Lecture 8. Contract Theory (contract theory)
Recommended reading : Microeconomics Table of Contents
1. Overview
2. Hidden action model
3. Hidden type model
1. Overview
⑴ Setting
① A situation in which one party offers a contract to another party, and the other party decides whether to accept it
② Principal: the person or firm that offers the contract (e.g., shareholders)
③ Agent: the person or firm that decides whether to accept the contract (e.g., CEO)
④ Contract theory is the simultaneous version of incomplete information
⑤ (Reference) Signalling is the sequential version of incomplete information
⑥ We want to understand how the agent’s asymmetric information affects outcomes
⑵ 1st best
① Definition: the contract proposed by the principal when there is no asymmetric information
② Feature 1. The principal tries to maximize total surplus
③ Feature 2. The principal designs the contract so that the agent’s surplus becomes 0
⑶ 2nd best
① Definition: the contract proposed by the principal when there is asymmetric information
② Feature: the principal proposes a contract to maximize their own surplus
2. Hidden action model: focusing on the owner–manager model
⑴ Hidden action model: also called the moral hazard model
① A case where the principal can observe only the outcome of the agent’s action
② The principal cannot observe the agent’s action itself
⑵ Terminology
① e: effort
② πg: gross profit (total profit)
③ s: wage / salary
④ πn: net profit (from the owner’s perspective)
⑤ c(e): cost as a function of effort e○ In general, assume c’(e) > 0 and c’‘(e) > 0
○ (Note) If you accept that people generally dislike working, these assumptions make sense⑥ U(s, e): the manager’s utility function over wage and effort
○ E(s) - c(e): net benefit (from the manager’s perspective)
○ (A/2) Var(s): risk premium from wage uncertainty (risk aversion)
○ Assume A > 0
⑶ 1st best
① Setting: the owner observes the manager’s effort e perfectly
② Equation 1. Without negotiation, the owner offers the following wage schedule
③ Equation 2. Participation constraint: the manager must obtain utility ≥ 0 to work
④ Equation 3. The owner chooses s* to maximize their net profit○ To maximize net profit, the owner sets s = c(e)
○ The inequality can be rewritten as follows
○ Since c’(e) is strictly increasing, we can find e* such that c’(e*) = 1
○ Setting s* = c(e*) makes the manager’s surplus 0 (i.e., the manager receives only the wage)
⑷ 2nd best
① Setting: instead of observing e directly, the owner can observe the following
② This can be viewed as the following game○ Stage 1. The owner chooses a and b
○ Stage 2. The manager decides whether to accept the contract (a constraint appears here)
○ Stage 3. If the manager accepts, they choose e③ Subgame-perfect equilibrium: solve by backward induction
○ Stage 3. s = a + bπg = a + be + bε
○ Stage 2. Participation constraint: since the manager must accept, U ≥ 0 must hold
○ Stage 1. The owner chooses a and b so that U = 0 while maximizing profit
○ (Reference) Stage 2 creates a constrained problem, but with some math it can be converted to an unconstrained problem
○ From the manager’s perspective, the required effort in the 2nd best is smaller than in the 1st best
○ Interpretation: because the manager has asymmetric information, they have bargaining power
3. Hidden type model: also called the adverse selection model
⑴ Overview
① Definition: the agent knows their own type but cannot change it, and reflects it in their behavior
② Example: borrowing money from a bank to raise funds
⑵ Basic assumptions
① Equation 1. Consumer’s surplus
○ T: total payment
○ v: satisfies v(q) ≥ 0, v’(q) ≥ 0, and v’‘(q) < 0② Equation 2. Parameter θ: must satisfy θH > θL
③ Equation 3. Seller’s surplus: also called the seller’s profit
⑶ 1st best: when the seller knows the consumer’s type θ
① Setting: the seller offers different contracts depending on the consumer’s type
② Condition: participation constraint (the condition for the consumer to participate)
③ Example: β = 0.5, θL = 15, θH = 20, v(q) = 2q^0.5, c = 5
⑷ 2nd best: when the seller does not know the consumer’s type θ
① Setting: since the seller cannot observe θ, the contract is designed so that the consumer self-selects the contract that matches their type
② Consumer’s constraints○ Condition 1. L’s participation constraint: for a type-θL consumer to participate
○ Condition 2. H’s participation constraint: for a type-θH consumer to participate
○ Condition 3. L’s incentive compatibility constraint: a type-θL consumer should not choose the contract intended for θH
○ Condition 4. H’s incentive compatibility constraint: a type-θH consumer should not choose the contract intended for θL③ The seller’s profit maximization problem
○ Condition 1 can be ignored
○ If Condition 2 does not bind (i.e., is not an equality), then a better solution exists○ Let TL ← TL + ε, TH ← TH + ε
○ If ε is small enough, Condition 1 and Condition 2 are not violated
○ It does not affect Condition 3 and Condition 4 (because ε cancels out)
○ Expected profit E(π) can be increased○ If Condition 4 does not bind (i.e., is not an equality), then a better solution exists
○ Let TH ← TH + ε
○ If ε is small enough, Condition 4 is not violated
○ If ε is small enough, Condition 2 is not violated
○ Condition 1 and Condition 3 are satisfied automatically○ It is enough to ignore Condition 3 when solving and then check whether it is satisfied by the solution
④ Conclusion: the problem becomes an unconstrained maximization problem
Entered: 2020.05.18 23:02