Leonardo Felli 7 January, 2002 Topics in Contract Theory Lecture 1 Contract Theory has become only recently a subfield of Economics. As the name suggest the main object of the analysis is a contract. Therefore the first natural question that needs to be answered is: What is a contract? 1
Topics in Contract Theory 2 Definition: A contract is the ruling of an economic transaction: the description of the performance that the contracting parties agree to complete at a (possibly future) date. Example: a contract for the purchase of a specific item, say a meal. It specifies: the restaurant s performance (number of courses, quality of food, cooking details, etc... ), the customer s performance (payment in full upon completion). Contracts involve not only the contracting parties, but also outsiders (enforcing agency: a court).
Topics in Contract Theory 3 We distinguish between implicit and explicit contracts. A contract is implicit or self-enforcing whenever the environment in which the contracting parties operate corresponds to an extensive form of a game whose (unique) subgame perfect Nash equilibrium exactly corresponds to the outcome the parties would like to implement. If you believe in Subgame Perfect equilibrium then there is no need for explicit communication. In the given environment two rational individuals will behave in the way required.
Topics in Contract Theory 4 If the outcome the parties would like to implement is not the subgame perfect Nash equilibrium of the environment in which they operate the parties might want to modify this environment. This is accomplished through and explicit contract. An explicit contract is a commitment device which requires: an explicit agreement between the parties, the intervention of a third party: a court. The role of the court is to force the parties to behave in a way that differs from the one that would arise in the absence of any agreement.
Topics in Contract Theory 5 An explicit contract therefore specifies a new extensive form corresponding to a new game for the parties. The usual way for the court to guarantee that the parties operate in this new environment is by modifying the parties payoffs, when necessary. By agreeing to bring in a court in the game the parties commit to a game that differs from the initial one they were in.
Topics in Contract Theory 6 To see how the presence of a court may work consider the following example: (Kreps, 1984) A buyer B and a seller S wish to trade an indivisible item at date 1. Preferences: buyer s valuation: v, seller s delivery cost: c. Let v > c trade is socially efficient.
Topics in Contract Theory 7 Let p be a reasonable price level (we abstract for the moment from bargaining) such that: v > p > c. B s and S s situation may be described by the following normal form: deliver not deliver pay p v p, p c p, p not pay p v, c 0, 0 The unique Nash equilibrium (dominant solvable) is: (B does not pay, S does not deliver).
Topics in Contract Theory 8 The situation does not change if any of the following two extensive forms are played: B pay p not pay p S S deliver not deliver not deliver deliver (v p, p c) ( p, p) (v, c) (0, 0) The unique SPE is: {B does not pay, S does not deliver at both nodes}.
Topics in Contract Theory 9 or deliver B B S not deliver pay p not pay p not pay p pay p (p c, v p) ( c, v) (p, p) (0, 0) The unique SPE is: {S does not deliver, B does not pay at both nodes}.
Topics in Contract Theory 10 Solution: to this inefficiency is an explicit contract enforced by a court that specifies (for example): the payment p that B is supposed to make contingent on S delivering the item, the punishment (implicit in the legal system) F B > p imposed by the court on B in the event that S delivers and B does not pay, the punishment (implicit in the legal system) F S > c imposed by the court on S in the event that B pays but S does not deliver.
Topics in Contract Theory 11 In this case the normal form describing the contracting parties problem once the contract is in place is: deliver not deliver pay p v p, p c F S p, p F S not pay p v F B, F B c 0, 0 The unique Nash equilibrium is now: (B pays p, S delivers). Notice that the particular contract considered does not require any inefficiency off-the-equilibrium-path: it is renegotiation proof. The latter property does not always hold.
Topics in Contract Theory 12 This example clearly shows the need for an enforcement mechanism. This mechanism may be due to: the parties being involved in a repeated relationship relationship/implicit contracting, (multiplicity might be a problem). the presence of a legal system that through a court enforces the parties agreement explicit contracting). Notice that according to this interpretation the court is essentially a commitment device available to the parties that can be used when the parties agree to call it in. An alternative interpretation is that the court itself is one of the players of the game.
