THE ROLE OF COMMITMENT IN DYNAMIC CONTRACTS: EVIDENCE...

THE ROLE OF COMMITMENT IN DYNAMICCONTRACTS EVIDENCE FROM LIFE INSURANCE

IGAL HENDEL AND ALESSANDRO LIZZERI

We use data on life insurance contracts to study the properties of long-termcontracts in a world where buyers cannot commit to a contract The data areespecially suited to test a theory of dynamic contracting since they include infor-mation on the entire prole of future premiums All the patterns in the data t thetheoretical predictions of a model with symmetric learning and one-sided com-mitment a la Harris and Holmstom The lack of commitment by consumers shapescontracts in the way predicted by the theory all contracts involve front-loading(prepayment) of premiums Front-loading generates a partial lock-in of con-sumers more front-loading is associated with lower lapsation Moreover contractsthat are more front-loaded have a lower present value of premiums over the periodof coverage This is consistent with the idea that front-loading enhances consumercommitment and that more front-loaded contracts retain better risk pools

I INTRODUCTION

The effect of agentsrsquo inability to commit to long-term planshas been shown to have far-reaching implications for economicarrangements The goal of this paper is to examine empiricallythe role of imperfect commitment in shaping contractual relationsusing data from the life insurance market

Lack of consumer commitment can generate inefciencies ininsurance markets because short-term contracts do not offer in-surance against reclassication risk bad news about the healthstatus of a consumer resulting in increased premiums Lack oflong-term insurance is considered an important market failure inhealth and life insurance markets and it has underscored muchof the policy debate concerning health reform (see Diamond[1992])

Several factors make life insurance suitable for studyingcommitment to long-term contracts Learning (about health) is aquantitatively important phenomenon Life insurance contractsare simple and explicit and issues such as asymmetric informa-tion and misreporting of accidents are not important in the caseof life insurance (see Section V) In contrast other contractual

We are very grateful to Glenn Daily We also thank Andrew Funderburk forresearch assistance Jonathan Levin Derek Neal and Steven Tadelis for theircomments and LIMRA Int and Compulife for generously providing us access totheir data Finally we would like to thank Edward Glaeser and anonymousreferees The nancial support of the National Science Foundation through grantsSES 9986287 and SES 9911496 is gratefully acknowledged

copy 2003 by the President and Fellows of Harvard College and the Massachusetts Institute ofTechnologyThe Quarterly Journal of Economics February 2003

299

relations such as labor contracts are plagued by unobservablesand by the presence of implicit agreements Finally very richdata on contracts are available1

We rst present a theoretical framework to understand howwe should expect lack of commitment to be reected in observedcontracts The model adapts to the life insurance market theHarris-Holmstrom [1982] model of symmetric learning that wasrst developed in a labor market context The main predictions ofthe model are then tested using data on the United States lifeinsurance industry

We assume that the initial health status of the insured isknown to all and that new information about her health type isrevealed over time symmetrically to all market participants Wealso assume that there is one-sided commitment ie insurancecompanies can commit to long-term contracts but buyers cannot

Because of learning short-term (spot) contracts involve re-classication risk future premiums will reect the future healthstatus of consumers Long-term contracts would solve the prob-lem if consumers could commit insurance companies would offerfuture premiums reecting the average future health of the poolLack of commitment makes such a contract infeasible healthyconsumers would drop out to obtain cheaper coverage from com-peting companies Hence to obtain insurance against reclassi-cation risk consumers must prepay some of the premiums Theprepayment or front-loading locks the consumer into the con-tract To be completely successful the front-loaded amount mustguarantee that no consumers drop coverage in the future thefuture premium must be no higher than the fair premium for thehealthiest type Such a contract would implement the same allo-cation that would be achieved under consumer commitment itprovides full insurance against reclassication risk

All contracts we observe in the life insurance market in theUnited States (see subsection IVA) are front-loaded and some ofthe contracts come close to achieving full insurance Howevermost of these contracts leave consumers subject to some reclas-

1 Previous empirical work on dynamic contracts includes ChiapporiSalaniersquo and Valentin [1999] who test a model of wage dynamics using data fromone large French rm They nd support for theories of learning with downwardrigidity of wages Murphy [1986] investigates a longitudinal sample of CEOs andcompares learning with incentives his evidence is mixed Dionne and Doherty[1994] present a dynamic model of adverse selection and with evidence from autoinsurance Crocker and Moran [1998] studies employee lock-in as a way to in-crease commitment to long-term health insurance

300 QUARTERLY JOURNAL OF ECONOMICS

sication risk The model provides an explanation for the varietyof contracts linking them to consumersrsquo heterogeneity in willing-ness to front-load More importantly given the variety of ob-served contracts the model will generate testable predictionsabout the cross section of contracts

Front-loading can be considered as the ex ante price thatconsumers pay for future caps on premiums a result of theirinability to commit Good risks can still reenter the market in thefuture while bad risks are protected against steeper premiumincreases if they remain in the front-loaded policy pool The keycomparative statics generated by the model are that contractsthat are more front-loaded lock in consumers to a greater extentThus more front-loaded contracts retain a better risk pool andhave a lower present value of premiums

As predicted by the model life insurance contracts are of-fered in several varieties involving different time proles ofpremiums which differ in how steeply premiums increase overtime Different premium proles represent different degrees offront-loading (commitment) We use the contract diversity to testthe implications of the model In accordance with the contracttheoretic predictions we nd that virtually every contract isfront-loaded and that there is a negative correlation betweenfront-loading and the present value premiums Remarkably mostof the (large) dispersion in the present value of premiums can beaccounted for by the extent of front-loading We conrm the roleof front-loading as a commitment device by looking at the relationbetween lapsation (a term commonly used in the actuarial liter-ature that means voluntary termination of coverage) and front-loading As predicted more front-loaded contracts have lowerlapsation Thus all the patterns in the data suggest that contractsare designed in a way that ts a model of symmetric learning withone-sided commitment Alternative commitment assumptions arerejected by the data Standard asymmetric information modelsdeliver predictions that are inconsistent with the data

Finally understanding and quantifying the inefciency fromthe lack of bilateral commitment in life insurance is of interest forthe policy debate on health insurance Lack of renewability andthe absence of insurance of reclassication risk are some of theconcerns raised about health insurance (eg in Clintonrsquos reformproposal new state regulation during the 1990s) However thehealth care market suffers from a variety of other problems (seeCutler [1993]) Isolating the problems due to the lack of bilateral

301THE ROLE OF COMMITMENT IN DYNAMIC CONTRACTS

commitment should be useful for understanding the source ofinefciency in health insurance (see Concluding Remarks)

II THE CONTRACTS AND THE DATA

II A The Contracts

The life insurance market is segmented in Term and CashValue contracts2 We focus on the market for Term insurance fortwo reasons (1) term is simpler since it involves pure insurancea xed sum is paid upon death of the insured if death should occurwithin the period of coverage In contrast Cash Value offers acombination of insurance and savings (2) we do not have data oncash values Term contracts represented over 37 percent of allordinary life insurance in 1997 according to LIMRA [1997]

All contracts are unilateral (see McGill [1967]) the insurancecompanies must respect the terms of the contract for the dura-tion but the buyer can look for better deals at any time Thecontract becomes void once the buyer stops paying premiums andthere are no cash ows upon termination These features t amodel of one-sided commitment3

The main distinction among term contracts is the premiumprole Annual Renewable Term (ART) contracts charge premi-ums that increase yearly Level Term contracts (LTn) offer pre-miums that increase only every n years ranging from 2 to 30years Term contracts are renewable without medical examina-tion up to a prespecied age (typically 70 or older) Future termsfor renewal are specied at the moment of underwriting4

Column 2 of Table I illustrates an aggregate ART Thiscontract has premiums that vary exclusively by age Column 3illustrates a ten-year Level Term contract (LT10) We discuss the

2 We focus on the individual market In employment-related insurancecommitment is enhanced by job lock-in

3 Severance payment would enhance buyersrsquo ability to commit to stay in thecontract However for legal reasons they are not used in any life insurancecontract Severance payments would be considered a stipulated ldquopenaltyrdquo andwould not be enforced by a Court ldquoA Common Law tradition prevents Courts fromenforcing terms stipulating damages that exceed the actual harm caused bybreachrdquo [Cooter and Ulen 1999] Insurance companies do not suffer any ex postdamage from the loss of a consumer

4 Premiums are guaranteed for a prespecied period of time ranging from ayear to twenty years Since no medical examination is needed to renew pricechanges are not insured specic Furthermore insurers rarely change the an-nounced premiums Consumer Reports (July 1993) found that only one increasedterm premiums between 1989 and 1993 Hence preannounced premiums are thefuture premiums


difference between these contracts in detail later For now notethat contracts differ substantially in the time prole of premiumsthe premiums for the ART start below those for the LT10 butincrease much more rapidly

The right side of Table I illustrates a Select and Ultimate(SampU) ART These contracts offer state contingent prices lowpremiums are contingent on the insured showing that he is stillin good health Thus premiums vary by issue age and durationsince underwriting (last date that good health was proved) Rowsshow the age at latest underwriting columns the years sinceunderwriting (captured by Policy Year) For instance at age 41the insured would pay $475 (rst row second column of matrix)for renewing his one-year-old policy if he does not requalify (passthe medical) If he had requalied he would have paid $385(second row rst column) The incentive to requalify increaseswith age at 59 the insured who does not requalify pays $1750instead of $1340 The big difference in premiums in the alterna-tive scenarios of a SampU ART shows that learning about the healthstatus of an individual is an important phenomenon and thatconsumers may be left to suffer signicant reclassication risk A59 year old could be charged between $1340 and $6375

The market shares of these contracts are ARTs (aggregate

TABLE ITYPES OF TERM CONTRACTS

Age ART LT10

SampU ART

Policy year

1 2 3 10 11 19 20

40 459 574 370 475 640 1485 2555 5680 637541 499 574 385 490 660 1565 2815 6375 704042 539 574 400 530 690 1705 3105 7040 7790

