Steven Shavell (1979) "On moral hazard and insurance," Quarterly Journal of Economics, Vol 93, No 4, pp. 541-562.

Introduction
	Economists traditionally have discussed two solutions to the problem of moral hazard: i) incomplete coverage against loss and ii) better observation of the care taken by the insured.  Shavell examines detailed model of moral hazard and concludes as the cost of taking care falls from very high levels, partial coverage becomes desirable.  Non-coverage, however, is never an optimal solution.  

Model

Insurers elect a non-negative premium p and non-negative coverage level q.  The probability of loss depends on the amount of care (x) the insured takes to prevent 'bad' state, and that x depends on q.  Thus we have a breakeven policy that p=p(x(q))q.

For the insured, all individuals are identical.  Using this simplification, we eliminate the possibility of adverse selection; and thus focus the model only on moral hazard.  Each individual wants to insure against a loss of dollar amount (l) which occurs with probability (p).  The insured maximize the following equation with respect to q:

EU(q)=[1-p(x(q))]*U[y-p(q)-x(q)] + p(x(q))*U[y-p(q)-x(q)-l+q]

FOC: x'p'[V-U] - x'[(1-p)U'+pV'] - p'[(1-p)U'+pV'] + pV'
	where U = U[y-p(q)-x(q)] and V = U[y-p(q)-x(q)-l+q]
	
substituting the fact that: p'=p'x'q+p

FOC: 	x'p'[V-U]		change in probability of loss
	-x'[(1-p)U'+pV']		change in level of care
	-p'x'q[(1-p)U'+pV']	change in premium due to change in premium rate per dollar of coverage
	-p[(1-p)U'+pV']		change in premium due to an increased level of coverage
	+pV'			change in level of coverage

Shavell states that only the third term reflects moral hazard while the fourth and fifth term correspond to the benefits and costs of purchasing additional coverage in the absence of moral hazard.  The first and second term offset (if x(q)>0) since the individual adjust the level of care to equate them.  With this result, Shavell plots out a graph of the amount of coverage (shown below) conditional on the cost of taking care.

Intuitively, this figure says that if when the cost of taking care is very low, everyone will do it.  Thus, there will be no moral hazard problem.  If the cost of taking care is very high, then individuals will not be responsive on the margin to increased cost sharing or observation.  In this case the insurance company should also fully insure.  It is in the middle ground where the insurer does not fully insure the customer.  In the middle region, the firm trades off increased insurance for increased moral hazard

Observation

Shavell also gives examples of when a firm would like to observe the care taken by an individual.  Ex ante observation is useful in the case of fire insurance (to check if the customer has installed alarms and smoke detectors).  On the other hand, for automobile insurance, ex-post observation would be more efficient since the amount of care taken could vary on a daily basis.  

Shavell concludes that under perfect accuracy observation:
•	The optimal policy is full insurance
•	Care is observed ex post unless the relative cost of ex ante observation is sufficiently small.

When there is no perfect accuracy but observation is costless, the author finds:
•	either ex-ante or ex post observations are of positive value
•	ex ante observation are more valuable as long as the probability distribution F(z;x) is the same for both.  F(z;x) is the probability distribution that the observed care (z) is equal to the actual care taken (x)