Topics in Contract Theory 13 It should therefore be endowed with a payoff function and an action space and should be explicitly considered in the analysis of the contractual situation (come back to it). It should be mentioned that using this line of argument one could obtain a rather extreme interpretation of a contract (a law) (Mailath, Morris and Postlewaite 2000). The view is that enforcement is the only relevant activity, a contract (a law) is at best cheap talk that allows the parties to coordinate on a particular equilibrium of the game. No new equilibrium is introduced by the parties agreeing on a contract or by the parliament passing a law.
Topics in Contract Theory 14 From now on we will assume that the two (or more) parties involved in the contractual relationship operate in a market economy with a well functioning legal system. Whatever contract the parties agree to it will be enforced by the court. The penalties for breaching the contract will be assumed to be sufficiently severe that no contracting party will ever consider the possibility of not honoring the contract. We will abstract from these penalties and leave the court in the background.
Topics in Contract Theory 15 Once we have established what a contract is and how it works the next natural question is: What could parties achieve in an economic environment in which they can costlessly negotiate a contractual agreement? The answer to this question is in an economic principle known as the Coase Theorem. Coase Theorem: (Coase 1960) In an economy where ownership rights are well defined and transacting is costless gains from trade will be exploited (a contract will be agreed upon) and efficiency achieved whatever the distribution of entitlements. That is rational agents write contracts that are individually rational and Pareto efficient.
Topics in Contract Theory 16 A contract is individually rational if each contracting party is not worse off by deciding to sign the contract then by deciding not to do it. This is the reflection of an other basic principle of a well functioning legal system known as: freedom of contract. This is equivalent to assume that the action space of the contracting parties always contains the option not to sign the contract. A contract is Pareto efficient if there does not exists an other feasible contract that makes at least one of the contracting party strictly better off without making any other contracting party worse off.
Topics in Contract Theory 17 To illustrate the Coase Theorem we consider the following simple model of a production externality. Consider two parties, labelled A and B. Party A generates revenue R A (e A ) (strictly concave) by choosing the input e A at a linear cost c e A (c > 0). A s payoff function is then: Π A (e A ) = R A (e A ) c e A Party B generate revenue R B (e B ) (strictly concave) by choosing the input e B at the linear cost c e B (c > 0). Party B also suffers from an externality γ e A (x > 0) imposed by A on B.
Topics in Contract Theory 18 B s payoff function is then: Π B (e B ) γ e A where Π B (e B ) = R B (e B ) c e B. Assume first that the parties choose the amounts of input e A and e B simultaneously and independently without any prior agreement. Party A s problem: max e A Π A (e A ) Party B s problem: max e B Π B (e B ) γ e A
Topics in Contract Theory 19 In equilibrium the amount of inputs chosen (ê A, ê B ) is such that: R A(ê A ) = c R B(ê B ) = c Consider now the social efficient amounts of input e A and e B. These solve the problem: max e A,e B Π A (e A ) + Π B (e B ) γ e A In other words (e A, e B ) are such that: R A(e A) = c + γ R B(e B) = c
Topics in Contract Theory 20 Comparing (ê A, ê B ) and (e A, e B ) we obtain using concavity of R A ( ): e B = ê B, e A < ê A In other words: Π A (e A)+Π B (e B) γ e A [Π A (ê A ) + Π B (ê B ) γ ê A ] = = [Π A (e A) Π A (ê A )] + γ (ê A e A) > 0 The joint surplus is reduced by the inefficiency generated by the externality. Assume now that the two contracting parties have the opportunity to get together and agree on a contract before the amounts of input are chosen.
Topics in Contract Theory 21 There exists strictly positive gains from trade. A reduction in the amount of input e A from ê A to e A will generate: a decrease in the net revenues from A s technology: Π A (e A) < Π A (ê A ) reduction in the negative externality γ e A < γ ê A The former effect is more than compensated by the latter one. This may create room for negotiation.