49 909 574 630 890 1080 2725 6375 13675 1478550 974 1064 690 945 1155 2895 7040 14785 1576551 1044 1064 735 1050 1295 3230 7790 15765 17230

58 2009 1064 1245 1750 2295 6420 14785 33165 3544559 2289 1064 1340 1785 2480 6945 15765 35445 38715

These are contracts offered in 71997 to a preferred nonsmoker male by Northwestern Mutual (ART andLT10) and Jackson National (SampU ART) for $500000 of coverage

ART 5 annual renewable term policyLT10 5 term policy with level premiums for ten yearsSampU ART 5 annual contract that allows for reclassication by showing good health


and SampU) 224 percent LT5 348 percent LT10 523 percent andLT20 110 percent (LIMRA [1997])

This initial look at contracts highlights some patterns thatwe will investigate more systematically in what follows Firstthere are both contingent (eg SampU ART) and noncontingent(eg aggregate ART) contracts Second the time prole of pre-miums differs across contracts some have a atter prole (aremore front-loaded) Furthermore contracts that have a low initialpremium have high premiums later on

II B The Data

The source of information on contracts is the Compulife Quo-tation System (July 1997) an information system compiled for theuse of insurance agents which contains quotes by the mainUnited States life insurers (240 rms) Premiums are reported byconsumer characteristics such as age smoking status genderetc It is an accurate source of pricing data since agents areforbidden to offer discounts by the antirebating laws For thepurpose of testing the predictions of the model the unique featureof Compulife is that it reports the whole prole of future premi-ums faced by a buyer at the moment of underwriting Thus weobserve the entire contract time prole not just the startingpremium The virtue of these data is that they are generated byusing the pricing rule of dozens of insurance companies Wepunch in the demographics and face value into the Compulifepricing software and we obtain the premiums for all future datesand contingencies This is ideal for testing the model since whenwe compare contracts we literally x all demographic character-istics and only vary the contract prole

In our sample we included insurers that satisfy a dual crite-rion of size and solvency They must be in categories IX and aboveof AM Best for size (in terms of sales of term policies) and ratingA or better in terms of solvency (according to AM Best) Thisguarantees product homogeneity and avoids nonrepresentativepolicies Fifty-ve insurers satisfy the criteria5

Table II presents descriptive statistics of the contracts in oursample The table presents the mean premium and standarddeviation by type of contract The third and fourth columns con-

5 Twenty-one of the rms in our sample held 468 percent of the 2937 billiondollars of in-force face amount of term insurance in 1996 [Bestrsquos Review July1996] We do not have sales information for the other companies in our samplebut clearly a large share of the market is covered


cern the rst-year premium for a 40 year old male nonsmokerwho just passed the medical screening The fth and sixth col-umns report the present value of premiums for twenty years ofcoverage When the contract is state contingent because premi-ums depend on whether the consumer does or does not requalifywe need to specify what contingency we look at The present valueis computed under the assumption that the buyer never requali-es As will be shown later this is a useful benchmark because itallows a theoretically meaningful comparison of contracts Weuse an 8 percent interest rate in computing present values6

Table II shows substantial variability in the present value ofpremiums in particular across contract types This variabilitywill be used to test the model7

III THEORY

Before presenting the formal analysis we informally discussthe main ideas of the model The key ingredients of the model are

6 We chose a high interest rate for two reasons to account for both timepreference and for the probability of surviving to a specic date and becauseresults are reinforced if the interest rate is lower We experimented with otherinterest rates and our qualitative results remain unchanged

7 Price dispersion may seem inconsistent with competition However Win-ter [1981] found that after controlling for contract characteristics only a smallfraction of the dispersion remains unexplained He concluded that observed pre-miums are consistent with the hypothesis of competition We will control for thesepolicy characteristics

TABLE IICONTRACT DESCRIPTIVES

Contracts Premium at age 40 PV 20 years of coverage

Type Observations Mean Std dev Mean Std dev

All types 125 6459 1850 160542 52447LT20 25 8666 1684 91874 17851LT10 42 6471 1419 176574 40907LT5 14 5939 1001 188893 38399Aggregate ART 16 6454 1546 128783 29847SampU ART 28 4732 963 201802 32518

Data Source Compulife Quotation System (July 1997) Contracts offered by the top 55 insurers in theUnited States according to size and solvency criteria

LTX 5 term policy with level premiums for X yearsAggregate ART 5 annual renewable term policy with premiums that depend only on ageSampU ART 5 annual renewable term with premiums that depend on age and time elapsed since last

medical examinationPV of the twenty years of coverage starting at age 40 and an 8 percent interest rate


symmetric learning (information about the health type of theinsured is revealed over time) one-sided commitment (insurerscommit buyers do not) and buyer heterogeneity in the cost offront-loading

Consider a two-period model where information about risktypes (health states) is revealed at the beginning of the secondperiod A contract that guarantees full insurance against re-classication risk has second-period premiums that are inde-pendent of the health state If buyers cannot commit to staywith a contract this constant premium should be smaller thanthe actuarially fair premium for the best health state Other-wise the healthier buyers would nd better rates on the spotmarket and drop coverage However this contract (that retainseven the healthiest buyers) suffers second-period lossesamounting to the difference between the second-period pre-mium (calculated to retain the healthiest buyer) and the cost ofcovering the entire pool Therefore this contract must front-load premiums a surcharge in the rst period to cover futureexpected losses

It is clear then that front-loading is benecial paying premi-ums up front reduces reclassication risk One of our empiricalndings is that almost all contracts display front-loading How-ever many contracts leave buyers to suffer considerable reclas-sication risk suggesting that many buyers are not willing tofront-load enough to fully insure There are several reasons whyconsumers may nd front-loading costly For instance with capi-tal market imperfections the tighter the credit constraints themore costly it is to front-load premiums

If it is costly for consumers to front-load equilibrium con-tracts offer imperfect insurance against reclassication risk inreturn for the front-loading policies offer a cap on the premiumrequired in the second period To capture the variety of degreesof front-loading offered in the life insurance market we as-sume that consumers are heterogeneous in the cost of front-loading Consumers with more resources in the rst period(those that face looser credit constraints) will front-load moreand face a lower cap on the premium in the second periodwhich translates into lower reclassication risk Front-loadingand the comparative static with respect to the premiumproles will provide the main testable implications of themodel


III A Model

Consider a market with insurance buyers and sellers Buyerswish to insure a stream of income for their dependents Whenconsidering a future period a buyer has two distinct sources ofutility If he will be alive and he consumes c $ 0 his futureutility is given by u(c) However if he will be dead his futureutility is given by the consumption of his dependents In this caseif his dependents consume c $ 0 his future utility is given byv(c) The functions u and v are both assumed to be strictlyconcave and twice differentiable

There are two periods In period 1 all agents have identicaldeath probabilities p We can think of p as representing a specichealth status in period 1 The model can be readily extended toallow several risk categories In period 1 there are 3 stages instage 1 insurance companies offer contracts At stage 2 buyerschoose a contract At stage 3 uncertainty about death is resolvedand consumption takes place In period 2 there are four stages Instage 0 uncertainty about the health status of each buyer isrealized A consumer with health status i dies with probability pi

in period 2 We order the health status so that p1 p2 pN We assume that p1 $ p (health worsens over time) The proba-bility of being in state i is denoted by pi Uncertainty (learning)about the second-period risk category creates reclassicationrisk since it potentially leaves the buyer facing uncertain premi-ums in the second period We assume that information is sym-metric at the end of stage 0 all insurance companies observe thehealth status of all buyers8 Stages 1 through 3 are the same as inthe rst period

We assume that there is perfect competition between insur-ance companies9 We also assume one-sided commitment insur-ance companies can commit to future premiums whereas con-sumers freely choose between staying with their period 1 contractand switching to one of the spot contracts offered in period 2Therefore the set of feasible rst-period contracts is the set ofunilateral contracts ie those contracts that terminate the mo-

8 We discuss this assumption in detail in Section V9 The market for term life insurance is quite competitive there are 400

competing companies [Life Insurance Factbook] and term insurance is increas-ingly viewed as a commodity (Record 1997) There is a regulatory presence whichmainly pertains to nancial solvency not pricing ldquoLife insurance rates for indi-vidual insurance are regulated only in a most indirect sense It is felt thatcompetition is an adequate regulator over any tendencies toward rate excessive-nessrdquo [Black and Skipper 1994]


ment the buyer stops paying the premiums there are no cashows upon termination10

A rst-period contract consists of rst-period premium Q1 andface amount F1 and a vector of premiums and face amounts(Q2

1F21) (Q2

NF2N) indexed by the second-period health status of

an individual Contract terms can depend on the information re-vealed in the second period A second-period contract consists of apremium and face amount (Q2

i F2i ) indexed by the second-period

health status in the second period rms can offer different con-tracts to buyers in different risk categories A rst-period contract isa long-term contract to which the company unilaterally commitswhereas a second-period contract is a short-term (spot) contract

Lack of consumer commitment per se need not preclude thepossibility of achieving full insurance In a world with no otherfriction all consumers would front-load sufciently to guaranteethat they have no incentive to drop out of the contract in thefuture An implication of such a solution is that all consumerswould be fully insured against reclassication risk Furthermorethere would be no contract variety all consumers would choosethe same contract

Explaining the existing variety of contracts and obtainingpredictions about them requires understanding why most con-sumers are reluctant to fully front-load For simplicity and to becloser to the formulation of Harris and Holmstrom we assumethat capital markets are completely absent11 The results wouldbe unchanged if we assumed a borrowing rate that is higher thanthe lending rate This kind of capital market imperfection seemsto be quite reasonable given the interest rates charged on unse-cured debt (eg median interest rate charged on credit cardbalances is 18 percent) Also note that to explain the variety ofbehavior we do not need all buyers to face such imperfections Wecapture consumer heterogeneity by assuming differences in theincome process12 If alive the consumer receives an income of y 2g in the rst period and y 1 g in the second period with g $ 0

