EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 - PowerPoint PPT Presentation

EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 Leonardo Felli CLM.G.4 22 November 2011

The Hold-Up Problem (Hart and Moore 1988) A buyer and seller want to trade one indivisible unit of a good at a future date. Denote q ∈ { 0 , 1 } the probability that trade occurs and p the trading price. Let v denote the buyer’s valuation for the good, and c the seller’s production cost. We assume that v and c are uncertain. Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 2 / 63

The Hold-Up Problem (2) In particular, v is such that: v ∈ { v , v } , v < v , Pr { v = v } = j The buyer can increase j by undertaking an ex-ante investment at the (strictly convex) costs ψ ( j ). The buyer’s ex-ante payoff is then: v q − p − ψ ( j ) Moreover, c is such that: c ∈ { c , c } , c < c , Pr { c = c } = i The seller can increase i by undertaking an ex-ante investment at the (strictly convex) costs φ ( i ). The seller’s ex-ante payoff is then: p − c q − φ ( i ) Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 3 / 63

The Hold-Up Problem (3) Assume that: c > v > c > v Timing: The parties write an ex-ante contract . The parties choose simultaneously their investments ( i , j ). Both parties learn the state of nature ( v , c ). The parties renegotiate the ex-ante contract if they want to. Trade may or may not occur. Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 4 / 63

The Hold-Up Problem (4) Notice that the gains-from-trade are positive only if v = v and c = c therefore ex-post efficiency requires: � 1 if v = v and c = c q ∗ = 0 otherwise Given this trading rule ex-ante efficiency requires that the ex-ante investments are such that: max i j ( v − c ) − φ ( i ) − ψ ( j ) i , j The first order conditions imply: i ∗ ( v − c ) = ψ ′ ( j ∗ ) j ∗ ( v − c ) = φ ′ ( i ∗ ) Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 5 / 63

The Hold-Up Problem (5) Assume now that ( v , c ) and ( i , j ) are observable but not verifiable to the parties to the contract . In other words, the parties ex-ante cannot write a contract contingent on ( v , c ) and ( i , j ) . That is the ex-ante contract can only specify a price (transfer) contingent on whether q ∈ { 0 , 1 } (a price for not trading and a price for trading): ( p 0 , p 1 ) The court can only assess whether trade occurred (the court cannot assess who did not perform ex-post). This is the incomplete contract assumption . Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 6 / 63

The Hold-Up Problem (6) Of course, once the state of nature ( v , c ) is realized the two parties can renegotiate, if they want, the terms of the ex-ante contract ( p 0 , p 1 ). Notice that at the renegotiation stage the contract incompleteness does not play any role: no need to contract on ( v , c ) and ( i , j ). Therefore Coase Theorem applies and this negotiation is efficient. Assume that the following extensive form applies to the renegotiation stage, known as contracting at will (double auction) . Both parties simultaneously and independently send one another new written offers: ( p B 0 , p B ( p S 0 , p S 1 ) , 1 ) Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 7 / 63

The Hold-Up Problem (7) Then both parties decide simultaneously and independently whether to approve the trade. Only if both approve then trade occurs: q = 1 (the court cannot force the parties to trade, freedom of contracts). The court observes q and enforces: if q = 1 the payment p 1 or any other payment p k 1 , k ∈ { B , S } , (dated more recently than the ex-ante contract) that either party received from the counterpart and is willing to show the court; if q = 0 the payment p 0 or any other payment p k 0 , k ∈ { B , S } , (dated more recently than the ex-ante contract) that either party received from the counterpart and is willing to show the court. Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 8 / 63

The Hold-Up Problem (8) Let (ˆ p 1 , ˆ p 0 ) be the enforced contract (ex-ante one or renegotiated). The buyer approves the trade if and only if: v − ˆ p 1 ≥ − ˆ v ≥ ˆ p 1 − ˆ p 0 , or p 0 The seller approves the trade if and only if: p 1 − c ≥ ˆ ˆ p 0 , or ˆ p 1 − ˆ p 0 ≥ c In other words in equilibrium q = 1 if and only if v ≥ ˆ p 1 − ˆ p 0 ≥ c Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 9 / 63

