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Incomplete Contracts

Incomplete Contracts. Renegotiation, Communications and Theory December 10, 2007. Introduction – “Show me the money!”. Long term contracts tend to be “incomplete” Difficult (and costly) to specify every contingency that might arise in a trading relationship

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Incomplete Contracts

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  1. Incomplete Contracts Renegotiation, Communications and Theory December 10, 2007

  2. Introduction – “Show me the money!” • Long term contracts tend to be “incomplete” • Difficult (and costly) to specify every contingency that might arise in a trading relationship • Communication (content and method) can be used during the contract process to affect outcome • Most contractual disputes revolve around contractual incompleteness • Examples: real estate contracts, M&A transactions (“MAC clauses) • Hart and Moore (1988) build a model of incomplete contracts and renegotiation that focuses on communication and contract revisions between two parties 2

  3. Introduction – “Show me the money!” 3

  4. Model Setup & Assumptions • Two parties who write an incomplete contract in Date 0 • Parties have the option to renegotiate or revise the contract • Expectations of revisions has an effect on the original contract • Parties are interested in creating an optimal “revision game” that is sensitive to each parties’ benefits and costs • After signing contract in Date 0 (but before Date 1), the buyer and seller make investments β, σ • Known to each other, but not publicly verifiable • Signing contract at Date 0 commits parties to these investments • After Date 0 (but before Date 2), v (buyer’s valuation) and c (seller’s costs) are realized 4

  5. Model Setup & Assumptions (cont’d) Date 0 Date 2 Date 1 v, c learned by parties Payments, Disputes? Contract Signed Trade? Investments (β, σ) Revision / Renegotiation • Realizations of v and c are determined by β and σ and the state of the world at Date 1, ω • Each party’s investments only affect his or her own payoff • Both buyer and seller must enact the trade at Date 2 • The Contract Process: 5

  6. The Transaction • At Date 2, trade happens (q = 1) or not (q = 0) • Price to the seller is p1 or p0 • Messages (m) exchanged by buyer and seller between Dates 1 and 2 • Contract can specify price functions p0 (m) or p1(m) • Trade occurs when the following are satisfied: • The above equations show that both buyer and seller prefer that the trade happens (q = 1) 6

  7. Messages and the Revision Process • Parties exchange messages between Dates 1 and 2 – “the revision process” • Parties can “tear up” the Date 0 contract and write a new one • Messages can be sent reliably and cannot be forged • Hart and Moore look at two message technologies • Case A – Impossible to publicly record messages (ie parties can deny the receipt of certain messages) • Case B – Messages can be publicly recorded and cannot be denied • The form of the optimal contract is very sensitive to each case 7

  8. Case A: Messages Cannot be Verified • The revision process can be thought of a game, consisting of two subgames • the message game between Date 1 and 2 and the dispute game after Date 2 • Proposition 1 (equilibrium trading rule) • ( ) are the prices specified at the Date 0 contract • The Trading Rule that will prevail at Date 2 are: • If v < c, q = 0, buyer pays seller • If v ≥ – ≥ c, q = 1, buyer pays seller • If v ≥ c > – , q = 1, buyer pays seller + c • If – > v ≥ c, q = 1, buyer pays seller + v • How does this look? 8

  9. Equilibrium Trading Rule Graphed Proposition 1: 9

  10. Trading Rule Intuition Part I 2 v < c, q = 0 1 1 v ≥ p1 – p0 ≥ c, q = 1 2 Result Game Insight / Messaging 10

  11. Trading Rule Intuition Part 2 3 v ≥ c > p1 –p0, q = 1 3 4 p1 – p0 > v ≥ c, q = 1 4 Result Game Insight / Messaging 11

  12. Case A: Conclusions • Unverified messages constrain the ability of the buyer and seller to renegotiate the Date 0 contract • The outcomes are determined by the graph in the previous slides • The trading mechanism can affect the buyer and seller’s decisions in equilibrium • Hart and Moore comment that the results are also sensitive to what the “courts” can retrospectively determine (which depends on the trading mechanism) 12

  13. Case B: Verifiable Messages • If a message is sent from outside prescribed set, a player who sends a message from outside this set (or does not send a message at all) can be penalized • Lead to “revised” contract prices,(p0ij, p1ij) • Messages, mappings are choice variables at date 0

  14. Final Trading Prices with Verifiable Messages As before, If v ≥ c whatever messages are sent, trade occurs If v < c  trade will not occur and the price will be p0ij

  15. Value Function of the Game If v ≥ c Expected Payoff to the seller: Where ρ= probability seller assigns to p1ij,π= probability buyer assigns to p1ij If v < c  Expected Payoff to the Seller

  16. Aside: General Intuition of the Minmax Theorem • Zero sum game • Each player responds by minimizing the maximum expected payoff of the other player • Minmax same as maxmin same as NE

  17. Expected Trade and Non-trade Prices with Verifiable Messages (1): The price that the buyer must pay for the good cannot fall if the seller’s cost rises or if the buyer’s valuation rises (2): If v and c rise by α, p1* rises by no more than α (3):Neither the buyer nor the seller can be worse off trading than not

  18. First Best Conditions (unverifiable) • If any one of these holds, first best can be achieved: • (1): Set p1-p0 = k, trade occurs, neither the buyer nor the seller influence the terms • of the trade • (2): Value for the buyer is independent of buyer’s investment decision, only seller • investment matters give seller all surplus, and he’ll invest optimally • (3): Cost for the seller is independent of seller’s investment decision, only buyer • investment matters give buyer all surplus, and he’ll invest optimally • (4) No uncertainty split surplus to make both parties better off trading

  19. Second Best Conditions • First best cannot be achieved • The distribution of the buyers valuation is a convex combination of two probability vectors, one which FOSD the other • Greater investment in β puts more relative weight on the preferred lottery • Create externality v now increasing in β • Decrease β, decrease v, which negatively affects the seller because either no trade or lower price trade

  20. Second Best (cont’d.) (3) Ensures a unique interior solution for β, σ (4) If v≥c, always achieve first best

  21. Case B Conclusions: Underinvestment • There exist (strict) conditions under which first best can be achieved (even with non-verifiable messages) • There exist conditions under which first best cannot be achieved (even with verifiable messages) • Second best can be achieved with verifiable messages • Even with verifiable messages, underinvestment

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