Some Propositions on Intergenerational Risk Sharing, Social Security and Self-Insurance

Transcription

1 MA Munih ersonal ee Arhie Some roositions on Intergenerational isk Sharing Soial Seurity and SelfInsurane Aoki Takaaki State Uniersity of New York at Buffalo Deartment of Eonomis August 6 Online at htt://mra.ub.unimuenhen.de/684/ MA aer No. 684 osted. Deember 8 / :58

2 Some roositions on Intergenerational isk Sharing Soial Seurity and SelfInsurane * Takaaki Aoki ** * JEL lassifiation: A D D8 H J. This aer is written by summarizing a art of results obtained in a series of my reious working aers: Aoki Takaaki 3 ** State Uniersity of New York at Buffalo Deartment of Eonomis 45 Fronzak Hall North Camus Buffalo NY 46 USA Tel: taoki@asu.buffalo.edu

3 Abstrat This artile desribes within a myoi intergenerational bargaining framework inororating two disrete eriods and binary states of risks some new asets regarding the mixture of intergenerational risk sharing and soial seurity. Here statedeendent utility under mortality risk roes to generate arents euliar indifferene ure regarding insurane ontrat and selfinsurane is shown to lay a ruial role on the deision regarding soial seurity holding and intergenerational transfer ontrat. This euliar aset gien for the first time in this artile also deries some noel features of insurane theory under lifetime unertainty where the urrent osition in soial seurity ontrat ould adersely affet arents deision regarding intergenerational risk sharing with hildren. In addition other basi results regarding the sensitiity to default risk and taxation in soial seurity are summarized.. Introdution The obetie of this artile is simly and learly to desribe some new eonomi asets of intergenerational risk sharing under lifetime unertainty within a myoi bargaining framework. Atkinson and Stiglitz 98 and Obstfeld and ogoff 996 are textbooks of ubli eonomis and international maroeonomis eseially the latter of whih ontains the desrition of a risk sharing with default risk and saing. Ehrlih and Beker 97 deelo a basi theory of demand for insurane that emhasizes the interation between market insurane selfinsurane and selfrotetion. For some other examles among related literatures Shiller 999 Ball and Mankiw Enders and Laan 98 and Gordon and Varian 988 examine the eonomi role of intergenerational risk sharing. Yaari 965 is a lassial artile whih ursuits the imliations of life insurane under the

4 mortality risk. Hayashi Altoni and otlikoff 996 is an emirial work on intragamily & intergenerational risk sharing aomanying the ossibility of selfinsurane. Analyses on bequest moties aear among many in Abel Hurd 989 Bernheim Shleifer and Summers 985 and Bernheim 99. In this artile we newly fous on a euliar shae of arents indifferene ure whih arises from a state deendent aset of utility under a life unertainty enironment using a model of two disrete eriods and binary states of mortality/inome risks. In setion a basi framework is set in whih two adaent generations arents in old adulthood and hildren in young adulthood are faing the deision regarding intergenerational risk sharing with/without an aailable oldage soial seurity for arents. In setion 3 some harateristis in arents differene ure are exlained where selfinsurane lays an imortant role on the insurane ontrat deision. On the basis of these analyses we laim some fundamental roositions regarding the otimal alloation of soial seurity and intergenerational risk sharing in setion 4 and some regarding the sensitiity to default risk and taxation in soial seurity in setion 5.. Basi setting At first we diide eah generation s lifetime roughly into three stages Y for the hild M for the young adulthood and O for the old adulthood eah of whih orresonds with eah disrete eriod 3 years. Assume that at the beginning of eriod t two adaent generations arents and hildren are now going to begin stage O and M resetiely. arents hold an aailable asset A and hildren s disosable inome isw. During this eriod there exist two tyes of binary risk the risk of death This is a tyial oerlaing generation model. These notations are also used in suersrits/subsrits. 3

