1 Dynamic Ine¢ ciency Due to Reclassi…cationRisk: Theory and Evidence1.1 Theory: Cochrane (1995, JPE): Time-Consistent HealthInsurance The health insurance market in the U.S. do not sell long-term health insur-ance contracts. Many think that the absence of e¤ective long-term healthinsurance is the most important issue concerning health care reforms. Lack of long term health insurance makes it possible that people who arein‡icted with long-term illnesses, such as AIDS, cancer, heart diseas e etc.will have to face ruinous increases in premiums; or lose health insruancealtogether. One may propose government regulations on the insurance companies sothat they can not raise premiums, or deny coverage in the event of suchcostly health risks. The problem is that individuals who turn out to behealthy are likely to opt out for cheaper insurance. This may lead themarket to unravel. Furthermore, one may even argue that long-term contracts are not evendesirable, because sometimes one may want to change an insurer, due tofor example marriage or job change. The question asked in this paper is: is there a decentralized way to repli-cate allocations achievable via long-term insurance contracts by piecinngtogether one-period contracts only? These short-term contracts must satisfy time-consistency, (or renegotiation-proof), and also participation constraints (i.e. each party must be willingto sign the next contract, no matter what happens).1.1.1 Optimal Health Insurance Contracts In the beginning of each period, wealth Wtis evaluated, premium ptcanbe made and the consumer earns income e: Then information about the consumer’s health is revealed, including cur-rent health cost xtas we ll as informa tion about future health costs. The optimal health insurance contract solves:maxfctgE01Xt=0tu(ct)s.t. E01Xt=011 + rtct= W0+ E01Xt=011 + rt(e  xt)where  = 1=(1 + r) : The …rst order condition to the consumer’s optimization problem will spec-ify a constant consumption level c at every date and in every state. Theconstant level of c onsumption is simplyc = rW0+ rE01Xt=0t(e  xt): (1) The optimal consumption pro…le can be state-contingent claims. At time0, the consumer sells claims to his income stream and buys contingentclaims to cover health expenses and consumption. Since insurers are as-sumed to be risk neutral, the time 0 value of contingent claims equals theirdiscounted expected present value. The problem with these time 0 contingent-claims contracts (among otherimpracticalities) is that they are not time consistent: they do not satisfy expost participation constraint. As soon as health status is revealed, healtyconsumers and the insurers of sick consumers will withdraw. Since both parties are free to abandon the contract at the end of anyperiod, a time-consistent contract must be equivalent to a series of one-period contracts. The rest of the paper then asks whether we can implement the Pareto-optimal or contingent-claims allocation using one-period contracts. The answer: One need only two contracts (or securities): one-period in-surance and riskless period-to-period saving (bank accounts). Speci…cally,– One-period insurance contract: the consumer pays a premium ptin thebeginning of the period, and the insurer then pays this period’s healthexpenses xtplus a potentially state- contingent severance payment yt,whose va lue is determined below. [The severance payment is thekey to the paper] How are the insurance premium and severance payment determined?pt= Et(xt)yt=(Et+1 Et)1Xj=111 + rjxt+j=(Et+1 Et)1Xj=111 + rjpt+jThat is, the consumer pays a premium equal to the expected health costsin that period Et(xt) ; and receives a severance payment that is equal tothe revision in the present value of health expenses. In order to show that with ptand ytspeci…ed above, one can show that ifthe consumer follows a simple consumption rule, in each period t; consumethe time t present value of resources, i.e.ct=r1 + r24Wt pt+ yt Et+11Xj=1jxt+j35+ 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 se e this, note that f or t = 0; the above consumption rule tells us thatc0=r1 + r24W0 p0+ y0 E11Xj=1jx1+j35+ e=r1 + r24W0 E0(x0)+ (E1 E0)P1j=111+rjx1+jE1P1j=1jx1+j35+ e=r1 + r24W0 E01Xj=011 + rjxj+1 + rre35=r1 + r24W0+ E01Xj=011 + rje  xj35 Show the rest recursively. The severance payment system described above mea ns that a sick con-sumer can pay higher premiums with no change to his consumption planbecause the severance payment e xactly o¤sets the innovation in his presentvalue of premiums as a result of being in a sick state. The contract also requires t hat a consumer who gets unexpectedly healthiermust ma ke an ex post severance payment to the insurer, equal to theunexpected decline in his health care expenses. This kind of payment from the consumer to the insurer may be hard tocollect. To deal with this problem, Cochrane proposes the establishment ofa health account. In each period, the consumer pays an amount qtintothe account in the …rst part of the period, then the account pays healthinsurance premiums, ptand pays or receives severance payments yt: Inparticular, if one is to set qtto be constant q across all periods, it mustbe thatq = rE01Xj=0jxj:1.2 Hendel and Lizzeri (2003, QJE): The Role of Com-mitment i n Dynamic Contracts: Evidence from LifeInsurance Hendel and Lizzeri (2003) examine how the life insurance market dealswith the reclassi…cation risk. Their starting point is the same theoretical point we discussed in Cochrane’spaper:– when one purchases an insurance, there is a committment problem:even when the insurer can commit to renew coverage, or not to raisepremium in the event of bad risk occurrence, the insured is not com-mitted not to switch insurer if a good risk occurrs. (Long term insur-ance with commitment on both sides are simply not enforceable by thecourt.) How does the form of the life insurance contracts adapt to overcome theproblem of lack of commitment? Theor etical Model Consider a market with insurance buyers and sellers. Buyers wish to insurea stream of income for their dependents. When an individual considers a future period, she has two distinctivesources of utility:– if she is