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  Risk Sharing with Moral Hazard

Risk Sharing with Moral Hazard:

If there is moral hazard, the agent will choose an unobservable action, which affects either the probability of a particular state of the world occurring or the level of output obtainable in each state of the world. Moral hazard exists when taking action to raise expected output involves some cost to the agent. This provides the agent an incentive to shirk.

Let’s assume that the probability is a function of the action (a) and that there is some cost (ca), in utility terms, that is subtracted from the agent’s expected utility or income.

The probability function is such that taking more action (working harder) lowers the probability of observing the “bad” state,

∂ρ/∂a < 0

This makes the agent’s expected utility function with moral hazard:

E{U } = ρ (a ) UA (w1) +[1− ρ(a)] UA (w 2) −  ca

The agent selects a to maximize his expected utility:

1657_expected utility.jpg

The equation adds a additional constraint to the principal’s maximization problem. This is referred to as the incentive-compatibility constraint, meaning any sharing rule has to be compatible with the agent’s incentive to shirk.

The new Lagrangean function by moral hazard will be:

713_langrangean function.jpg

Sharing Rules:

If the principal is risk neutral and the agent is risk averse, the last two expressions imply that the optimal sharing rule guarantees the agent a fixed payment, plus a variable payment which is a function of the amount of the observed output. The fixed payment represents less than full insurance provided by the principal. (If the agent were not risk averse, s/he would be willing to pay a fixed amount and absorb all the risk. That would solve the moral hazard problem and generate a Pareto optimal outcome.)

With a risk-averse agent, some insurance may be Pareto superior to none, but moral hazard means that Pareto optimality cannot be achieved with full insurance, even if the principal is risk neutral. A sharing rule that gives the agent more if x2 is observed than if x1 is observed [(∂ρ/∂a) < 0] reduces that moral hazard. Such a sharing rule is referred to as being second best. It is not Pareto optimal because both parties could be made better off if the agent’s effort could be specified and enforced at no cost. But it does maximize the principal’s utility, subject to the incentive-compatibility constraint and a minimum utility constraint for the agent.

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