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1 Dynamic Ine ciency Due to Reclassi cation Risk Theory and Evidence 1 1 Theory Cochrane 1995 JPE Time Consistent Health Insurance The health insurance market in the U S do not sell long term health insurance contracts Many think that the absence of e ective long term health insurance is the most important issue concerning health care reforms Lack of long term health insurance makes it possible that people who are in icted with long term illnesses such as AIDS cancer heart disease etc will have to face ruinous increases in premiums or lose health insruance altogether One may propose government regulations on the insurance companies so that they can not raise premiums or deny coverage in the event of such costly health risks The problem is that individuals who turn out to be healthy are likely to opt out for cheaper insurance This may lead the market to unravel Furthermore one may even argue that long term contracts are not even desirable because sometimes one may want to change an insurer due to for example marriage or job change The question asked in this paper is is there a decentralized way to replicate allocations achievable via long term insurance contracts by piecinng together one period contracts only These short term contracts must satisfy time consistency or renegotiationproof and also participation constraints i e each party must be willing to sign the next contract no matter what happens 1 1 1 Optimal Health Insurance Contracts In the beginning of each period wealth Wt is evaluated premium pt can be made and the consumer earns income e Then information about the consumer s health is revealed including current health cost xt as well as information about future health costs The optimal health insurance contract solves max E0 1 X t 1 s t E0 ct 1 r t 0 where 1 1 r fctg 1 X t 0 1 X W0 E0 t 0 t u c t t 1 e 1 r xt The rst order condition to the consumer s optimization problem will specify a constant consumption level c at every date and in every state The constant level of consumption is simply c r W0 r E0 1 X t e xt 1 t 0 The optimal consumption pro le can be state contingent claims At time 0 the consumer sells claims to his income stream and buys contingent claims to cover health expenses and consumption Since insurers are assumed to be risk neutral the time 0 value of contingent claims equals their discounted expected present value The problem with these time 0 contingent claims contracts among other impracticalities is that they are not time consistent they do not satisfy ex post participation constraint As soon as health status is revealed healty consumers and the insurers of sick consumers will withdraw Since both parties are free to abandon the contract at the end of any period a time consistent contract must be equivalent to a series of oneperiod contracts The rest of the paper then asks whether we can implement the Paretooptimal or contingent claims allocation using one period contracts The answer One need only two contracts or securities one period insurance and riskless period to period saving bank accounts Speci cally One period insurance contract the consumer pays a premium pt in the beginning of the period and the insurer then pays this period s health expenses xt plus a potentially state contingent severance payment yt whose value is determined below The severance payment is the key to the paper How are the insurance premium and severance payment determined pt Et xt yt Et 1 Et 1 1 X j 1 Et xt j j 1 1 r 1 j X 1 Et pt j j 1 1 r That is the consumer pays a premium equal to the expected health costs in that period Et xt and receives a severance payment that is equal to the revision in the present value of health expenses In order to show that with pt and yt speci ed above one can show that if the consumer follows a simple consumption rule in each period t consume the time t present value of resources i e 2 r 4 ct Wt 1 r pt yt Et 1 1 X j 1 jx t j 3 5 e then in fact the consumer consumes a constant level c exactly as given in 1 i e the fully insurance optimal consumption plan To see this note that for t 0 the above consumption rule tells us that 2 r 4 c0 W0 1 r p0 y0 2 E1 1 X j 1 jx 1 j 3 5 e 3 j P1 1 r 4 W0 E0 x0 E1 E0 j 1 1 r x1 j 5 e P1 j 1 r E1 j 1 x1 j 2 3 1 j X 1 1 r 5 r 4 xj W0 E0 e 1 r r j 0 1 r 2 3 1 j X r 4 1 W0 E0 e xj 5 1 r j 0 1 r Show the rest recursively The severance payment system described above means that a sick consumer can pay higher premiums with no change to his consumption plan because the severance payment exactly o sets the innovation in his present value of premiums as a result of being in a sick state The contract also requires that a consumer who gets unexpectedly healthier must make an ex post severance payment to the insurer equal to the unexpected decline in his health care expenses This kind of payment from the consumer to the insurer may be hard to collect To deal with this problem Cochrane proposes the establishment of a health account In each period the consumer pays an amount qt into the account in the rst part of the period then the account pays health insurance premiums pt and pays or receives severance payments yt In particular if one is to set qt to be constant q across all periods it must be that q r E0 1 X j 0 jx j 1 2 Hendel and Lizzeri 2003 QJE The Role of Commitment in Dynamic Contracts Evidence from Life Insurance Hendel and Lizzeri 2003 examine how the life insurance market deals with the reclassi cation risk Their starting point is the same theoretical point we discussed in Cochrane s paper when one purchases an insurance there is a committment problem even when the insurer can commit to renew coverage or not to raise premium in the event of bad risk occurrence the insured is not committed not to switch insurer if a good risk occurrs Long term insurance with commitment on both sides are simply not enforceable by the court How does the form of the life insurance contracts adapt to overcome the problem of lack of commitment Theoretical Model Consider a market with insurance buyers and sellers Buyers wish to insure a stream of income for their dependents When an individual considers a future period she has two distinctive sources of utility if she is still alive she obtains u c from consumption c …

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