The Hold-Up Problem (9) Result (Hart and Moore 1988) The outcome of the renegotiation process is ex-post efficiency: v ≥ c q = 1 iff In other words: if v = v or c = c then q = 0 , if instead v = v and c = c then q = 1 . Proof: As we have seen above if v ≥ ˆ p 1 − ˆ p 0 ≥ c then trade occurs. Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 10 / 63

The Hold-Up Problem (10) Consider now the state of nature v = v and c = c and let’s compute the renegotiation prices. The ex-ante contract ( p 1 , p 0 ) is such that only three cases are possible: case 1: v ≥ p 1 − p 0 ≥ c , case 2: p 1 − p 0 > v > c , case 3: v > c > p 1 − p 0 . Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 11 / 63

The Hold-Up Problem (11) In case 1: v ≥ p 1 − p 0 ≥ c trade occurs at price p 1 . In this case necessarily: p S p S 1 > p 1 , 0 > p 0 , p 1 > p B p 0 > p B 1 , 0 . But then neither the buyer nor the seller has an incentive to reveal the offers that they received. Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 12 / 63

The Hold-Up Problem (12) In case 2: p 1 − p 0 > v > c , although trade is efficient, the buyer finds it too expensive and hence at the price p 1 will not approve trade. However the buyer will show the seller’s offer and approve trade if and only if v − p S 1 ≥ − p 0 In equilibrium the seller’s offer is such that: p S 1 = p 0 + v Since in this case the seller’s payoff is: p S 1 − c = p 0 + v − c > p 0 p 1 = p S p 1 − ˆ Therefore ˆ 1 and ˆ p 0 = p 0 and trade occurs: v = ˆ p 0 > c . Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 13 / 63

The Hold-Up Problem (13) In case 3: v > c > p 1 − p 0 , although trade is efficient, the seller finds trade to be too costly for the price difference. However the seller will show the buyer’s offer and approve trade if and only if p B 1 − c ≥ p 0 In equilibrium the buyer’s offer is such that: p B 1 = p 0 + c Since in this case the buyer’s payoff is: v − p B 1 = v − c − p 0 > − p 0 p 1 = p B p 1 − ˆ Therefore ˆ 1 and ˆ p 0 = p 0 and trade occurs: v > ˆ p 0 = c . Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 14 / 63

The Hold-Up Problem (14) Whenever the state of nature is such that v = v and c = c then (ˆ p 1 , ˆ p 0 ) are such that: v ≥ ˆ p 1 − ˆ p 0 ≥ c and trade occurs q = 1. We can now consider the ex-ante efficiency of the parties’ investment decision. Notice that the buyer’s payoff in equilibrium is: i j [ v − (ˆ p 1 − ˆ p 0 )] − p 0 − ψ ( j ) While the seller’s payoff in equilibrium is: i j [(ˆ p 1 − ˆ p 0 ) − c ] + p 0 − φ ( i ) Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 15 / 63

Underinvestment The equilibrium investments i ∗∗ and j ∗∗ are then characterized by the solution to the following problems: i j [ v − (ˆ p 1 − ˆ p 0 )] − p 0 − ψ ( j ) max j max i j [(ˆ p 1 − ˆ p 0 ) − c ] + p 0 − φ ( i ) i The first order conditions of these problems are then: i ∗∗ [ v − (ˆ p 0 )] = ψ ′ ( j ∗∗ ) p 1 − ˆ j ∗∗ [(ˆ p 0 ) − c ] = φ ′ ( i ∗∗ ) p 1 − ˆ We can evaluate ex-ante efficiency by comparing the latter FOC and the ex-ante efficiency conditions i ∗ ( v − c ) = ψ ′ ( j ∗ ) , j ∗ ( v − c ) = φ ′ ( i ∗ ) Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 16 / 63

Underinvestment (2) Result (Hart and Moore 1988) When contracts are incomplete ex-ante inefficiency may arise. The parties’ investments choices are such that j ∗∗ ≤ j ∗ , i ∗∗ ≤ i ∗ with at least one of the inequality being strict. Proof: Notice first that from the condition that guarantees trade: v ≥ ˆ p 1 − ˆ p 0 ≥ c we have that only one of the following three cases may arise: v > ˆ p 1 − ˆ p 0 > c , v = ˆ p 1 − ˆ p 0 > c , v > ˆ p 1 − ˆ p 0 = c Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 3 22 November 2011 17 / 63