge set of inefficient SPE of the game.
Topics in Contract Theory 31 Result 12. Consider any x A as in (1) and choose any odd number n. Then there exists an SPE of the A subgames with (continuation) payoffs Π A = δa n(x A c A ) Π B = δb n (1 x A c B ) (3) Moreover, let any x B as in (2) and choose any even number n. Then there exists an SPE of the A subgames with (continuation) payoffs Π A = δa n(x B c A ) Π B = δb n (1 x B c B ) (4) The symmetric result holds for B subgames. Second inefficiency: When δ A (1 c A c B ) c A and δ B (1 c A c B ) c B there exist both efficient and inefficient equilibria with arbitrarily long delays.
Topics in Contract Theory 32 Robustness Finite horizon version of the game Γ N : Remark 1. Let any finite N 1 be given. Then the unique SPE outcome of Γ N is neither player pays his participation cost in any period and hence agreement is never reached. Sequential payment of costs: Define Γ S the bargaining game with the following: Stage I player i first decides whether to pay c i, the other player j observes i n s decision and decides whether to pay c j ;
Topics in Contract Theory 33 Remark 2. The game Γ S always has an SPE in which neither player ever pays his participation cost, and hence agreement is never reached. Robustness of the first inefficiency to a random turn in making offers. Assume that Stage I of the game is modified so that: in odd periods A makes offer x OA with probability p while B makes offer x OB with probability (1 p); in even periods B makes offer x EB with probability p while A makes offer x EA with probability (1 p).
Topics in Contract Theory 34 Result 13. This new game has an SPE in which an agreement is reached in finite time if and only if δ i and c i for i {A, B} are such that: c A p and c B p and p δ B (1 c A c B ) c B (1 p) and q δ A (1 c A c B ) c A (1 q); or c A p and c B (1 p) or c B p and c A (1 p).
If p > 1 2 then: Topics in Contract Theory 35 c A 1 p (1 p)... SPE. (1 p) p 1 c B
If p = 1 2 then: Topics in Contract Theory 36 c A p = 1 2 1 SPE p = 1 2 1 c B
Topics in Contract Theory 37 Selecting Efficient Equilibria Natural question: since the players bargain with complete information will they find a way to agree to play the efficient equilibrium? In other words will the players renegotiate out of inefficient equilibria? First approach to renegotiation: in a Coasian fashion we attempt simply to select the efficient equilibria. Minimal consistency requirement: it should be done in every proper subgame. Definition 1. An SPE (σ A, σ B ) is Consistently Pareto Efficient (CPE SPE) if and only if it yields a Pareto-efficient outcome in every possible subgame.
Topics in Contract Theory 38 Result 14. The set of CPE SPE for this game is empty. Proof: Assume there exists an CPE SPE. By Definition 1, agreement must be immediate in every period (subgame). Adapting Shaked and Sutton (1984), we obtain: and x A = x H A = x L A = 1 δ B + δ A c A 1 δ A δ B x B = x H B = x L B = δ A[1 δ B (1 c B ) c A ] 1 δ A δ B The two equalities above imply that: x B = δ A (x A c A ) (5) and 1 x A = δ B (1 x B c B ) (6)
Topics in Contract Theory 39 Both A and B need to be willing to pay their costs in Stage I of every period. This implies: x B c A δ A (x A c A ) (7) and 1 x A c B δ B (1 x B c B ) (8) a contradiction of (5) and (6). There is no way to select consistently the efficient outcomes in the game. The only robust equilibrium of this bargaining game with transaction costs is the one in which an extreme inefficiency is reached: no agreement. This is obviously a serious failure of the Coase Theorem.