10 These are exactly the kind of contracts offered in the U S life insurancemarket see McGill [1967] or Daily [1989]

11 Note that this assumption does not mean that consumers are constrainedin their front-loading Consumers have enough resources to sufciently front-loadtheir premiums but absent capital markets front-loading requires giving upcurrent consumption which introduces the trade-off between consumptionsmoothing and insurance against reclassication risk

12 Others sources of heterogeneity in front-loadingare discussed in the workingpaper [Hendel and Lizzeri 2002] These include heterogeneous and uncertain needsfor life insurance and heterogeneous beliefs probability of requalication


Thus all consumers have the same permanent income but dif-ferent consumers have different income growth as representedby the parameter g If the buyer is dead then his dependentsreceive no income except for the face amount of the insuranceBecause of the absence of credit markets individuals with higherg are more tightly constrained Heterogeneity in g is what drivesthe different choice of contracts in our model

IIIB Solving the Model

The approach we follow to obtain the equilibrium contracts isthe following In a competitive equilibrium allocations mustmaximize consumersrsquo expected utility subject to a zero protconstraint and a set of no-lapsation constraints that captureconsumersrsquo inability to commit Solving this constrained maxi-mization problem delivers the set of premiums and face amountsthat must be available to a consumer in a competitive marketHowever there are several ways in which this set of premiumsand face amounts can be made available to consumers The con-strained maximization problem below is constructed via fullycontingent contracts that involve no lapsation We later describehow to obtain the same allocation with noncontingent contractsthat are quite common in the life insurance market

In a competitive equilibrium premiums and face amountsthat are fully contingent on the health state Q1 F1 (Q2

1 F21) (Q2

N F2N ) must maximize consumersrsquo expected

utility

(1) pv~F1 1 ~1 2 pu~ y1 2 g 2 Q1

1 ~1 2 p Oi51

N

p i piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i

subject to the zero prot condition

(2) ~1 2 pQ1 2 pF1 1 ~1 2 p Oi51

N

p i~1 2 piQ2i 2 piF2

i 5 0

and the no lapsation constraints for i 5 1 N and for allQ2

i F2i such that (1 2 pi)Q2

i 2 piF2i 0

(3) piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i

$ piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i


The constraint in equation (2) requires that insurance com-panies break even on average This must hold under competitionThe no-lapsation constraints dened by equation (3) are theadditional constraints imposed by lack of consumer commitmentThey require that at every state in the second period the buyerprefers staying in the long-term contract rather than switching toa competing insurance company In other words an equilibriumcontract must be such that there does not exist another contractthat is protable and that offers the buyer higher utility in anystate of period 2

We say that a consumer gets full event insurance in state i ofdate 2 (the denition for date 1 is analogous) if

(4) v9~F2i 5 u9~ y2 1 g 2 Q2

i

We call Q2i (FI) the fair premium for full insurance in state i

namely the premium that guarantees zero protsThe following proposition which is proved in the Appendix

provides a characterization of equilibrium contracts

PROPOSITION 1 In the equilibrium set of contracts

(i) All consumers obtain full event insurance in period 1 andin all states of period 2

(ii) For every g there is an s such that Q2i 5 Q2

i (FI) for i 51 s 2 1 and Q2

i 5 Q2s for i 5 s N where Q2

i

Q2i (FI) for i 5 s 1 1 N

(iii) There is a g such that if g g then there isfront-loading

(iv) More front-loaded contracts provide more insuranceagainst reclassication risk contracts with higher Q1involve a lower s Buyers with higher g choose contractswith less front-loading

Part (i) follows from the fact that under competition eventinsurance is offered at a fair ratemdash hence buyers will choose tofully insure event risk13

Part (ii) says that there is a premium cap at health state s or

13 Parts (ii) and (iii) of Proposition 1 are adaptations of Harris and Holm-strom [1982] to our environment Part (i) was also shown by De Garidel [1999]Cochrane [1995] studies a similar problem He shows that lack of commitmentdoes not preclude full reclassication risk insurance His result is seemingly atodds with ours as well as with Harris and Holmstrom [1982] However there isno such inconsistency under his assumptions consumers are eager to front-loadsufciently to obtain full insurance In our context this means that s 5 1 whichobtains when p1 is much larger than p and g is small He only looks at a subcase


worse the premiums are capped at a common price which ischeaper than the spot rate for each of those states In the optimalcontract consumers transfer income from the rst period to thebad states in the second period The reason why premiums in thebad states are all the same is the following given that the no-lapsation constraints are nonbinding in those states transferringresources across these states is costless leading to an eliminationof all variability in premiums at the optimum If in contrast theconsumer is healthier than in state s the premiums are actuari-ally fair These are the states where the no-lapsation constraintsare binding In those states buyers must be offered spot marketrates

Part (iii) says that contracts are front-loaded so long asincome does not increase too rapidly

Part (iv) says that insurance companies should offer a menuof contracts differing in the trade-off between front-loading andreclassication risk Less front-loaded contracts appeal to buyerswith lower rst-period income ie consumers who nd front-loading more costly g captures the marginal rate of substitutionbetween a dollar today and a dollar in the future Differentcontracts are offered to cater to consumers with different willing-nesses to front-load in exchange for different degrees of reclassi-cation risk insurance

It can be shown that for g 0 the model predicts thatpremiums are upwardly rigid This is a direct counterpart of thedownwardly rigid wages in Harris and Holmstrom Thus themodel predicts that the most front-loaded contracts have atpremiums which is exactly what we observe in the data

III C Contract Equivalence and Empirical Implementation ofthe Model

Proposition 1 obtains equilibrium allocations via fully con-tingent contracts that involve no lapsation However as we sawin Section II both contingent and noncontingent contracts arecommon in the life insurance market We now show that noncon-tingent contracts can also be optimal We then show how weaccount for lapsation which is quite common in life insurance

where no tension between insurance and consumption smoothing arises See alsoPauly Kunreuther and Hirth [1995]


and we discuss how we compute the present value of premiumsfor contingent contracts

To gain an idea of how equivalence is possible note that abuyer who chooses an LT20 or an aggregate ART (which arenoncontingent) is free to drop out of the contract at any date to(for instance) purchase an SampU ART

More formally x an equilibrium contingent contract C Anoncontingent contract NC is also an equilibrium contract if ineach state the consumer for whom the C contract is optimal canobtain the same utility with the NC contract and the insurancecompanies offering the contracts obtain the same prots witheither contract Clearly this requires that the consumer dropsout of NC in the appropriate states

From part (ii) of Proposition 1 the terms of a C contract areconstant Q2

i 5 Q2s and F2

i 5 F2s for i 5 s N In states i 5

1 s 2 1 the premiums and face amounts equal thoseoffered in the spot market the buyer pays the actuarially fairpremium Fix the terms of the noncontingent contract NC to bethe same as for contract C in the rst period and (Q2

s F2s ) in the

second period Thus the two contracts are equivalent by con-struction in the rst period and in the bad states of the secondperiod However in states i 5 1 s 2 1 contract C hasbetter terms But in those states the consumer can obtain thesame terms by dropping out of contract NC and purchasing spotcontracts Because by Proposition 1 in each state i 5 1 s 2 1 contract C has the same terms as a spot contract theconsumer obtains the same utility via NC and C Insurers areindifferent between selling the two kinds of contracts as well instate s through N consumers stay in the contract under bothcontracts and the terms are exactly the same In states 1 to s 21 the two alternative contracts induce different behavior by theconsumer but equal (zero) prots for the rms in those statespremiums in contract C are actuarially fair Thus in those statesrms are indifferent between retaining and not retaining theconsumers This shows that the two ways of achieving the equi-librium allocation are equivalent

This equivalence result will help us in two ways rst byexplaining within our model the existence of noncontingent con-tracts in the life insurance market Second the result helps uscompare contingent and noncontingent contracts All we have todo is transform each contingent contract into its equivalent non-contingent contract along the lines of the previous discussion


The importance of having comparable noncontingent contractscomes from the next proposition which is at the core of ourempirical analysis It presents the main comparative staticsacross noncontingent contracts Due to the equivalence result wewill be able to apply it to all contracts

PROPOSITION 2 Consider two contracts offered in the rst periodthat are not contingent on the health state in period 2 Inequilibrium the contract with the higher rst-period pre-mium has a lower present value of premiums it is chosen byconsumers with lower income growth and has lower lapsation(retains a healthier pool of consumers)

Proof The contract with the higher rst-period premiummust have a lower second-period premium since otherwise noconsumer would choose it Thus in the second period this con-tract will retain a healthier pool of consumers This implies thatthe average cost of this contract (payments to dependents ofdeceased policy holders) is lower Under competition this impliesthat the present value of the premiums must also be lower Thefact that the contract with the higher rst-period premium ischosen by consumers with lower income growth follows fromProposition 1

The importance of Proposition 2 is that it provides a way totest the theory by comparing noncontingent contracts that dis-play different degrees of front-loading The equivalence resultdiscussed above provides a way to extend the comparison to theentire set of contracts by telling us how to evaluate contingentcontracts Whenever we compute the present value of premiumsfor a contingent contract we do so by looking only at the nonre-qualifying states in the terminology of the model we consider theportion of the contract (Q1 F1) (Q2

s F2s ) where s is the state

obtained in Proposition 1 The resulting contract is exactly thenoncontingent contract that implements the same allocation (asdiscussed before in Proposition 2) An example for the SampU ARTcontract described in Table I this means computing the presentvalue of the rst row of the matrix Note that absent the equiva-lence result we would not be able to compute the present valueof premiums since we do not have data on requalifying probabil-ities these probabilities depend on the underwriting stringency(the state s in terms of our model) for individual contracts The


model allows us to compare contracts even in the absence of suchinformation

To be consistent with the theory in the empirical analysis ofcontingent contracts we should consider requalication to be alapsation ie when a consumer requalies we consider him tohave started a new contract This is consistent with industrypractice which treats requalication as a new contract and it isthe theoretically correct way to compare contracts Of course inour empirical analysis we make sure that our ndings are not anartifact of our denitions we also test the predictions by lookingonly across contracts that are directly comparable

