EC476 Contracts and Organizations, Part III: Lecture 3

Transcription

1 EC476 Contracts and Organizations, Part III: Lecture 3 Leonardo Felli 32L.G January 2015

2 Failure of the Coase Theorem Recall that the Coase Theorem implies that two parties, when faced with a potential externality, if they can costlessly trade and ownership rights are well defined, will be able to achieve efficiency. We are now going to consider a first environment in which the Theorem fails. This is a situation where parties bargain under bilateral asymmetric information (with no externalities). We will show that in this situation efficiency cannot be achieved. Clearly, since efficiency cannot be achieved even in the absence of externalities efficiency is not achievable when externalities are present. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

3 Bilateral Asymmetric Information Recall that when proving at the Coase Theorem we had to focus on a specific extensive form of the parties negotiation. In general, this does not imply that we cannot find an extensive form that will achieve efficiency. Fortunately an other fundamental principle of contract theory will help in this case: Revelation Principle. Using the revelation principle we will be able to conclude that efficiency cannot be achieved whatever bargaining protocol governs the negotiation between the two parties under bilateral asymmetric information. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

4 Bilateral Trade (Chatterjee and Samuelson, 1983): Consider the following simple model of bilateral trade (double auction). Two players, a buyer and a seller: N = {b, s}. The seller names an asking price: p s. The buyer names an offer price: p b. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

5 Bilateral Trade (2) The action spaces: A s = {p s 0}, A b = {p b 0}. The seller owns and attaches value v s to an indivisible unit of a good. The buyer attaches value v b to the unit of the good and is willing to pay up to v b for it. The valuations for the unit of the good of the seller and the buyer are their private information of each player. Player i {b, s} believes that the valuation of the opponent v i takes values in the unit interval. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

6 Bilateral Trade (3) The type spaces: T s = {0 v s 1}, T b = {0 v b 1} Player i {b, s} also believes that the valuation of the opponent is uniformly distributed on [0, 1]: µ s = 1, µ b = 1. The extensive form of the game is such that: If p b p s then they trade at the average price: p = (p s + p b ). 2 If pb < p s then no trade occurs. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

7 Bilateral Trade (4) The payoffs to both the seller and the buyer are then: (p s + p b ) if p u s (p s, p b ; v s, v b ) = b p s 2 v s if p b < p s and v u b (p s, p b ; v s, v b ) = b (p s + p b ) if p b p s 2 0 if p b < p s Players strategies: p s (v s ) and p b (v b ). We consider strictly monotonic and differentiable strategies. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

8 Seller s Best Reply Consider now the seller s best reply. This is defined by the following maximization problem: max p s E vb {u s (p s, p b ; v s, v b ) v s, p b (v b )} Consider now the seller s payoff, substituting p b (v b ) we have: (p s + p b (v b )) if p u s = b (v b ) p s 2 v s if p b (v b ) < p s Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

9 Seller s Best Reply (2) or (p s + p b (v b )) u s = 2 v s if v b p 1 b (p s) if v b < p 1 b (p s) The seller s maximization problem is then: max p s p 1 b (ps) v b =0 1 (p s + p b (v b ) v s dv b + dv b v b =p 1 b (ps) 2 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

10 Seller s Best Reply (3) Recall that by Leibniz s rule: ( ) β(y) G(x, y)dx y α(y) = = G(β(y), y) β (y) G(α(y), y)α (y) + + β(y) α(y) G(x, y) y dx Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

11 Seller s Best Reply (4) Therefore the first order conditions are: v s dp 1 b (p s) dp s 1 2 [ ps + p b (p 1 b or from p s = p b (p 1 b (p s)): which gives: (p s)) ] dp 1 b (p s) + dp s + 1 p 1 b (ps) 1 2 dv b = 0 (v s p s ) dp 1 b (p s) + 1 [ ] 1 vb dp s 2 p 1 b (ps) = 0 (v s p s ) dp 1 b (p s) + 1 [ 1 p 1 dp s 2 b (p s) ] = 0 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