5 mortality risk s for arents and that of disosable inome s for hildren. The risk of death exists is reealed exatly at the middle oint of stage O when they are alie s with robability or die s with robability. The inome risk is reealed exatly at the middle oint of stage M when they earn the higher inome s with robability ' or the lower inome W W s with '. Therefore the reelation of mortality risk for arents and that of inome risk for hildren exatly oinide with eah other in time. Eah generation i holds an egoisti utility whih deends exliitly only on its own onsumtion only during stage M and O not during Y and takes a form of statedeendent utility: u i u i i i i i i i i i u + b u + b { u + bu } if s M M O O i i i i i i u + b u + b u if s M M O s u i / s if s ¹ or u i ln if s. Here b is a onstant time referene for eah half eriod s is a onstant relatie risk aersion oeffiient i M is a real onsumtion of generation i during the first half eriod of stage M i O is a real onsumtion of generation i during the seond half eriod of stage O or et. Utility funtion u is inreasing and onae and assumes ordinary Inada onditions. The real interest rate for eah half eriod is denoted by r. 3 Children s life strategies during stage M number of hildren to bear N human aital inestment for eah hild H asset lan A are exogenously gien exet for the intergenerational transfer ontrats with arents S B. For arents there are two otions of old adulthood insurane for mortality risk intergenerational transfer with hildren S B and soial 3 b / + r is assumed ust for oneniene but without loss of generality. 4

6 seurity. 4 5 If these transfer ontrats are atuarially fair they neessarily satisfy: l : S B or l :. At first assume that s and s are unorrelated. Then the assoiated indiret utilities for arents and hildren regarding a transfer ontrat shedule S B are reresented as: S B b r A º u A B + bu + r S + B.4a 6 æ[ u ö + bu + r W S ] S B b r W º maxç.4b 7 è+ [ u + bu + r W + B ] ø 3. Some euliar asets of intergenerational risk sharing See Figure. The dotted line a and d are the indifferene ures of arents whih draws the ontour lines b S B r A for distint onstants s. On the other hand b e and f are those of hildren whih draws the ontour lines b S B r W for distint s. Under the settings of setion there exists a set of intergenerational transfer ontrat S B suh that: both arents and hildren are willing to onlude the ontrat. artiiation onstraints and eah ontrat is aretootimum. areto otimality onditions Furthermore the omat set denoted by X satisfying the aboe onditions and is loated inside the area S > B > 4 B S and denote Suort Bequest and e eit ayment S B and denotes reeit with ourrene of s s the lefthand side of resetiely. Here and the righthand side does ayment with both sides measured by resent alue at the beginning of the eriod. 5 In this artile we do not set any substantial distintion between soial seurity and market insurane. 6 If soial seurity rogram is also aailable then: S + B + b r A º u A B + bu + r S + + B +.4a 7 Under no orrelation between s and s is W ' W + ' W º. W is defined as the exeted disosable inome that 5

7 and B > S. X is a bold segment line GF where oint F is a tangent oint of arents indifferene ure a and that of hildren e and oint G is a tangent oint of hildren s indifferene ure b and that of arents d. Any indifferene ures of arents a or d are shown to be tangent with two lines B A and ertainty line l : B S. Children if they do not onlude any transfer ontrat will be undoubtedly at oint O that is S B being tangent with the atuarially fair line l : S B. In general as illustrated in the indifferene ure f any arbitrary indifferene ure of hildren is tangent with the onstant remium line l ': B S k at the intersetion of l ' and l : B S. See Figure. Now we examine arents osition within a gien transfer ontrat sheme S B. If arents are not gien any transfer ontrat their osition is illustrated as / s / s oint D : S B ba / + b axis S where a is tangent with the horizontal exatly at oint D. 8 Here B is not an amount of bequest but is some onditional ost on death s to be additionally disarded as a result of artially selfinsuring mortality risk s. 9 Thus oint D is an otimal selfinsurane selfontrat whih arents would hoose when the soial seurity is not aailable. Instead if arents make the atuarially fair and flexible soial seurity ontrat their osition is oint E : ba / + b ba / + b where their indifferene ure is tangent with l : at oint E. See oint I and J both loated on l. I is the oint where a whih asses oint D intersets with l. Therefore I is a 8 In this ase arents maximization roblem is equialent with maximizing their indiret utility assoiated with the transfer ontrat S B.4a with regard to B keeing S fixed at. 9 This ost is aid along axis B. Note that arents maximization roblem is equialent with maximizing their assoiated indiret utility.4a with regard to S B satisfyingl. Here intergenerational transfer S B is relaed with the notation for soial seurity ontrat. 6

