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  Risk Sharing Without Moral Hazard

Mathematical Model: Risk Sharing Without Moral Hazard

The existence of moral hazard means that the Pareto optimal sharing of risk cannot be achieved.

1) Assumptions:

• Two states of the world: State 1 (bad) and State 2 (good)

• Agent produces output (x1 or x2) and then shares the output with the principal.

• In state 1, principal will get (x1 − w1) and agent will get w1.

• In state 2, principal will get (x2 − w2) and agent will get w2.

• If no moral hazard, we could determine the optimal sharing rule (from the principal’s view point) by maximizing the principal’s utility.

• We have to guarantee the minimum utility or else the agent would never enter into the contract.

• The agent’s expected utility from the contract with the principal has to be at least as high as his opportunity cost associated with working in the next best alternative employment.

2) Analysis:

max E{Up}= PUP(x1 - w1) + (1-P)Up (x2 - w2) ............(i)

s.t. E{UA}= PUA(w1)+ (1-P)UA (w2) ≥U¯A

The Lagrangean is:

L = ρUP (x1 − w1) + (1− ρ)UP (x2 − w2) + λ[ρUA (w1) + (1− ρ)UA (w2) −UA] ..........(ii)

The first-order conditions are:

1219_first order conditions.jpg

The last expression shows that the optimal sharing rule equates the marginal rates of substitution (MRS) between states of the world of the principal and the agent.

If the principal is risk neutral and the agent is risk averse, the optimal sharing rule has the principal guarantee the agent a fixed income, regardless of the state of the world. This is equivalent to having the principal pay the agent a fixed wage.

If the agent is risk neutral and the principal is risk averse, the optimal sharing rule has the agent guarantee the principal a fixed income, regardless of the state of the world. This is equivalent to having the agent pay a fixed rent to the principal.

Finally, if both parties are risk averse, the optimal sharing rule pays some share to each party in each state of the world, where the actual shares depend on the parties’ relative attitudes toward risk.

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