12 Buyer s Best Reply The buyer s best reply is instead defined by: max p b E vs {u b (p s, p b ; v s, v b ) v b, p s (v s )} Consider now the buyer s payoff obtained substituting p s (v s ): v u b = b (p s(v s ) + p b ) if v s ps 1 (p b ) 2 0 if v s > ps 1 (p b ) we then get max p b p 1 v s=0 s (p b ) [ v b (p ] s(v s ) + p b ) dv s 2 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

13 Buyer s Best Reply (2) Therefore the first order conditions are: [v b (p s(ps 1 ] (p b )) + p b ) dp 1 s (p b ) + 2 dp b 1 2 p 1 s (p b ) v s=0 dv s = 0 or [v b p b ] dp 1 s (p b ) 1 [ ] p 1 s (p b ) vs = 0 dp b 2 0 which gives: (v b p b ) dp 1 s (p b ) 1 dp b 2 p 1 s (p b ) = 0 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

14 Equilibrium Characterization To simplify notation we re-write p 1 b ( ) = q b( ) and ps 1 ( ) = q s ( ). The two differential equations that define the best reply of the seller and the buyer are then: [q s (p s ) p s ] q b (p s) [1 q b(p s )] = 0 [q b (p b ) p b ] q s(p b ) 1 2 q s(p b ) = 0 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

15 Equilibrium Characterization (2) Solving the second equation for q b (p b ) and differentiating yields: q b (p b) = 1 [3 q s(p b )q ] s (p b ) 2 [q s(p b )] 2 Substituting this expression into the first differential equation we get: [q s (p s ) p s ] [3 q s(p s )q s ] [ (p s ) [q s(p s )] p s q ] s(p s ) 2q s(p = 0 s ) Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

16 Equilibrium Characterization (3) This is a second-order differential equation in q s ( ) that has a two-parameter family of solutions. The simplest family of solution takes the form: q s (p s ) = α p s + β Then the values α = 3/2 and β = 3/8 solve the second-order differential equation. The definition of q s ( ) and q b ( ) imply that: p s = 2 3 v s + 1 4, p b = 2 3 v b Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

17 Equilibrium Characterization (4) This is the (unique) Bayesian Nash equilibrium of this game. Notice now that it is efficient to trade whenever: v b v s However in this double auction game trade occurs whenever: p b p s or 2 3 v b v s Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

18 Inefficient Trade In other words, in equilibrium trade occurs whenever: v b v s v b v s = v b 1... Trade v b = v s (0, 0) 1 v s Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

19 Bilateral Trade (Myerson and Satterthwaite 1983): The key question is whether the inefficiency we found in the double auction can be eliminated by choosing a different trading mechanism. The revelation principle provides the right tool to get an answer to the question above. Obviously, there is no principal, but the two parties at an ex-ante stage, before they learn their private information act as the mechanism designer. Assume further that this is a pure bilateral contract transfers cannot involve a third party. In jargon the mechanism has to be budget balancing. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

20 Unavoidable Inefficincy Theorem (Myerson and Satterthwaite 1983) When parties to a contract bargain in a situation of bilateral asymmetric information efficiency of trade cannot be achieved. There always exist configurations of the parameter values that yield no trade when trade is efficient. The reason is that the inefficiency is necessary to induce separation of the different types of buyer and seller. In other words, the incentive compatibility constraints of the parties imply that efficiency is no longer guaranteed. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

21 Transaction Costs 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 from other sources of failure of the Coase Theorem such as asymmetric information. Theorem (Strong version of the Coase Theorem) The Coase theorem guarantees efficiency: 1. regardless of the way in which property rights are assigned, and 2. whenever the mutual gains from trade exceed the necessary transaction costs. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

22 Transaction Costs (2) We are going to show that 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 generate an inefficiency usually known as a hold-up problem. The hold-up problem yield an outcome that is constrained inefficient. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