8 reseration atuarially fair ontrat whih assures a minimum utility same as an otimal selfinsuring ontrat D. On the other hand J is a oint on l at whih an indifferene ure of arents takes a minimum in S exatly at J. Now we onsider some fixed atuarially fair ontrat on l reresented as oint :. Assume that is loated on between oint O and J. In this ase arents an be een better off than at by disarding some additional ost ' say onditionally on death along the axis in B as a kind of selfinsuring ontrat. Let oint ' be the tangent oint of arents indifferene ure and S. Then the otimal additional ost ' whih arents should disard onditionally on death is alulated as distane '. If is loated in the uerright of J along l then arents do not hae to ay any additional ost along in B. The oerall lous of a mixed ontrat shedule say whih should inlude that additional and onditional ost in orresondene with eah gien ontrat would be a semisegment of line DJEQ as drawn in a bold line in Figure. we denote this set whih an be otimally attained as a result of making use only of an atuarially fair ontrat set by X a. 3 Without any ontrats onluded arents would stand at oint D while hildren would at a different ointo. This aset makes it for both arents and hildren imossible to set initially some alue for the state ontingent laim between two states of s or equialently to set the initial relatie rie between S and B. This is a totally different oint from ArrowDebreu stateontingent exhange eonomy in whih state ontingent laim or state rie enables them to arrie at a marketlearing and areto otimal ³ ³. oint Q on l is infinitely far in the uerright side. 3 Clearly oinides with ' if is loated on a segment lineoj and oinides with itself if is on a semisegment line JQ. 7

9 equilibrium. 4 Therefore in this myoi bargaining frame work an automati rie adustment roess to a unique equilibrium oint on X annot be exeted as far as any additional restrition e.g. regarding the altruisti weight in utility between arents and hildren or any other euliar agreements or algorithms are not introdued. 5 This is one imortant eonomi feature of intergenerational ontrat ure X. On the other hand with a fixed leel of aailable soial seurity for examle J in whih selfinsurane is not neessary ArrowDebreu stateontingent exhange eonomy an be well defined. In this ase an equilibrium areto otimal and market learing ontrat does not deend on the existene of altruism between arents and hildren sine in general the weight of altruism does not transform the shae of extended ontrat ure whih is drawn ust by relaxing artiiation onstraints Some roositions regarding the mixture of intergenerational risk sharing and soial seurity Now we omare within the urrent framework atuarially fair soial seurity and intergenerational transfer ontrat from arents iewoint. Eseially one imortant question is: Do arents hoose only an atuarially fair soial seurity or only an intergenerational transfer ontrat with hildren or both of them? Although it deends on where an aailable soial seurity and an aailable intergenerational transfer Y are 4 If both arents and hildren agree with standing initially at oint O there does exist a X GF. ometitie equilibrium on 5 One examle is onesided or twosided altruisti utility of the form U u + y u. 6 With a fixed soial seurity this extended ontrat ure not X is drawn by a set of oints where arents indifferene ure is tangent with the ure generated by shifting hildren s indifferene ure in arallel along with l by etor. 8