Summary of predictions In the next section we look at thetime prole of premiums of the various policies to test the impli-cations of the model The model predicts the following 1 Optimalcontracts involve front-loading 2 Contracts with higher front-loading retain higher proportions of insureds in the long run iehave lower lapsation 3 Since better risk types have higher in-centive to lapse any given contract higher lapsation impliesworse pools 4 A worse pool (lower front-loading) translates intohigher present value of premiums for a xed period of coverage

IV TESTING THE IMPLICATIONS OF THE MODEL

To test the model we use the variety of offered contracts Thedifferent types of contracts are characterized by different pre-mium proles or slopes (namely steepness of premium increasesover time) The slope of the premiums determines the extent offront-loading The steeper the premiums the less front-loadedthe contract We want to establish rst the extent of front-loadingof contracts offered and then the relation between front-loadingand lapsation as well as with the quality of the insured pool

IV A Front-loading

Table III shows the yearly premiums paid over twenty yearsby a 40 year old purchasing half a million dollars of coverageunder three different contracts The numbers are averages overall contracts of each type in our sample Column 1 is for aggregateARTs Column 3 is for twenty-year level term contracts (LT20)Columns 5 and 7 describe two extreme scenarios for SampU ARTs(recall from Table I that the complete description of this type ofcontract is a matrix) in column 5 the buyer never satised the


criterion for requalifying in column 7 he was lucky and requali-ed year after year

Observe the substantial difference in payment proles acrosscontracts By denition premiums in LT20s are constant in nomi-nal terms Due to discounting and aging the level term contractpresents a high degree of front-loading The rst payment in thetwenty-year policies is 83 percent higher than in the yearly selectcontracts this gap represents a lower bound on the magnitude offront-loading

The initial overpayment creates consumer lock-in or commit-ment to the contract When the insured reaches age 50 he cannotbe lured by a SampU ART since the remaining premiums are lowerin LT20 contracts than the premiums he would pay in SampU ARTs(even if he were to keep requalifying as a select customer)

We now argue that even ARTs are front-loaded To checkthis in Table III we report the ratio of the yearly premium overthe probability of death the ratio was normalized to equal 100 at

TABLE IIIPREMIUM PROFILES AND PREMIUM TO MORTALITY RATIOS

SampU ART

Age

Aggregate ART LT20 NoRequal Requal

AvgPrm PrmDeath AvgPrm PrmDeath AvgPrm PrmDeath AvgPrm

1 2 3 4 5 6 740 645 100 866 100 473 100 47342 739 60 866 52 700 77 51344 852 50 866 38 926 74 55346 1000 44 866 29 1202 73 61848 1184 39 866 21 1593 71 70850 1395 37 866 17 2500 91 81352 1611 34 866 14 3073 90 93754 1877 30 866 10 3825 84 109156 2223 27 866 8 4690 78 129358 2746 27 866 6 5795 78 1583

AvgPrm 5 average premium of all policies of a specic type in our sample paid at different ages by aconsumer who buys (qualies for) insurance at age 40

PrmDeath 5 average premiums divided by probability of death at each age We use the 198590 selectand ultimate actuarial table by the Society of Actuaries Select and Ultimate tables reect mortalityexperience for insured population who passed a medical We are using death probabilities of a person passingthe medical at age 40

PrmDeath is rescaled to 100 at age 40NoRequal 5 premium prole of an SampU ART buyer who failed to requalify (show he is in good health)

as good risk in later yearsRequal 5 premium prole of a buyer who requalied (passing a medical) year after year


age 4014 This ratio should be constant absent front-loading Incontrast the ratio declines over time For example for a 40 yearold and an aggregate ART policy the ratio halves within fouryears and it keeps declining to below a third of its initial levelHence contracts do not break even period by period

More surprisingly even SampU ARTs obvious candidates forinvolving no long-term insurance involve some degree of front-loading Column 6 conrms this Absent front-loading the ratio(column 6) should increase over time since the denominatorreects the entire population while the numerator reects theworsening pool that fails to requalify year after year In contrastthe ratio mildly declines

Note that on average the rst year premium of aggregateARTs is 36 percent higher than that of select contracts despite thefact that both are yearly contracts This reects the fact that in anaggregate contract the insurance company is offering a lower capon future premiums in the event that the consumer is in poorhealth The aggregate contract offers more insurance of reclassi-cation risk at the cost of higher initial payments

To summarize all the available contracts involve front-load-ing as suggested by the theory Let us contrast this nding withthe alternative assumptions on commitment If consumers couldcommit front-loading would not be necessary to achieve the ef-cient allocation If front-loading is costly we would not observe itAt the other extreme suppose that insurance companies cannotcommit to future terms of the contract Then front-loading wouldnot provide any insurance against reclassication risk Thus inneither of these scenarios would we observe front-loading

IV B The Negative Relation between Front-Loading and thePresent Value of Premiums

We saw in the previous subsection that contracts differ in theextent of front-loading We now show that there is a systematic

14 The death probabilities used for Table III are taken from select andultimate tables compiled by the Individual Life Insurance Experience Committee[1999] Select and ultimate tables reect the mortality experience of insuredindividuals namely those who passed a medical examination as a function ofboth age and time elapsed since they passed the medical Reported numbersunderstate the extent of front-loading since most contracts do not retain the bestdraws One caveat concerns the trend of decreased mortality The mortality tablesrefer to past populations the contracts may thus reect an expected decline inmortality However the declines in the ratios that we observe in Table III are toobig to be explained by this


relation between present value of premiums and the extent offront-loading along the lines of Proposition 2

We look at the relation between the slope of the premiumsand the present value of premiums over twenty years of cover-age15 To do this we proxy the slope of the premium prole by theratio of the rst over the eleventh premium Q(1st)Q(11th) TableIV shows basic statistics for contracts offered to 40 year old malenonsmokers who qualify as preferred risks Note the wide rangeof premium slopes and present values

A higher Q(1st)Q(11th) ratio means that more is paid up-front (there is more front-loading) The prediction of the model isthat contracts with a higher Q(1st)Q(11th) should have lowerpresent value of premiums16 We now investigate whether therelation between slopes and present values is indeed negative inthe data We also ask how much of the variability in presentvalues can be explained by contract slopes

The rst column of Table V presents the regression of the logof the present value of the premiums on the Q(1st)Q(11th) ratioand other contract characteristics The front-loading variable ishighly signicant and it explains most of the variability in thepresent value of premiums Excluding this variable from theregression drops the R2 from 744 percent to 166 percent (column

15 Recall that the present value is calculated conditional on the consumernot requalifying As argued in subsection IIIC this is the theoretically meaning-ful measure

16 Recall from subsection IIB that a contract with a lower present value ofpremiums does not dominate a contract with a higher one Consumers who choosea contract with a low Q(1st)Q(11th) ratio (hence with a lower present value ofpremiums) will drop coverage in the future if they remain healthy since in thiscase they obtain lower rates by initiating a new contract The theory says that thepool of consumers who stays with a contract in the long run is not the same forcontracts with different slopes The evidence presented in subsection IVC isconsistent with this

TABLE IVCONTRACT SLOPES AND COST OF COVERAGE

Mean Std dev Min Max

Q(1st)Q(11th) 043 031 011 100PV 16055 5245 6871 28754ln(PV) 962 036 883 1027

Q(1st)Q(11th) 5 is the ratio of the rst to the eleventh premium we use it to capture the slope ofpremiums

PV 5 present value of the cost of twenty years of coverage at r 5 008


2) The effect is economically signicant as well one standarddeviation increase in Q(1st)Q(11th) translates into a 28 percentdecline in the cost of coverage

The third column presents a similar regression after omittingthe LT20s from the sample We do so to check whether thesecontracts which have no variability in Q(1st)Q(11th) were re-sponsible for the negative relation between premiums and slopeThe negative relation is still strong among the rest of the con-tracts as well Columns 4 repeats the exercise for a sample of ve-and ten-year contracts

Column 5 includes only the noncontingent contracts in thesample The purpose of this column is to test the robustness of theresults within noncontingent contracts Recall that to comparecontingent contracts we have to appeal to the equivalence resultin subsection IIIC One might thus wonder about the robustnessof these results once we only compare contracts that do notrequire such a theoretical interpretation By comparing noncon-tingent contracts we test Proposition 2 directly without relyingon the equivalence result The negative relation is conrmed

TABLE VREGRESSION PRESENT VALUES ON SLOPE OF PREMIUMS

log(PV)

(1) (2) (3) (4) (5)

Q(1st)Q(11th) 2106 mdash 2135 2105 2073(21679) mdash (2877) (2484) (2284)

Guarant 001 2002 001 0004 001(295) (2396) (174) (103) (133)

Renew 20002 20003 2002 20001 0001(2122) (2118) (2100) (2001) (039)

Convert 001 001 006 007 0004(319) (285) (341) (273) (156)

Spec Cond 021 011 021 021 033(322) (097) (314) (183) (328)

Constant 963 920 962 938 922(621) (333) (535) (367) (297)

R2 744 166 561 449 538N 125 125 100 57 41

DependentVariable log(PV) is the log of the present value (r 5 8) of the cost to the consumer of twentyyears of coverage starting at age 40 Convert 5 age until policy can be converted to another Guarant 5 yearspremiums are guaranteed Renew 5 last age the policy can be renewed Spec Cond 5 special underwritingconditions like ldquononpreferredriskrdquo Column (1) includes all the contracts in the sample Column (2) excludescontracts slope as an explanatory variable Column (3) includes all contracts but LT20s in the sampleColumn (4) includes only LT5s and LT10s Column (5) includes only noncontingent contracts


Moreover to assess the economic magnitude of the estimatedeffect of front-loading we compare the predicted gap in costbetween the average LT20 and the average aggregate ART Theformer by denition has a slope of 1 while the latter has a slopeof 044 (see Table III) By multiplying the differences in slopes bythe coefcient in the regression (2073) we nd that on averageby decreasing premium slopes from the average aggregate ARTsto a at contract the present value of premiums declines by 377percent

The same picture arises when we look at different ages Inthe working paper [Hendel and Lizzeri 2002] we compare con-tracts across ages We also included xed rm effects to makesure the cross-contract differences are not due to rm differences