23 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% ) = 10, Ex-ante Payoff to party B = (90% ) = 70, Social surplus = 60. Coasian negotiation opportunity is left unexploited. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

24 Coasian Solution 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. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

25 Coasian Solution (2) In the Numerical Example above 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 = 1, A accepts this transfer if: or x % x 0, Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

26 Coasian Solution (3) There always exists an equilibrium in which x = 10. Ex-ante payoff to party A π A = 10% = 1 No negotiation (contingent or not) will take place. A Coasian solution is not available. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

27 Examples of Transaction Costs Ex-ante transaction costs: time to arrange and/or participate in 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. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

28 Examples of Transaction Costs (2) These costs can be monetized through the hiring of an outside party: typically a lawyer. The problem does not disappear if the lawyer needs to be paid independently of the success of the negotiation: no contingent fees. Indeed, monetizing the costs may increase the magnitude of the inefficiency: moral hazard. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

29 Simple Coasian Negotiation 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. Each party faces ex-ante costs: (c A, c B ). Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

30 Complement Transaction Costs 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. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

31 Complement Transaction Costs (2) Timing: Simult. Decisions on (c A, c B ) Contract Negotiated Contract enforced t = 0 t = 1 t = 2 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

32 Complement Transaction Costs (3) We solve the model backward. We start with a simple bargaining rule: Let λ be the bargaining power of A. The division of surplus at t = 1 is then (λ, 1 λ). Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

33 Complement Transaction Costs (4) Result 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 ) Result 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 ) Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

34 Complement Transaction Costs (5) Proof: 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 1 λ < c B. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

35 Substitute Transaction Costs Assume now 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 Both results above hold. In the second result only one type of inefficiency may occur. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

36 Substitute Transaction Costs (2) Proof: 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. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

37 Substitute Transaction Costs (3) 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 strategic complements. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

38 The Impossibility of a Coasian Solution Is there a Coasian solution to this hold-up problem? 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. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

39 A Potential Coasian Solution Assume: λ < c A Simult. Decision Simult. Decision on (ca 1, c1 B ) on (c A, c B ) A Accepts/Rejects B makes offer Tranfers to A Negotiation Contract enforced A/B does not pay Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

40 A Potential Coasian Solution (2) Result 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. 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. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

41 A Potential Coasian Solution (3) Result 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 ex-ante costs (ca 1, c1 B ) at t = 2 and party A has accepted the transfer σ B 0. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

42 A Potential Coasian Solution (4) 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. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

43 A Potential Coasian Solution (5) 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. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

44 A Potential Coasian Solution (6) We can then construct 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. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

45 A Potential Coasian Solution (7) In this equilibrium necessarily σ B = c A + c 1 A λ Therefore the agreement is successfully negotiated. Observations: All equilibria of the model are constrained inefficient: costs paid are inefficiently high. The equilibrium described in last result is not renegotiation-proof. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

46 Continuous Costs Consider now the case with continuous ex-ante costs (Holmström 1982): the more detailed the agreement is, the higher the surplus. The surplus is a monotonic and concave function of costs: the costs are complements: x(c A, c B ) 2 x(c A, c B ) c A c B > 0. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

47 Continuous Costs (2) 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. Result Given λ, every equilibrium is such that c A < ce A c B < ce B. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

48 Continuous Costs (3) Proof: The equilibrium costs (ca, 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 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

49 Continuous Costs (4) The (constrained) efficient level of costs (ca E, ce B ) are such that: max c A,c B x(c A, c B ) c A c B The first order conditions of this problem are: x(c E A, ce B ) c A = 1 and x(c E A, ce B ) c B = 1 Concavity of x(, ) and the fact that: 2 x(c A, c B ) c A c B > 0. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50

50 More Pervasive Inefficiency imply c A < ce A c B < ce B. One key difference between this result and the one we found for discrete costs is that this result holds for every λ (0, 1). In other words, when costs are continuous the inefficiency is more pervasive. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January / 50