10 loated on l and on X resetiely some asets regarding this question an be extrated by setting one simle assumtion regarding hildren s behaior that they would aet any intergenerational transfer whih is offered from arents if it assures at least the same utility as at oint Y in terms of hildren s assoiated indiret utility.4b. 7 Denote arents maximized utilities whih an be attained by onluding only soial seurity only intergenerational transfer Y both of them by V V Y and V + Y resetiely. 8 At first we laim a following roosition and orollary. roosition : See Figure. Then: i Assume that Y oinides with G the oint whih attains arents maximum utility on X. Then for any arbitrary whih is loated on the segment line of l OZ it holds that V Y V + Y and V V + Y. ii Assume that Y oinides with F the oint whih attains arents minimum utility on X. Then for any arbitrary on l suh that ' O O'Z ' it holds that V Y V + Y and V ³ V + Y. Corollary : Consider an already onluded mandatory intergenerational transfer Y on X. Then any arbitrary soial seurity on l S ³ B ³ surely enhanes arents indiret utility without any neessity to disard any additional and onditional ost if is not extremely large in amount. This always holds whether there exists some orrelation between arents state s and hildren s state s or not. 7 Therefore we assume imliitly that hildren do not enter any other transfer ontrat inluding soial seurity. 8 With eah of these three otions arents may ay if neessary an additional and onditional ost along the axis in B selfinsurane as exlained in Setion 3. For rigorous formulation of arents roblems to be soled see Aendix. 9

11 Corollary is lear from hildren s indifferene ure under ositie/negatie orrelation between s and s as shown in Figure 3 and 4. roosition has quite interesting eonomi imliations. First arents together with a mandatory intergenerational transfer would almost always hoose to take any arbitrarily gien soial seurity. Seond but if reersely any soial seurity is mandatory while a fixed intergenerational transfer is not it may not be the ase. If a nonmandatory intergenerational transfer Y oinides with G the maximum utility oint arents are ery likely to take both of any arbitrary and the intergenerational transfer Y G on the other hand if a nonmandatory intergenerational transfer Y oinides with F the minimum utility oint arents are ery likely to take only soial seurity for any arbitrary. This imlies that it is quite natural to think that for any arbitrary but mandatory whih is not extremely large in amount there exists some oint Y on X suh that arents are indifferent to whether to aet an intergenerational ontrat or not. From ontinuity and monotoniity of arents indiret utilities on X we hae a roosition and a orollary as follows. roosition : See Figure. For any arbitrary soial seurity whih is loated on l suh that O O'Z' ' there always exists at least one intergenerational transfer Y as a funtion of on X suh that V V + Y. Corollary : Assume Y is not ointg. Then Y moes slightly along X GF in the diretion to G for a slight ositie hange in. Just for urely mathematial interest we laim following two lemmas. Lemma : X A segment line GF has a negatie tangent sloe of with regard to

12 whih is less than. Also a segment line JD whih is a art of X a has a negatie tangent sloe of with regard to whih is less than. Lemma : Denote a tangent oint of arents indifferene ure with a onstant remium line l ' k by oint k E. So k E is the same oint as E. Then the lous of the set of oint k E has a negatie tangent sloe of with regard to whih is less than. The roof of next roosition is diretly deried from Lemma. roosition 3: Assume an already onluded mandatory intergenerational transfer Y on X and a flexible atuarially fair soial seurity on l. Then the otimal soial seurity ˆ Y as a funtion of Y whih gies the maximum of arents indiret utility V ˆ Y + Y dereases in its size O ˆ as Y moes along X from G to F. Lastly we examine the simlest ase in whih only atuarially fair soial seurity on l is aailable for arents. Assume that only atuarially fair soial seurity on l is aailable for arents. Then as moes along l from O to Q that is as moes along X a DJEQ arents' marginal utility of soial seurity dereases. Eseially at oint J the marginal utility disontinuously ums into a lower leel and it beomes zero at oint E. This aset shows that if soial seurity is some oint between O and J the marginal utility benefit of soial seurity is relatiely high beause of the dereasing ost of selfinsurane. Together with intergenerational transfer howeer this kind of disontinuity does not aear. 5. Other results regarding the sensitiity to default risk and taxation in soial seurity

