Moral Hazard: Part 1 April 9, 2018
Introduction In a standard moral hazard problem, the agent A is characterized by only one type. As with adverse selection, the principal P wants to engage in an economic transaction with A, i.e. delegating a task; buying or selling a commodity... The crucial difference is that P does not observe the action performed by A, i.e. how much effort A exerts; how much time A devotes to the task... The P only observes the outcome, which is jointly determined by the action of A and by a random component. P wants to design the best way in order to incentivize A to: engage in the economic transaction; perform the action which is optimal from the P s perspective.
A simple example Mark is buying a house insurance. There are two possible outcomes: in G (good) nothing happens; in B (bad) the house burns down. The probability of G is p (0, 1). The utility of Mark is u (y) if G and u (y L) if B, with u > 0 and u < 0.
A simple example Mark can purchase an insurance coverage C at unit price π. The insurance coverage pays C to Mark if B happens. Independently of the state, Mark pays the premium πc. The utility with coverage C is u (y πc) if G and u (y πc L + C) if B, with u > 0 and u < 0.
A simple example The insurance company has the following revenues: πc if G; πc C if B. Assume perfect competition. Then zero profit condition implies that: p (πc) + (1 p) (πc C) = 0, which requires π = 1 p. It is easy to show that the consumer would optimally choose full coverage, i.e. C = L. The optimal choice problem of Mark is: or, setting π = 1 p max pu (y πc) + (1 p) u (y πc L + C) C max pu (y (1 p) C) + (1 p) u (y (1 p) C L + C) C The FOC requires that p (1 p) u (y (1 p) C ) = (1 p) pu (y + pc L).
A simple example Assume now that p is endogenous and depends on Mark s choices, i.e. Mark can take precautions (devices) or avoid making a barbeque in the living room or using candles. Moreover, effort is costly for Mark: utility is now u (y) e; effort is not observable by the insurance company (moral hazard). Assume that p (e) satisfies: p > 0 and p < 0; p (0) = a 0. Assume again perfect competition in the insurance market. Then, if companies expect the effort level e: π = 1 p (e) and C = L
A simple example Note, however, that the optimal effort of Mark when C = L is e = 0. max u (y πl) e e Then, p (0) = a and π = 1 a. The insurance premium is so large that Mark might be better off without an insurance: Assume a = 0. Then π = 1 and Mark s utility is u (y L)! No demand for insurance means that the market breaks down! What happens? Insurance companies cannot punish those consumers that exert little effort and, thus, cannot incentivize good behavior!
Other examples A firm hiring a worker who may spent his time working or on Facebook A bank financing an entrepreneur who can invest in his project, or run away with the money. An insurance company providing car-insurance to drivers who can drive safely, or recklessly A government agency regulating a natural monopoly who could invest in reducing its costs, or not The shareholders of a company delegating the managment to a CEO who can instead act in its own interest
The basic model P wants to delegate the production of q units of a good to A The output q could be low or high, {q, q} and depends on: the effort e {0, 1} exerted by A; a stochastic component. Effort e is costly for A: the cost function is ψ (1) = ψ > 0 = ψ (0). Effort makes the high output more likely: P ( q = q e = 1 ) = p1 > p 0 = P ( q = q e = 0 ) Utility functions: P s utility is V = S(q) t with S > 0 and S < 0 A s utility is U = u(t) ψ (e) with u > 0 and u < 0 Asymmetric information: Effort exerted by A is not observable by P
The basic model: parenthesis Remember that P ( q = q e = 1) = p 1 > p 0 = P ( q = q e = 0) Thus, the distribution of production quantity conditional on high effort first order stochastically dominates the distribution of production quantity conditional on low effort. Thus, the P (due to S > 0) strictly prefers that A exerts high efforts. Finally, define π π 1 π 0.
The basic model: the contract A contract between P and A specifies the transfer t ( q) conditional on the realized outcome q: { t (q) = t q = q t ( q) = t ( q) = t q = q. Output q is ex-post observable and verifiable by a court of law / judge. We assume that the contract is enforced by the law. Important: effort e is neither observed by P nor by a court of law / judge. Thus, the contract cannot depend on it!
The basic model: the timing First. P offers a contract t ( q) = {t, t} to A. Second. A decides whether to accept the contract or not. Third. A exerts the chosen level of effort e = {0, 1}. Fourth. The outcome q realizes. Fifth. The contract is executed: A gives q to P; P transfers t ( q) to A.
The basic model: complete information We assume that P (and court of law) observe the level of effort. P offers a contract t ( q) conditional on A exerting high level of effort in order to maximize S ( q) t ( q). (any deviation from exerting high effort can be severely punished) A chooses whether to accept or not based on utility maximization. Her outside option is equal to 0. By backward induction, A accepting the contract means she will exert high effort. A will accept a contract t ( q) only if π1u ( t) + (1 π 1) u (t) ψ 0 (the partecipation constraint of A)
The basic model: complete information The problem of P can be written as: max π 1 (S ( q) t) + (1 π 1 ) (S (q) q) t, t s.t.π 1 u ( t) + (1 π 1 ) u (t) ψ 0 Clearly, the constraint has to hold with equality: P needs to pay just enough to ensure A accepts the high-effort contract. Thus: max π 1 (S ( q) t)+(1 π 1 ) (S (q) t)+µ [π 1 u ( t) + (1 π 1 ) u (t) ψ], t, t where µ is the Lagrangean multiplier.
The basic model: complete information The FOCs are: π 1 = µπ 1 u ( t ) (1 π 1 ) = µ (1 π 1 ) u (t ) Thus, t = t = t! A gets the same transfer independently of the outcome q and is fully insured! The optimal transfer is given by the participation constraint: u (t ) = ψ or, inverting, t = u 1 (ψ)
The basic model: complete information The indirect utility of P is given by: V 1 = π 1 S ( q) + (1 π 1 ) S (q) u 1 (ψ) P has the alternative to incentivize A to participate but to exert no effort. The indirect utility of P is then: V 0 = π 0 S ( q) + (1 π 0 ) S (q) Incentivizing high effort gives the benefit of π (S ( q) S (q)). The cost of doing so is u 1 (ψ). So, P chooses the high-effort contract iff: π (S ( q) S (q)) u 1 (ψ).