Incomplete ContractsRenegotiation, Communications and TheoryDecember 10, 2007

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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

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Introduction – “Show me the money!”

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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

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Model Setup & Assumptions (cont’d)

• 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:

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Date 0

Contract Signed

Date 1

v, c learned by parties

Date 2

Trade? Payments, Disputes?

Investments (β, σ) Revision / Renegotiation

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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)

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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

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• 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

Case A: Messages Cannot be Verified

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Equilibrium Trading Rule Graphed

• Proposition 1:

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Trading Rule Intuition Part I

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1 Valuation (v) is less than costs (c), so trade will not happen (ie q=1), so price will be p0

Messages sent don’t matter (and either party can choose not to reveal them) so price ends up at equilibrium p0

2 Valuation is more than the difference between p1 and p0 so buyer and seller agree to trade (q=1) and price is p1

Messages also don’t matter, buyer can choose to send no messages or reveal none from the seller so equilibrium is p1

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Result Game Insight / Messaging

1 v < c, q = 02 v ≥ p1 – p0 ≥ c, q = 1

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Trading Rule Intuition Part 2

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3 Seller can always obtain p0 but buyer wants the sale and pays p0 + c for trade to go through

Seller can ignore messages and get p0, so buyer needs q=1, so he offers + c on the last day of renegotiation to ensure q = 1

4 Seller offers p0 + v and the buyer take the offer because that is still less than or equal to his valuation

On the last day of renegotiation, seller can send a message of p0 + v and the buyer accepts because its still within rule 4

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Result Game Insight / Messaging

3 v ≥ c > p1 – p0, q = 14 p1 – p0 > v ≥ c, q = 1

• 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)

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Case A: Conclusions

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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,(p0

ij, p1ij)

• Messages, mappings are choice variables at date 0

Final Trading Prices with Verifiable Messages

If v ≥ c whatever messages are sent, trade occurs

If v < c trade will not occur and the price will be p0

ij

As before,

Value Function of the Game

If v ≥ c Expected Payoff to the seller:

If v < c Expected Payoff to the Seller

Where ρ= probability seller assigns to p1ij, π= probability buyer

assigns to p1ij

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

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

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 sellerinvestment matters give seller all surplus, and he’ll invest optimally

(3): Cost for the seller is independent of seller’s investment decision, only buyerinvestment matters give buyer all surplus, and he’ll invest optimally

(4) No uncertainty split surplus to make both parties better off trading

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

Second Best (cont’d.)

(3) Ensures a unique interior solution for β, σ(4) If v≥c, always achieve first best

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