IV C Front-Loading and Lapsation

The role of front-loading as a device to provide more insur-ance depends on locking in consumers We now attempt to pro-vide further corroboration for the theoretical prediction that lessfront-loaded contracts suffer from more health-related lapsation

Table VI presents lapsation data from LIMRA [1996] for theperiod 1993ndash1994 The sample includes all the contracts in forceat the beginning of the 1993ndash1994 policy year for 33 top UnitedStates insurers (namely all LIMRA members) In total the sam-ple contains over 7 million policies that were in force in 1993(approximately one-third of all policies in the United States in1993) Lapse rates are percentages of face amount (or number of

TABLE VILAPSATION BY AGE AND CONTRACT TYPE

Contract year

of faceamount of policies of face amount

ART Term ART Term ART Term ART TermAges 20ndash39 Ages 40ndash59

1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39

Data Source LIMRArsquos Long-Term Ordinary Lapse SurveymdashUnited States [1996]Term includes all term level contracts from 2 to 30 years of lengthTerm 5 term contracts other than ATRs


policies) in force at their 1993 policy anniversaries that lapse onor before their 1994 anniversary

These data are not ideal since they do not provide a completebreakdown of lapsation by type of contract It only allows us tocontrast ARTs with term contracts of longer length (2 to 30 yearlevel contracts) The model predicts that longer term contractswhich involve more front-loading should suffer lower rates oflapsation Moreover we expect the difference in lapsing to bemore pronounced when there is more health-related learning(namely for older age groups)

The rst two columns represent lapsation measured as thepercent of the face amounts ldquoContract Yearrdquo measures the age ofthe contract For example ldquoContract Yearrdquo 5 1 means contractsissued during 1993 while ldquoContract Yearrdquo 5 2 means contractsissued in 1992

The numbers are in line with Proposition 2 Aside fromcontracts just issued on which not much lock-in has occurred yetTerm (longer) contracts have lower lapsation and a steeper de-cline in lapsation over time A similar picture appears in the nexttwo columns that measure lapsation as a percent of the number ofpolicies

The second part of the table separates lapsation by agegroups We expect more health-related information to be revealedin the 40ndash59 age-group than in the 20ndash39 Thus we expect theeffect of front-loading in reducing lapsation to be more pronouncedfor the older group As expected we nd that the difference inlapsation between ARTs and Term is bigger for the older group

IV D Accidental Death Insurance A Test Case

Accidental death insurance which is a special type of lifeinsurance contract provides an ideal way to further test whetherthe main forces behind the model are responsible for the shape ofthe available contracts in the industry

An accidental death insurance policy is essentially the sameproduct as a term policy with the exception that it only pays ifdeath is accidental it does not pay if death is due to illnessAccidental death rates are quite at between the ages of 25 to 60and rms are unlikely to learn much about the characteristics ofthe consumers Since learning is a key force behind our resultswe expect the predictions of the model to have less bite in theAccidental insurance market In other words there is no reasonto predict that front-loading will be used to improve long-run


insurance We found that both the probability of an accident andpremiums are at between ages 25 to 60 namely there is nofront-loading (see Jaffe [1998] and quotes available on the web)

Thus a very similar product which involves no learningabout risk characteristics displays no front-loading and no rela-tion between the present value of premiums and front-loadingthis can be viewed as an experiment whereby removing the keyassumption about learning makes the main theoretical predic-tions disappear

V ALTERNATIVE EXPLANATIONS

In this section we explore and attempt to rule out alternativeexplanations for the empirical ndings

Adverse selection One possible concern is the presence ofasymmetric information Cawley and Philipson [1999] found noevidence of adverse selection in term life insurance This can beexplained by the fact that buyers have to pass a medical exami-nation and answer a detailed questionnaire (See Society of Ac-tuaries [1995] for a detailed description of underwriting practicesand the stringency of medical screening) Misrepresentation orconcealment of material information would render the policyvoid Furthermore insurance companies have an informationclearing system where they share information about buyers

Even absent asymmetric information about the current riskcategory in the context of long-term contracts there is anotherpotential source of adverse selection Buyers may have superiorinformation about their living habits which inuence futuredeath probabilities This is plausible However if this asymmetricinformation was an important force more front-loaded contractswould attract buyers with worse future health who are seeking alower cap on their future premiums But then more front-loadingwould be associated with higher cost of coverage We nd theopposite our numbers suggest that the prevailing force shapingcontracts is not asymmetric information otherwise front-loadedcontracts would be more costly

Our discussion of adverse selection is only valid for term lifeinsurance There is evidence that annuity markets suffer fromadverse selection [Brugiavini 1990 Friedman and Warshawsky1990] Note that there is no medical exam for annuities

Fixed underwriting costs Underwriting involves xedcosts Because more front-loaded contracts have lower lapsation


these xed costs are incurred less frequently This would explainthe lower present value of premiums of more front-loadedcontracts

If this hypothesis were correct then as a fraction of the faceamount the savings from lower lapsation would decrease as theface amount increases since the xed underwriting costs wouldbecome relatively less important17 Thus xed costs cannot ac-count for the observed relation between front-loading and thepresent value of premiums

Cross-rm differences in underwriting standards Ifrms specialize in specic types of contracts with those rmsoffering more front-loaded contracts performing stricter under-writing then we would obtain a negative relation between thepresent value of premiums and front-loading We have no reasonto believe that this specialization takes place However to ruleout this explanation we resort to within-rms contract compari-sons performed in two ways First we ran the regression in TableV with rm xed effects and we found that results are unchangedfrom the previous analysis Second we conrmed that the rela-tion between front-loading and cost of coverage holds for everycompany individually that offers several contract types (see foot-note 17)

Correlation mortality-income Since mortality and in-come are negatively correlated the lower cost of more front-loaded contracts could be caused by higher income consumerspurchasing those contracts This explanation can be evaluated bylooking at larger face amounts If income levels (through thecorrelation with mortality) were creating the link between front-loading and present value then that relation should disappear asthe face amount becomes large since only the very rich will buya $10 million policy This does not happen (see footnote 17)

VI CONCLUDING REMARKS

We have shown that term life insurance contracts t thetheoretical predictions of a model with symmetric learning andone-sided commitment They are not in line with alternative

17 We compared present values for different levels of coverage from$100000 to $10 million dollars for all the insurers in our sample that offercomparable ARTs and LT20s For every company the same pattern arises Theratio of present values of the ART over the LT20 is larger than 1 moreover theratio does not decline in face amount (see details in the working paper)


assumptions on commitment They also cannot be explained bystandard asymmetric information models The life insurance in-dustry reacts to the absence of consumer commitment to long-term contracts by front-loading premiums Front-loading createsconsumer lock-in which in turn reduces reclassication risk

Our analysis suggests that in the life insurance industry theextent of the distortions due to dynamic information revelationare not large According to our numbers some of the commonlypurchased term contracts come very close to capturing all thegains from long-term insurance However this achievementcomes at a cost because it requires consumers to pay moremoney up front than would be optimal from the point of view ofintertemporal consumption smoothing It is worth noting thatthe industry achieved this partial solution to the problem ofreclassication risk without need of regulation (eg no imposi-tion of guaranteed renewability)

Understanding and quantifying the inefciency from the lackof bilateral commitment in life insurance is also of interest inlight of the policy debate over health insurance In the healthinsurance market front-loaded contracts are not offered andreclassication risk is a major concern The health care marketsuffers from a variety of other problems What makes the perfor-mance of these two industries so different There are severaldifferences between these markets rst while life insurance isabout insuring an income stream health insurance is abouthealth care treatment Thus the amount needed to be front-loaded to generate long-term insurance is proportional to incomein the life insurance market while independent of income in thehealth insurance market Hence health insurance front-loadingis likely to be unaffordable to low income households Secondtreatment cost risk is nondiversiable since it affects all patients(for more on this argument see Cutler [1993]) Finally healthinsurance involves a quality of service that is susceptible toopportunism on the part of insurance companies (in particularunder managed care) Hence consumers are likely to be morereluctant to lock in to a health insurer who can later oppor-tunistically reduce quality

An interesting connection exists between life insurance andsome credit contracts (eg mortgages) Borrowers cannot easilycommit not to prepay if interest rates drop However there is animportant difference interest rate risk is aggregate risk thatbanks offering mortgages cannot easily diversify In contrast


reclassication risk is an idiosyncratic risk that is fully diversiableThus we expect more insurance against reclassication risk thanof interest rate risk Indeed no more than a couple of ldquopointsrdquo arefront-loaded in common mortgage contracts only moderately re-ducing the present value of long-run payments This is in contrastto the life insurance case where there is a wide range of front-loading with a large impact on long-run payments

Most life insurance contracts are in nominal terms It ispuzzling that contracts in real terms are not more common Apossible explanation is that buyers prefer decreasing levels ofcoverage in real terms the present value of the income to beinsured declines as the buyer ages Furthermore ination-ad-justed contracts would need more front-loading simply becausethe future stakes subject to adverse retention would be biggerthan in a nominal contract This may also explain consumersrsquopreference for nominal contracts Finally this puzzling feature iscommon to a wide variety of securities ination-indexed bondsare not very common

APPENDIX

Proof of Proposition 1

First note that we can replace the set of constraints (3) withthe following simpler set of constraints

(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N

To see this note that if (Q1 F1) (Q21 F2

1) (Q2N F2

N )maximize (1) subject to (2) and (5) then there is no state i and no(Q2

i F2i ) that makes positive prots and gives buyers a higher

utility in that state Thus (3) is satised Conversely supposethat (Q2

i F2i ) are such that (5) is violated Then it is clear that (3)

is violated as well since a competing insurance company can offerterms that are slightly better for buyers than (Q2

i F2i ) and that

still make positive protsLet m be the Lagrange multiplier for the constraint in (2) and

li the Lagrange multiplier for the ith constraint in (5)The rst-order conditions for an optimum are

(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m


(8) 2~1 2 ppiu9~y 1 g 2 Q2i 1 ~1 2 ppim 1 li 5 0 i

(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i

Part (i) follows from rst combining equations (6) and (7) andthen combining equations (8) and (9)