13 In this setion we limit our analysis only on soial seurity and examine the sensitiity both of arents and hildren to default risk and taxation on the demand for soial seurity wherein now hildren s inome risk s arises and W W >. 9 Sensitiity to default risk We introdue another risk s 3 for the default risk of soial seurity system where s 3 reresents nondefault and s 3 reresents default. Also assume that the robability of default s 3 is h and s 3 has no orrelation with s and s resetiely. At first onsider the demand for soial seurity by arents during stage O. The ay off of arents for eah realization of two releant risks s and s 3 is as following. arents reeie for { s s3 } with robability h for s s } with h zero for s s } with h { 3 { 3 for s s } with h resetiely. In ase of default arents still hae a { 3 liability if they die that is if s s } ours. Now we hae two definitions for { 3 atuarially fair ondition: Conditional atuarially fair ondition on nondefault l : 5. and unonditional atuarially fair ondition h l : h 5.. Furthermore arents assoiated indiret utility inluding default risk is redefined as: h b r A º u A + h bu + r + + hbu + r 5.3 Now we examine the sensitiity of arents demand for an atuarially fair soial seurity in the sense of 5. and 5. when h deiates slightly from zero by a ositie bit. In artiular our interest is in the sensitiity of an otimal ontrat E and a reseration 9 Therefore ondition i of.4 has been relaxed. It seems aroriate to assume that there exists no orrelation among s s and s 3 so far as there does not our any strong soial systemi risk. Otherwise these three risks may hae a onsiderable strong ositie orrelation with eah other. 5.3 is a modified ersion of.4a.

14 ontrat I to h. In order to do this we denote the tangent oint of either l or h l with the indifferene ure based on this modified assoiated indiret utility 5.3 h by E : E h E h. Also we denote the oint on either l or h l whih with the indifferene ure based on a modified assoiated indiret utility 5.3 attains the same utility as at oint D of h following roosition. h h D say by I : h h I I. 3 Now we laim a roosition 4: arents demand sensitiity to default risk Assume that soial seurity has default risk with onditional atuarially fair ondition 5.. Then we hae: i ' h > ' h > if s > ' h ' h if s and E E E ' h < ' h < if s <. Furthermore we hae: ii ' ' > E E irresetie of the alue of s. E I > I Instead of 5. assume that soial seurity has default risk with unonditional atuarially fair ondition 5.. Then it always holds that: iii ' h > ' h > and i ' ' > irresetie of the alue of s. I > I E E Next onsider the otimal demand for soial seurity by hildren during stage M. The ay off of hildren for eah realization of two releant risks s ands 3 is as following. Children reeie for s s with robability ' h 3 for s s } with ' h for s s } with ' h zero { 3 { 3 for s s } with ' h resetiely. In ase of default hildren still hae a { 3 liability if they hae a higher inome that is if s s } ours. Now we hae { 3 two definitions for atuarially fair ondition: Conditional atuarially fair ondition on E h : E E oinides with oint E : S E BE. 3 h I is the oint whih with the default risk assures the same minimum utility as when no soial seurity is aailable. I h : I I oinides with oint I : S I BI. 3

15 nondefault l : ' ' 5.4 and unonditional atuarially fair ondition 3 h l : ' ' 5.5. Furthermore hildren s modified assoiated indiret 3 h utility inluding default risk is defined as: 4 ' b h r W æ'[ u ö + bu + r W ] ç º maxç+ ' h[ u ] bu r W ç è+ ' h[ u + bu + r W ] ø 5.6 Now we examine the sensitiity of hildren s otimal demand for an atuarially fair soial seurity h h under 5.4 and 5.5 to h when h deiates slightly from by a ositie bit. Then: roosition 5: Children s demand sensitiity to default risk Assume that soial seurity has default risk with onditional atuarially fair ondition 5.4. Then i ' < ' <. Instead of 5.4 assume that soial seurity has default risk with unonditional atuarially fair ondition 5.5. Then ii ' > ' <. In either ase of 5.4 or 5.5 ' W W and ' W W where hildren fully insure their inome risk. Sensitiity to taxation We turn our fous to taxation on soial seurity both for arents and hildren. Consider two kinds of tax: a lumsum atuarially fair tax and an exerise tax only on ayment. Let T and T be onditional taxes imosed on the realization of reeit and ayment resetiely. Lumsum atuarially fair tax is desribed as T T where T T for arents and 'T ' T for hildren 5.7. An exerise tax on is a modified ersion of.4b. 4