Topics in Contract Theory 22 Normalize for simplicity the total size of the surplus that is available to share between the two contracting parties to have size 1 (simple normalization). To establish a negotiation well defined and enforced ownership rights need to be specified. Entitlements/ownership rights define the outside option of each party to the contract. In other words it defines the payoff each party is entitled to without need for the other party to agree. Denote w A and w B the entitlements of party A, respectively B where: w A + w B < 1.
Topics in Contract Theory 23 We assume the following extensive form for the costless negotiation between the two parties: Infinite horizon, alternating offers bargaining game with discounting and outside options. Denote: δ the parties common discount factor, x the share of the pie to party A, (1 x) the share of the pie to party B.
Odd periods: Topics in Contract Theory 24 Stage I A makes an offer x A to B, Stage II B observes the offer and has three alternative choices: he can accept the offer, then x = x A and the game terminates; he can reject the offer and take his outside option w B and the game terminates; he can reject the offer and do not take his outside option, then the game moves to Stage I of the following period.
Even periods: Topics in Contract Theory 25 Stage I B makes an offer x B to A, Stage II A observes the offer and ha three alternative choices: he can accept the offer, then x = x B and the game terminates; he can reject the offer and take his outside option w A and the game terminates; he can reject the offer and do not take his outside option, then the game moves to Stage I of the following period.
Topics in Contract Theory 26 Payoffs: If parties agree on x in period n + 1: π A (σ A, σ B ) = δ n x, π B (σ A, σ B ) = δ n (1 x), If they do not agree and either party takes his outside option in period n + 1: π A (σ A, σ B ) = δ n w A, π B (σ A, σ B ) = δ n w B. Result 1. (Deal Me Out) For any discount factor δ < 1, and any pair (w A, w B ), w A + w B < 1, the bargaining game has a unique subgame perfect equilibrium, where agreement between the parties is immediate and the outside options are never exercised.
Topics in Contract Theory 27 Proof: (sketch) Denote x H i, respectively xl i, i {A, B}, the highest, respectively the lowest, possible share that A can receive in a subgame that starts with i making the offer. We then have that: x H B max{w A, δ x H A } 1 x L A max{w B, δ ( 1 x L B) } Moreover: x L B max{w A, δ x L A}, 1 x H A max{w B, δ ( 1 x H B) }
Topics in Contract Theory 28 Solving these inequalities we obtain: x H A = x L A = x A, x H B = x L B = x B We also obtain that: If then w A δ 1 + δ, w B δ 1 + δ x A = 1 1 + δ, x B = δ 1 + δ If then w A δ 1 + δ, w B δ(1 w A ) x A = 1 δ(1 w A ), x B = w A
Topics in Contract Theory 29 If then w A δ(1 w B ), w B δ 1 + δ x A = 1 w B, x B = δ(1 w B ) If w A δ(1 w B ), w B δ(1 w A ) then x A = 1 w B, x B = w A These offers characterize a pair of strategies (σ A, σ B ). It is easy to show that these strategies constitute the unique subgame perfect equilibrium of the bargaining game.
Topics in Contract Theory 30 Notice that the efficient agreement is reached independently of the size of the entitlements. In particular if each party is entitle to the choice of his input, then: Π A (ê A ) w A = Π A (e A ) + Π B(e B ) γ e A w B = Π B (ê B ) γ ê A Π A (e A ) + Π B(e B ) γ e A If instead party B is entitled to preclude party A from operating his technology, then: w A = 0, w B = Π B (ê B ) Π A (e A ) + Π B(e B ) γ e A
Topics in Contract Theory 31 In either case the result above implies that we would get the efficient outcome: (e A, e B ). However, the share that accrue to each party depends on the entitlements w A and w B. The equilibrium contract specifies a transfer between the two parties and A s choice of input e A. Also the transfer depend on the entitlements w A and w B.