To prove part (ii) note rst that equations (8) and (9) implythat

(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i

If constraint i in equation (5) is binding (1 2 pi)Q2i 5 piF2

i Substituting from equation (11) we obtain

(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i

From equation (12) we see that since u9 and v9 are decreasingfunctions if i and j are two binding constraints with i j (so thatpi pj) then Q2

i Q2j

Suppose now that constraint k in equation (5) is nonbindingThen equation (8) simplies to

(13) u9~ y 1 g 2 Q2k 5 m

Thus if constraints k and l are nonbinding Q2k 5 Q2

l If in contrast constraint i in equation (5) is binding

(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m

where the inequality holds because if constraint i is binding li 0Thus if constraint i is binding and k is not Q2

i Q2k We now

want to show that if i is binding and k is not then i k Since kis not binding substituting from equation (11) we obtain

(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2

k and pi pk render equations (12) and (15)incompatible proving that indeed pi pk

To prove part (iii) we need to show that the rst-periodpremium Q1 must be larger than actuarially fair premiumQ1(FI) If any of the no-lapsation constraints (5) is not bindingthen Q1 Q1(FI) is immediate from the zero prot condition forthe insurance company (equation (2))


Suppose instead that all the no-lapsation constraints are bind-ing Substituting from equation (6) into equation (8) we obtain

(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i

Thus Q1 Q2N (FI) 2 2g This inequality clearly requires

that Q1 Q1(FI) if g is small enough since pN pPart (iv) is obvious as g increases more and more of the

no-lapsation constraints become binding This means that as ggrows s becomes larger When s is larger Q1 declines

UNIVERSITY OF WISCONSIN AND NATIONAL BUREAU OF ECONOMIC RESEARCH

NEW YORK UNIVERSITY

REFERENCES

Bestrsquos Review ldquoThe Industry Respondsrdquo September 1996Black Keneth and Harold Skipper Jr Life Insurance (Englewood Cliffs NJ

Prentice Hall 1994)Brugiavini Agar ldquoLongevity Risk and the Life Cyclerdquo PhD thesis London

School of Economics London 1990Cawley John and Tomas Philipson ldquoAn Empirical Examination of Information

Barriers to Trade in Insurancerdquo American Economic-Review LXXX (1999)827ndash846

Chiappori Pierre Andre Bernard Salanie and Julie Valentin ldquoEarly Startersversus Late Beginnersrdquo Journal of Political Economy CVII (1999) 731ndash760

Cochrane John ldquoTime Consistent Health Insurancerdquo Journal of Political Econ-omy CIII (1995) 445ndash473

Crocker Keith and John Moran ldquoContracting with Limited Commitment Evi-dence from Employment Based Insurance Contractsrdquo University of Michigan1998

Cooter Robert and Thomas Ulen Law and Economics (Reading MA Addison-Wesley 2000)

Cutler David ldquoWhy Doesnrsquot the Market Fully Insure Long-Term Carerdquo NBERworking paper No 4301 1993

Daily Glenn The Individuals Investorrsquos Guide to Low-Load Insurance Products(Chicago IL International Publishing Corporation 1989)

De Garidel Tomas ldquoPareto Improving Asymmetric Information in a DynamicInsurance Marketrdquo University College London 1999

Diamond Peter ldquoOrganizing the Health Insurance Marketrdquo Econometrica LX(1992) 1233ndash1254

Dionne Georges and Neil Doherty ldquoAdverse Selection Commitment and Rene-gotiation Extension to and Evidence from Insurance Marketsrdquo Journal ofPolitical Economy CII (1994) 210ndash235

Friedman Benjamin and Mark Warshawsky ldquoThe Cost of Annuities Implica-tions for Savings Behavior and Bequestsrdquo Quarterly Journal of EconomicsCV (1990) 135ndash154

Harris Milton and Bengt Holmstrom ldquoA Theory of Wage Dynamicsrdquo Review ofEconomic Studies XL (1982) 315ndash333

Hendel Igal and Alessandro Lizzeri ldquoThe Role of Commitment in DynamicContracts Evidence from Life Insurancerdquo working paper 2002

Individual Life Insurance Experience Committee 1985ndash90 Basic Select and Ul-timate Mortality Tables 1999

Jaffe Jay M ldquoAccidental Death Experience A Review of Recent Experience forthe Practicing Actuaryrdquo working paper 1998

Life Insurance Fact Book American Council of Life Insurance several issuesLIMRA ldquoLong-Term Ordinary Lapse SurveymdashU Srdquo Connecticut 1996


LIMRA ldquoBuyerrsquos Studyrdquo Connecticut 1997McGill Dan Life Insurance (Homewood IL Richard D Irwin 1967)Murphy Kevin ldquoIncentives Learning and Compensation A Theoretical and

Empirical Investigation of Managerial Labor Contractsrdquo Rand Journal ofEconomics XVII (1986) 59ndash76

Pauly Mark Howard Kunreuther and Richard Hirth ldquoGuaranteed Renewabilityin Insurancerdquo Journal of Risk and Uncertainty X (1995) 143ndash156

Society of Actuaries Report on Preferred Underwriting 1995Winter Ralph ldquoOn the Rate Structure of the American Life Insurance Marketrdquo

Journal of Finance XXXVI (1981) 81ndash96


relations such as labor contracts are plagued by unobservablesand by the presence of implicit agreements Finally very richdata on contracts are available1

We rst present a theoretical framework to understand howwe should expect lack of commitment to be reected in observedcontracts The model adapts to the life insurance market theHarris-Holmstrom [1982] model of symmetric learning that wasrst developed in a labor market context The main predictions ofthe model are then tested using data on the United States lifeinsurance industry

We assume that the initial health status of the insured isknown to all and that new information about her health type isrevealed over time symmetrically to all market participants Wealso assume that there is one-sided commitment ie insurancecompanies can commit to long-term contracts but buyers cannot

Because of learning short-term (spot) contracts involve re-classication risk future premiums will reect the future healthstatus of consumers Long-term contracts would solve the prob-lem if consumers could commit insurance companies would offerfuture premiums reecting the average future health of the poolLack of commitment makes such a contract infeasible healthyconsumers would drop out to obtain cheaper coverage from com-peting companies Hence to obtain insurance against reclassi-cation risk consumers must prepay some of the premiums Theprepayment or front-loading locks the consumer into the con-tract To be completely successful the front-loaded amount mustguarantee that no consumers drop coverage in the future thefuture premium must be no higher than the fair premium for thehealthiest type Such a contract would implement the same allo-cation that would be achieved under consumer commitment itprovides full insurance against reclassication risk

All contracts we observe in the life insurance market in theUnited States (see subsection IVA) are front-loaded and some ofthe contracts come close to achieving full insurance Howevermost of these contracts leave consumers subject to some reclas-

1 Previous empirical work on dynamic contracts includes ChiapporiSalaniersquo and Valentin [1999] who test a model of wage dynamics using data fromone large French rm They nd support for theories of learning with downwardrigidity of wages Murphy [1986] investigates a longitudinal sample of CEOs andcompares learning with incentives his evidence is mixed Dionne and Doherty[1994] present a dynamic model of adverse selection and with evidence from autoinsurance Crocker and Moran [1998] studies employee lock-in as a way to in-crease commitment to long-term health insurance









II A The Contracts













Age ART LT10

SampU ART

Policy year

1 2 3 10 11 19 20

40 459 574 370 475 640 1485 2555 5680 637541 499 574 385 490 660 1565 2815 6375 704042 539 574 400 530 690 1705 3105 7040 7790

49 909 574 630 890 1080 2725 6375 13675 1478550 974 1064 690 945 1155 2895 7040 14785 1576551 1044 1064 735 1050 1295 3230 7790 15765 17230

58 2009 1064 1245 1750 2295 6420 14785 33165 3544559 2289 1064 1340 1785 2480 6945 15765 35445 38715






II B The Data








III THEORY

















III A Model










1F21) (Q2















1 F21) (Q2


utility

(1) pv~F1 1 ~1 2 pu~ y1 2 g 2 Q1

1 ~1 2 p Oi51

N

p i piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i


(2) ~1 2 pQ1 2 pF1 1 ~1 2 p Oi51

N


i 5 0



i 2 piF2i 0

(3) piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i

$ piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i




(4) v9~F2i 5 u9~ y2 1 g 2 Q2

i









i




















i 5 Q2s and F2



s F2s ) in the
















IV A Front-loading








SampU ART

Age



1 2 3 4 5 6 740 645 100 866 100 473 100 47342 739 60 866 52 700 77 51344 852 50 866 38 926 74 55346 1000 44 866 29 1202 73 61848 1184 39 866 21 1593 71 70850 1395 37 866 17 2500 91 81352 1611 34 866 14 3073 90 93754 1877 30 866 10 3825 84 109156 2223 27 866 8 4690 78 129358 2746 27 866 6 5795 78 1583






















Q(1st)Q(11th) 043 031 011 100PV 16055 5245 6871 28754ln(PV) 962 036 883 1027








log(PV)

(1) (2) (3) (4) (5)


Guarant 001 2002 001 0004 001(295) (2396) (174) (103) (133)

Renew 20002 20003 2002 20001 0001(2122) (2118) (2100) (2001) (039)

Convert 001 001 006 007 0004(319) (285) (341) (273) (156)

Spec Cond 021 011 021 021 033(322) (097) (314) (183) (328)

Constant 963 920 962 938 922(621) (333) (535) (367) (297)

R2 744 166 561 449 538N 125 125 100 57 41









Contract year



1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39




































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES


































II A The Contracts













Age ART LT10

SampU ART

Policy year

1 2 3 10 11 19 20

40 459 574 370 475 640 1485 2555 5680 637541 499 574 385 490 660 1565 2815 6375 704042 539 574 400 530 690 1705 3105 7040 7790

49 909 574 630 890 1080 2725 6375 13675 1478550 974 1064 690 945 1155 2895 7040 14785 1576551 1044 1064 735 1050 1295 3230 7790 15765 17230