16 ayment is desribed as T T T where z and we assume taxdeduted atuarially fair onditions z 5.8 for arents ' ' z 5.9 for hildren. z is defined as a roortional tax rate on. T The exeted tax inome by the goernment is ET º T T 5. for arents and ET º ' T ' T 5. for hildren. The assoiated indiret utilities with regard to remain almost the same as.4a for arents and.4b for hildren: b r A º u A + bu + r +.4a b r W æ'[ u + bu + r W ] º maxç è + ' [ u + bu + r W + ö.4b ] ø Clearly a lumsum atuarially fair tax is better than an exerise tax on ayment for both arents and hildren in the sense that keeing the exeted tax inome ET at onstant a lumsum atuarially fair tax ould always attain better assoiated indiret utility with regard to than an exerise tax only on ayment. We roeed to the sensitiity analysis to an exerise taxation on ayment as desribed in 5.8 for arents and 5.9 for hildren. 5.8 and 5.9 are in a sense equialent with unonditional atuarially fair onditions inororating default risk 5. for arents and 5.5 for hildren resetiely if we set h z. Here we an interret h as a onditional rofit margin or a subsidy margin on the realization of reeit. Denote arents otimal demand for soial seurity with the ondition 5.8 by z z and hildren s otimal demand for soial seurity with the ondition 5.9 by z z resetiely. Then: roosition 6: arents demand sensitiity to taxation Assume that the goernment imoses an exerise tax on ayment for arents soial seurity with taxdeduted 5

17 atuarially fair ondition 5.8. Then i ' z < irresetie of the alue of s. Furthermore ii ' z > if s > ' z if s and ' z < if s <. roosition 7: Children s demand sensitiity to taxation Assume that the goernment imoses an exerise tax on ayment for hildren s soial seurity with taxdeduted atuarially fair ondition 5.9. Then i ' < irresetie of the alue of s. 6. Final remarks In a ontinuoustime ase arents roblem to be soled an be reresented as follows: Define arents transfer ontrat inororating the risk of death S t B t for t T. T 3 years is the length of stage O and S t B t is measured as a resent alue at time not t and B t is ontinuously differentiable for all t T. Also let t be their onsumtion lan at time t measured as a resent alue at time t and A be the resent alue of total aailable wealth measured at time. Define the robability that arents are alie at time t as t where and T feasibility onstraint is written as: t ò {S t' + t'ex rt'} dt' A B t for all t T. 6.. Then the budget Equialently in a differential form: t [B' t + S t]ex rt for all t T 6. From 6. we hae A and B T B. An atuarially fair ondition of the transfer T ontrat S t B t is: ò t S t dt ò B t ' t dt 6. T Here S º ò t S t dt is an exeted suort and B º ò T T B t ' t dt is an exeted bequest. In a differential form t S t B t ' t 6.. This is a ontinuoustime ersion of an atuarially fair ondition in a twoeriod ase l : S B. Assume that 6

18 arents transfer shedule S t B t is redetermined. Then they sole: X T max t u t t dt t ò exb 6.3 s.t. 6.. Howeer as a matter of fat the maximization roblem is not left for arents but their onsumtion is automatially determined at t [B' t + S t]ex rt for all t T 6.. Therefore as in a twoeriod ase there exists some ossibility of selfinsurane in whih arents must ay an additional ost onditionally on death. On the other hand hildren s indiret utility at time t with transfer ontrat S t B t t T roes to hae an indifferene ure whih is tangent with a ontinuoustime atuarially fair line t S t B t ' t 6.. From all the aboe oints our analysis made in the reious 4 setions with two disrete eriods does not lose any generality een in a ontinuoustime ase. Thus this artile has ust summarized using a simle model of two disrete eriods and binary states of mortality/inome risks some fundamental roositions regarding the mixture of intergenerational risk sharing and soial seurity. Here for the first time statedeendent utility under mortality risk roes to generate arents euliar indifferene ure regarding insurane ontrat and selfinsurane is shown to lay a ruial role on the deision regarding soial seurity holding and intergenerational transfer ontrat. eferene Atkinson Anthony B. and Joseh E. Stiglitz 98 Leture on ubli Eonomis MGrawHill. Obstfeld Maurie and enneth ogoff 996 Foundation of International Maroeonomis The MIT ress Ehrlih Isaa and Gary Beker 97 Market Insurane SelfInsurane and Selfrotetion The Journal of olitial Eonomy Vol. 8 Issue 4 Jul.Aug Ball Laurene and N. Gregory Mankiw Intergenerational isk Sharing in the Sirit of Arrow Debreu and awls with Aliations to Soial Seurity Design NBE Working aer No.87 May. Enders Walter and Harey E. Laan 98 Soial Seurity Taxation and Intergenerational isk Sharing International Eonomi eiew Vol.3 Issue 3 Ot Gordon oger H. and Hal. Varian 988 Intergenerational isk Sharing Journal of ubli Eonomis Vol

