EC476 Contracts and Organizations, Part III: Lecture 3 Leonardo Felli 32L.G.06 26 January 2015
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 2015 2 / 50
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 2015 3 / 50
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 2015 4 / 50
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 2015 5 / 50
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 p b < p s then no trade occurs. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 6 / 50
Bilateral Trade (4) ◮ The payoffs to both the seller and the buyer are then: ( p s + p b ) if p b ≥ p s u s ( p s , p b ; v s , v b ) = 2 v s if p b < p s and v b − ( p s + p b ) if p b ≥ p s u b ( p s , p b ; v s , v b ) = 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 2015 7 / 50
Seller’s Best Reply ◮ Consider now the seller’s best reply. ◮ This is defined by the following maximization problem: max E v b { u s ( p s , p b ; v s , v b ) | v s , p b ( v b ) } p s ◮ Consider now the seller’s payoff, substituting p b ( v b ) we have: ( p s + p b ( v b )) if p b ( v b ) ≥ p s u s = 2 v s if p b ( v b ) < p s Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 8 / 50
Seller’s Best Reply (2) ◮ or ( p s + p b ( v b )) if v b ≥ p − 1 b ( p s ) u s = 2 if v b < p − 1 v s b ( p s ) ◮ The seller’s maximization problem is then: � 1 � p − 1 ( p s ) ( p s + p b ( v b ) b max v s dv b + dv b 2 p s v b = p − 1 v b =0 ( p s ) b Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 9 / 50
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 ) ∂ G ( x , y ) + dx ∂ y α ( y ) Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 10 / 50
Seller’s Best Reply (4) ◮ Therefore the first order conditions are: dp − 1 � dp − 1 b ( p s ) b ( p s ) − 1 p s + p b ( p − 1 � v s b ( p s )) + dp s 2 dp s � 1 1 + 2 dv b = 0 p − 1 ( p s ) b ◮ or from p s = p b ( p − 1 b ( p s )): ( v s − p s ) dp − 1 b ( p s ) + 1 � 1 � v b ( p s ) = 0 p − 1 dp s 2 b ◮ which gives: ( v s − p s ) dp − 1 b ( p s ) + 1 1 − p − 1 � � b ( p s ) = 0 dp s 2 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 11 / 50
Buyer’s Best Reply ◮ The buyer’s best reply is instead defined by: max E v s { u b ( p s , p b ; v s , v b ) | v b , p s ( v s ) } p b ◮ Consider now the buyer’s payoff obtained substituting p s ( v s ): v b − ( p s ( v s ) + p b ) if v s ≤ p − 1 s ( p b ) u b = 2 if v s > p − 1 0 s ( p b ) ◮ we then get � p − 1 ( p b ) � v b − ( p s ( v s ) + p b ) � s max dv s 2 p b v s =0 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 12 / 50
Buyer’s Best Reply (2) ◮ Therefore the first order conditions are: � dp − 1 v b − ( p s ( p − 1 � s ( p b )) + p b ) s ( p b ) + 2 dp b � p − 1 ( p b ) − 1 s dv s = 0 2 v s =0 ◮ or [ v b − p b ] dp − 1 s ( p b ) − 1 � p − 1 ( p b ) � s v s = 0 0 dp b 2 ◮ which gives: ( v b − p b ) dp − 1 s ( p b ) − 1 2 p − 1 s ( p b ) = 0 dp b Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 13 / 50
Equilibrium Characterization ◮ To simplify notation we re-write p − 1 b ( · ) = q b ( · ) and p − 1 s ( · ) = q s ( · ). ◮ The two differential equations that define the best reply of the seller and the buyer are then: b ( p s ) + 1 [ q s ( p s ) − p s ] q ′ 2 [1 − q b ( p s )] = 0 s ( p b ) − 1 [ q b ( p b ) − p b ] q ′ 2 q s ( p b ) = 0 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 14 / 50
Equilibrium Characterization (2) ◮ Solving the second equation for q b ( p b ) and differentiating yields: � 3 − q s ( p b ) q ′′ � b ( p b ) = 1 s ( p b ) q ′ 2 [ q ′ s ( p b )] 2 ◮ Substituting this expression into the first differential equation we get: � 3 − q s ( p s ) q ′′ s ( p s ) � � 1 − p s − q s ( p s ) � [ q s ( p s ) − p s ] + = 0 s ( p s )] 2 [ q ′ 2 q ′ s ( p s ) Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 15 / 50
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 p b = 2 3 v b + 1 4 , 12 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 16 / 50
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 + 1 12 ≥ 2 3 v s + 1 4 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 17 / 50
Inefficient Trade In other words, in equilibrium trade occurs whenever: v b ≥ v s + 1 4 ✻ v b v s = v b ............................................................. . . 1 . . � � . . . . � � . . . . . � � . . . Trade . . � � . . . . � � . . . . . � � . . . . � � . . . . . � � . . . . � � . . . . v b = v s + 1 . � � . . . 4 . . . . . . ✻ . . . . . ✲ . . q (0 , 0) v s 1 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 26 January 2015 18 / 50
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 2015 19 / 50
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