58 2009 1064 1245 1750 2295 6420 14785 33165 3544559 2289 1064 1340 1785 2480 6945 15765 35445 38715






II B The Data








III THEORY

















III A Model










1F21) (Q2















1 F21) (Q2


utility

(1) pv~F1 1 ~1 2 pu~ y1 2 g 2 Q1

1 ~1 2 p Oi51

N

p i piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i


(2) ~1 2 pQ1 2 pF1 1 ~1 2 p Oi51

N


i 5 0



i 2 piF2i 0

(3) piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i

$ piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i




(4) v9~F2i 5 u9~ y2 1 g 2 Q2

i









i




















i 5 Q2s and F2



s F2s ) in the
















IV A Front-loading








SampU ART

Age



1 2 3 4 5 6 740 645 100 866 100 473 100 47342 739 60 866 52 700 77 51344 852 50 866 38 926 74 55346 1000 44 866 29 1202 73 61848 1184 39 866 21 1593 71 70850 1395 37 866 17 2500 91 81352 1611 34 866 14 3073 90 93754 1877 30 866 10 3825 84 109156 2223 27 866 8 4690 78 129358 2746 27 866 6 5795 78 1583






















Q(1st)Q(11th) 043 031 011 100PV 16055 5245 6871 28754ln(PV) 962 036 883 1027








log(PV)

(1) (2) (3) (4) (5)


Guarant 001 2002 001 0004 001(295) (2396) (174) (103) (133)

Renew 20002 20003 2002 20001 0001(2122) (2118) (2100) (2001) (039)

Convert 001 001 006 007 0004(319) (285) (341) (273) (156)

Spec Cond 021 011 021 021 033(322) (097) (314) (183) (328)

Constant 963 920 962 938 922(621) (333) (535) (367) (297)

R2 744 166 561 449 538N 125 125 100 57 41









Contract year



1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39




































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES































Age ART LT10

SampU ART

Policy year

1 2 3 10 11 19 20

40 459 574 370 475 640 1485 2555 5680 637541 499 574 385 490 660 1565 2815 6375 704042 539 574 400 530 690 1705 3105 7040 7790

49 909 574 630 890 1080 2725 6375 13675 1478550 974 1064 690 945 1155 2895 7040 14785 1576551 1044 1064 735 1050 1295 3230 7790 15765 17230

58 2009 1064 1245 1750 2295 6420 14785 33165 3544559 2289 1064 1340 1785 2480 6945 15765 35445 38715






II B The Data








III THEORY

















III A Model










1F21) (Q2















1 F21) (Q2


utility

(1) pv~F1 1 ~1 2 pu~ y1 2 g 2 Q1

1 ~1 2 p Oi51

N

p i piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i


(2) ~1 2 pQ1 2 pF1 1 ~1 2 p Oi51

N


i 5 0



i 2 piF2i 0

(3) piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i

$ piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i




(4) v9~F2i 5 u9~ y2 1 g 2 Q2

i









i




















i 5 Q2s and F2



s F2s ) in the
















IV A Front-loading








SampU ART

Age



1 2 3 4 5 6 740 645 100 866 100 473 100 47342 739 60 866 52 700 77 51344 852 50 866 38 926 74 55346 1000 44 866 29 1202 73 61848 1184 39 866 21 1593 71 70850 1395 37 866 17 2500 91 81352 1611 34 866 14 3073 90 93754 1877 30 866 10 3825 84 109156 2223 27 866 8 4690 78 129358 2746 27 866 6 5795 78 1583






















Q(1st)Q(11th) 043 031 011 100PV 16055 5245 6871 28754ln(PV) 962 036 883 1027








log(PV)

(1) (2) (3) (4) (5)


Guarant 001 2002 001 0004 001(295) (2396) (174) (103) (133)

Renew 20002 20003 2002 20001 0001(2122) (2118) (2100) (2001) (039)

Convert 001 001 006 007 0004(319) (285) (341) (273) (156)

Spec Cond 021 011 021 021 033(322) (097) (314) (183) (328)

Constant 963 920 962 938 922(621) (333) (535) (367) (297)

R2 744 166 561 449 538N 125 125 100 57 41









Contract year



1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39




































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES





























II B The Data








III THEORY

















III A Model










1F21) (Q2















1 F21) (Q2


utility

(1) pv~F1 1 ~1 2 pu~ y1 2 g 2 Q1

1 ~1 2 p Oi51

N

p i piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i


(2) ~1 2 pQ1 2 pF1 1 ~1 2 p Oi51

N


i 5 0



i 2 piF2i 0

(3) piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i

$ piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i




(4) v9~F2i 5 u9~ y2 1 g 2 Q2

i









i




















i 5 Q2s and F2



s F2s ) in the
















IV A Front-loading








SampU ART

Age



1 2 3 4 5 6 740 645 100 866 100 473 100 47342 739 60 866 52 700 77 51344 852 50 866 38 926 74 55346 1000 44 866 29 1202 73 61848 1184 39 866 21 1593 71 70850 1395 37 866 17 2500 91 81352 1611 34 866 14 3073 90 93754 1877 30 866 10 3825 84 109156 2223 27 866 8 4690 78 129358 2746 27 866 6 5795 78 1583






















Q(1st)Q(11th) 043 031 011 100PV 16055 5245 6871 28754ln(PV) 962 036 883 1027








log(PV)

(1) (2) (3) (4) (5)


Guarant 001 2002 001 0004 001(295) (2396) (174) (103) (133)

Renew 20002 20003 2002 20001 0001(2122) (2118) (2100) (2001) (039)

Convert 001 001 006 007 0004(319) (285) (341) (273) (156)

Spec Cond 021 011 021 021 033(322) (097) (314) (183) (328)

Constant 963 920 962 938 922(621) (333) (535) (367) (297)

R2 744 166 561 449 538N 125 125 100 57 41









Contract year



1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39




































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES





























III THEORY

















III A Model










1F21) (Q2















1 F21) (Q2


utility

(1) pv~F1 1 ~1 2 pu~ y1 2 g 2 Q1

1 ~1 2 p Oi51

N

p i piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i


(2) ~1 2 pQ1 2 pF1 1 ~1 2 p Oi51

N


i 5 0



i 2 piF2i 0

(3) piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i

$ piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i




(4) v9~F2i 5 u9~ y2 1 g 2 Q2

i









i




















i 5 Q2s and F2



s F2s ) in the
















IV A Front-loading








SampU ART

Age



1 2 3 4 5 6 740 645 100 866 100 473 100 47342 739 60 866 52 700 77 51344 852 50 866 38 926 74 55346 1000 44 866 29 1202 73 61848 1184 39 866 21 1593 71 70850 1395 37 866 17 2500 91 81352 1611 34 866 14 3073 90 93754 1877 30 866 10 3825 84 109156 2223 27 866 8 4690 78 129358 2746 27 866 6 5795 78 1583






















Q(1st)Q(11th) 043 031 011 100PV 16055 5245 6871 28754ln(PV) 962 036 883 1027








log(PV)

(1) (2) (3) (4) (5)


Guarant 001 2002 001 0004 001(295) (2396) (174) (103) (133)

Renew 20002 20003 2002 20001 0001(2122) (2118) (2100) (2001) (039)

Convert 001 001 006 007 0004(319) (285) (341) (273) (156)

Spec Cond 021 011 021 021 033(322) (097) (314) (183) (328)

Constant 963 920 962 938 922(621) (333) (535) (367) (297)

R2 744 166 561 449 538N 125 125 100 57 41









Contract year



1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39




































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES
































III A Model










1F21) (Q2















1 F21) (Q2


utility

(1) pv~F1 1 ~1 2 pu~ y1 2 g 2 Q1

1 ~1 2 p Oi51

N

p i piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i


(2) ~1 2 pQ1 2 pF1 1 ~1 2 p Oi51

N


i 5 0



i 2 piF2i 0

(3) piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i

$ piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i




(4) v9~F2i 5 u9~ y2 1 g 2 Q2

i









i




















i 5 Q2s and F2



s F2s ) in the
















IV A Front-loading








SampU ART

Age



1 2 3 4 5 6 740 645 100 866 100 473 100 47342 739 60 866 52 700 77 51344 852 50 866 38 926 74 55346 1000 44 866 29 1202 73 61848 1184 39 866 21 1593 71 70850 1395 37 866 17 2500 91 81352 1611 34 866 14 3073 90 93754 1877 30 866 10 3825 84 109156 2223 27 866 8 4690 78 129358 2746 27 866 6 5795 78 1583






















Q(1st)Q(11th) 043 031 011 100PV 16055 5245 6871 28754ln(PV) 962 036 883 1027








log(PV)

(1) (2) (3) (4) (5)


Guarant 001 2002 001 0004 001(295) (2396) (174) (103) (133)

Renew 20002 20003 2002 20001 0001(2122) (2118) (2100) (2001) (039)

Convert 001 001 006 007 0004(319) (285) (341) (273) (156)

Spec Cond 021 011 021 021 033(322) (097) (314) (183) (328)

Constant 963 920 962 938 922(621) (333) (535) (367) (297)

R2 744 166 561 449 538N 125 125 100 57 41









Contract year



1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39




































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES





























1F21) (Q2















1 F21) (Q2


utility

(1) pv~F1 1 ~1 2 pu~ y1 2 g 2 Q1

1 ~1 2 p Oi51

N

p i piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i


(2) ~1 2 pQ1 2 pF1 1 ~1 2 p Oi51

N


i 5 0



i 2 piF2i 0

(3) piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i

$ piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i




(4) v9~F2i 5 u9~ y2 1 g 2 Q2

i









i




















i 5 Q2s and F2



s F2s ) in the
















IV A Front-loading








SampU ART

Age



1 2 3 4 5 6 740 645 100 866 100 473 100 47342 739 60 866 52 700 77 51344 852 50 866 38 926 74 55346 1000 44 866 29 1202 73 61848 1184 39 866 21 1593 71 70850 1395 37 866 17 2500 91 81352 1611 34 866 14 3073 90 93754 1877 30 866 10 3825 84 109156 2223 27 866 8 4690 78 129358 2746 27 866 6 5795 78 1583






