19 Yaari Menahem E. 965 Unertain Lifetime Life Insurane and the Theory of the Consumer The eiew of Eonomi Studies Vol. 3 Issue Aoki Takaaki 3 Essay on the Intergenerational isk Sharing Soial Seurity and Taxation mimeo Unublished The First Version Term aer for Eo 743 State Uniersity of New York at Buffalo Det. of Eonomis Jan. 3. Selfinsurane and Some Asets on the Intergenerational isk Sharing and Soial Seurity mimeo Unublished The seond Version: Se. 3 the third Version: Jan. 4 the last ersion: June 5. Hayashi Fumio Joseh Altoni and Laurene otlikoff 996 isksharing between and within Families Eonometria Vol.64 Issue Mar Abel Andrew B. 987 Oeratie Gift and Bequest Moties The Amerian Eonomi eiew Vol. 77 Issue De Abel Andrew B. 985 reautionary Saing and Aidental Bequest The Amerian Eonomi eiew Vol.75 Issue 4 Se Hurd Mihael D. 989 Mortality isk and Bequests Eonometria Vol.57 Issue 4 Jul Bernheim Douglas B. Andrei Shleifer and Lawrene H. Summers 985 The Strategi Bequest Motie The Journal of olitial Eonomy Vol. 93 Issue 6 De Bernheim Douglas B. 99 How Strong Are Bequest Moties? Eidene Based on Estimates of the Demand for Life Insurane and Annuities The Journal of olitial Eonomy Vol. 99 Issue 5 Ot Drazen Allan 978 Goernment Debt Human Caital and Bequests in a LifeCyle Model The Journal of olitial Eonomy Vol.86 Issue 3 Jun Shiller obert J. 998 Soial Seurity and Institutions for Intergenerational Intragenerational and International isk Sharing Carnegieohester Series in ubli oliy Vol. 5: 655 June

20 S d E b a G e O D X F B f O' H l : S B Atuarially Fair Line l : S B Certainty Line l ': B S k Constant remium Line 9 Figure No Correlation between s and s

21 S Z' Q l : S B Z E l ": B S k' d a O I L'' L X J ' a L' G X D ' F Z'' b A l ': B S k e B f O' H Figure No orrelation between s and s l : S B

22 S b d G a e X O D F B O' l : S B l : S B Figure 3 ositie Correlation betweens ands

23 S d a b G e O D X F B O' O' l : S B Figure 4 Negatie Correlation betweens ands

24 Aendix igorous definitions of V V Y and V + Y Define an atuarially fair soial seurity where is on : l : and an intergenerational transfer Y : S B Y Y on X. Then: V º max ' ³ + ' b r A A. 5 V V Y º S B b r A A. + Y º Y Y max S B ' ³ + S + B + ' b r A s.t. S B b r W S B b r W º Y Y A.3 6 and l : roof of roosition : See Figure. Grahially V an be determined as arents indiret utility of the oint where a horizontal line rosses X a that is arents indifferene ure is tangent with a horizontal line learly on X a V Y simly as that of oint Y on X and V + Y as that of the oint where arents indifferene ure is tangent with the ure generated by shifting hildren s indifferene ure in arallel along with l by etor. i Case: Y G Let Z be the oint where hildren s shifted indifferene ure whih is tangent both with arents indifferene ure d and an atuarially fair line l is tangent 5 A. an be rewritten as: V º max b r A A. ³ 6 A.3 an be rewritten as: V + Y º max + ' b r A A.3 ' ³ s.t. b r W SY BY b r W º and l : 3