Q(1st)Q(11th) 043 031 011 100PV 16055 5245 6871 28754ln(PV) 962 036 883 1027








log(PV)

(1) (2) (3) (4) (5)


Guarant 001 2002 001 0004 001(295) (2396) (174) (103) (133)

Renew 20002 20003 2002 20001 0001(2122) (2118) (2100) (2001) (039)

Convert 001 001 006 007 0004(319) (285) (341) (273) (156)

Spec Cond 021 011 021 021 033(322) (097) (314) (183) (328)

Constant 963 920 962 938 922(621) (333) (535) (367) (297)

R2 744 166 561 449 538N 125 125 100 57 41









Contract year



1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39




































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES































1 F21) (Q2


utility

(1) pv~F1 1 ~1 2 pu~ y1 2 g 2 Q1

1 ~1 2 p Oi51

N

p i piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i


(2) ~1 2 pQ1 2 pF1 1 ~1 2 p Oi51

N


i 5 0



i 2 piF2i 0

(3) piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i

$ piv~F2i 1 ~1 2 piu~ y2 1 g 2 Q2

i




(4) v9~F2i 5 u9~ y2 1 g 2 Q2

i









i




















i 5 Q2s and F2



s F2s ) in the
















IV A Front-loading








SampU ART

Age



1 2 3 4 5 6 740 645 100 866 100 473 100 47342 739 60 866 52 700 77 51344 852 50 866 38 926 74 55346 1000 44 866 29 1202 73 61848 1184 39 866 21 1593 71 70850 1395 37 866 17 2500 91 81352 1611 34 866 14 3073 90 93754 1877 30 866 10 3825 84 109156 2223 27 866 8 4690 78 129358 2746 27 866 6 5795 78 1583






















Q(1st)Q(11th) 043 031 011 100PV 16055 5245 6871 28754ln(PV) 962 036 883 1027








log(PV)

(1) (2) (3) (4) (5)


Guarant 001 2002 001 0004 001(295) (2396) (174) (103) (133)

Renew 20002 20003 2002 20001 0001(2122) (2118) (2100) (2001) (039)

Convert 001 001 006 007 0004(319) (285) (341) (273) (156)

Spec Cond 021 011 021 021 033(322) (097) (314) (183) (328)

Constant 963 920 962 938 922(621) (333) (535) (367) (297)

R2 744 166 561 449 538N 125 125 100 57 41









Contract year



1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39




































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES





























(4) v9~F2i 5 u9~ y2 1 g 2 Q2

i









i




















i 5 Q2s and F2



s F2s ) in the
















IV A Front-loading








SampU ART

Age



1 2 3 4 5 6 740 645 100 866 100 473 100 47342 739 60 866 52 700 77 51344 852 50 866 38 926 74 55346 1000 44 866 29 1202 73 61848 1184 39 866 21 1593 71 70850 1395 37 866 17 2500 91 81352 1611 34 866 14 3073 90 93754 1877 30 866 10 3825 84 109156 2223 27 866 8 4690 78 129358 2746 27 866 6 5795 78 1583






















Q(1st)Q(11th) 043 031 011 100PV 16055 5245 6871 28754ln(PV) 962 036 883 1027








log(PV)

(1) (2) (3) (4) (5)


Guarant 001 2002 001 0004 001(295) (2396) (174) (103) (133)

Renew 20002 20003 2002 20001 0001(2122) (2118) (2100) (2001) (039)

Convert 001 001 006 007 0004(319) (285) (341) (273) (156)

Spec Cond 021 011 021 021 033(322) (097) (314) (183) (328)

Constant 963 920 962 938 922(621) (333) (535) (367) (297)

R2 744 166 561 449 538N 125 125 100 57 41









Contract year



1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39




































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES







































i 5 Q2s and F2



s F2s ) in the
















IV A Front-loading








SampU ART

Age



1 2 3 4 5 6 740 645 100 866 100 473 100 47342 739 60 866 52 700 77 51344 852 50 866 38 926 74 55346 1000 44 866 29 1202 73 61848 1184 39 866 21 1593 71 70850 1395 37 866 17 2500 91 81352 1611 34 866 14 3073 90 93754 1877 30 866 10 3825 84 109156 2223 27 866 8 4690 78 129358 2746 27 866 6 5795 78 1583






















Q(1st)Q(11th) 043 031 011 100PV 16055 5245 6871 28754ln(PV) 962 036 883 1027








log(PV)

(1) (2) (3) (4) (5)


Guarant 001 2002 001 0004 001(295) (2396) (174) (103) (133)

Renew 20002 20003 2002 20001 0001(2122) (2118) (2100) (2001) (039)

Convert 001 001 006 007 0004(319) (285) (341) (273) (156)

Spec Cond 021 011 021 021 033(322) (097) (314) (183) (328)

Constant 963 920 962 938 922(621) (333) (535) (367) (297)

R2 744 166 561 449 538N 125 125 100 57 41









Contract year



1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39




































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES







































IV A Front-loading








SampU ART

Age



1 2 3 4 5 6 740 645 100 866 100 473 100 47342 739 60 866 52 700 77 51344 852 50 866 38 926 74 55346 1000 44 866 29 1202 73 61848 1184 39 866 21 1593 71 70850 1395 37 866 17 2500 91 81352 1611 34 866 14 3073 90 93754 1877 30 866 10 3825 84 109156 2223 27 866 8 4690 78 129358 2746 27 866 6 5795 78 1583






















Q(1st)Q(11th) 043 031 011 100PV 16055 5245 6871 28754ln(PV) 962 036 883 1027








log(PV)

(1) (2) (3) (4) (5)


Guarant 001 2002 001 0004 001(295) (2396) (174) (103) (133)

Renew 20002 20003 2002 20001 0001(2122) (2118) (2100) (2001) (039)

Convert 001 001 006 007 0004(319) (285) (341) (273) (156)

Spec Cond 021 011 021 021 033(322) (097) (314) (183) (328)

Constant 963 920 962 938 922(621) (333) (535) (367) (297)

R2 744 166 561 449 538N 125 125 100 57 41









Contract year



1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39




































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES
































SampU ART

Age



1 2 3 4 5 6 740 645 100 866 100 473 100 47342 739 60 866 52 700 77 51344 852 50 866 38 926 74 55346 1000 44 866 29 1202 73 61848 1184 39 866 21 1593 71 70850 1395 37 866 17 2500 91 81352 1611 34 866 14 3073 90 93754 1877 30 866 10 3825 84 109156 2223 27 866 8 4690 78 129358 2746 27 866 6 5795 78 1583






















Q(1st)Q(11th) 043 031 011 100PV 16055 5245 6871 28754ln(PV) 962 036 883 1027








log(PV)

(1) (2) (3) (4) (5)


Guarant 001 2002 001 0004 001(295) (2396) (174) (103) (133)

Renew 20002 20003 2002 20001 0001(2122) (2118) (2100) (2001) (039)

Convert 001 001 006 007 0004(319) (285) (341) (273) (156)

Spec Cond 021 011 021 021 033(322) (097) (314) (183) (328)

Constant 963 920 962 938 922(621) (333) (535) (367) (297)

R2 744 166 561 449 538N 125 125 100 57 41









Contract year



1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39




































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES











































Q(1st)Q(11th) 043 031 011 100PV 16055 5245 6871 28754ln(PV) 962 036 883 1027








log(PV)

(1) (2) (3) (4) (5)


Guarant 001 2002 001 0004 001(295) (2396) (174) (103) (133)

Renew 20002 20003 2002 20001 0001(2122) (2118) (2100) (2001) (039)

Convert 001 001 006 007 0004(319) (285) (341) (273) (156)

Spec Cond 021 011 021 021 033(322) (097) (314) (183) (328)

Constant 963 920 962 938 922(621) (333) (535) (367) (297)

R2 744 166 561 449 538N 125 125 100 57 41









Contract year



1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39




































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES































log(PV)

(1) (2) (3) (4) (5)


Guarant 001 2002 001 0004 001(295) (2396) (174) (103) (133)

Renew 20002 20003 2002 20001 0001(2122) (2118) (2100) (2001) (039)

Convert 001 001 006 007 0004(319) (285) (341) (273) (156)

Spec Cond 021 011 021 021 033(322) (097) (314) (183) (328)

Constant 963 920 962 938 922(621) (333) (535) (367) (297)

R2 744 166 561 449 538N 125 125 100 57 41









Contract year



1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39




































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES

































Contract year



1 118 142 150 212 143 182 144 1082 141 114 148 141 153 132 164 953ndash5 134 80 124 74 125 74 162 646ndash10 101 50 94 52 92 51 141 45111 70 39 65 38 70 41 98 39




































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES




























































APPENDIX



(5) ~1 2 piQ2i 2 piF2

i 0 for i 5 1 N


1) (Q2N F2






i F2i ) and that



(6) u9~ y 2 g 2 Q1 5 m

(7) v9~F1 5 m



(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES




























(9) ~1 2 ppiv9~F2i 2 ~1 2 pp im 2 li 5 0 i

(10) ~~1 2 piQ2i 2 piF2

i l i 5 0 i



(11) F2i 5 ~v921~u9~ y 1 g 2 Q2

i



(12) ~1 2 piQ2i 5 pi~v921~u9~ y 1 g 2 Q2

i


i Q2j


(13) u9~ y 1 g 2 Q2k 5 m



(14) u9~ y 1 g 2 Q2i 5 m 1 li~1 2 pp i m


i Q2k We now


(15) ~1 2 pkQ2k pk~v921~u9~ y 1 g 2 Q2

k

Thus Q2i Q2





(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES




























(16) u9~ y 1 g 2 Q2i 5 u9~ y 2 g 2 Q1 1 li~1 2 pp i





NEW YORK UNIVERSITY

REFERENCES



























THE ROLE OF COMMITMENT IN DYNAMIC CONTRACTS: EVIDENCE...

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