25 with l. For some soial seurity hildren s shifted indifferene ure is denoted by ' for the original indifferene ure b. Grahially it is lear that for any arbitrary on l b suh that O OZ arents indifferene ure whih is tangent with hildren s shifted indifferene ure b ' is loated in the uer side of both oint and indifferene ure d or oint G. So we hae V Y G V + Y G with equality when Z and V V + Y G with equality when E. ii Case: Y F Denote the oint where hildren s indifferene ure e rosses the ertainty line l : by O '. Also denote some onstant remium line whih asses O ' by l ": k '. So e is tangent with arents indifferene ure a at oint F and also is tangent with l " at oint O '. Z " is the oint where hildren s shifted indifferene ure whih is tangent with both arents indifferene ure a and a onstant remium line l " is tangent with l ". For some soial seurity denote hildren s shifted indifferene ure whih orresonds with original indifferene ure e by e '. Grahially it is lear that for any arbitrary on l suh that O O'Z" that arents indifferene ure whih is tangent with hildren s shifted indifferene ure e ' is loated in the uer side of arents indifferene ure a or oint F. So we hae V Y F V + Y F with equality when O O'Z". Now we roeed to the roof of V ³ V + Y F. For later oneniene we rewrite V and V + Y as V and V Y using the amount of reeit for soial seurity. Clearly when O zero reeit zero ayment we hae V V Y F 7 beause arents are indifferent between D and F. So at first we show that 7 That is V O V O + Y F. 4

26 5 F Y V V > A. and next show that F Y V an neer ath and oerass V for any arbitrary suh that " O'Z O. First ste: Sine gien an intergenerational transfery arents need not ay an additional ost on death i.e. one onstraint '³ is binding. we an rewrite A.3 as: max Y V º A.3 s.t. Y Y B S º and l : Here denotes a mixed transfer ontrat shedule. lugging the seond equation of onstraints into other equations of A.3 we hae a following Lagrangian and orresonding first order onditions. ø ö ç è æ L l l l A.4 From the eneloe theorem also using A.3 we obtain: ' Y V Y V º X + ø ö ç è æ + º l A.5 Similarly we obtain the following equation quite easily: ' V V º A.6

27 6 Denote the oint at whih arents indifferene ure whih rosses a solution oint of A.3 : W W W say intersets with a X by : a a a W W W. Clearly at any arbitrary oint on a X we hae so A.5 and A.6 atually share the same alue at W and W a that is X. Now we hae only to show that X is dereasing as inreases moes along arents' indifferene ure from a W to W. Here denote arents indifferene ure whih asses a W and W by g. emember arents indiret utility: r u A u A r º b b.4a It follows diretly that: 8 < < A.7 From the definition we also hae; + º X A.8 So from A.7 and A.8 it follows that: < + º X < + º X A.9 * > d d As grahially lear inreases moes along arents indifferene ure from a W to W also inreases. 8 For the first order ondition we hae >. Furthermore if is loated in the right side of DJ on a X then <.

28 where * denotes that is on g : W W say. a d d Now sine X * X * + X A. d d d and also d > from A.9 and A. we get X < * d learly W a A.. If D so we hae omleted the first ste that is hae roed A.. W F Seond ste: Assume that for some we hae V V + Y F in : another exression V V Y F. For this denote solution oints of mixed ontrat shedule for A.3 and A.3 by W again and S : S S resetiely. W and S attain the same indiret utility for arents so S should oinide with W a in the aboe notation. Then we an use the same inequality A. in order to roe: V S W > V Y F a W A. From the ontinuity of V and V Y F with regard to now we hae ust roed V ³ V Y F that is V ³ V + Y F for any arbitrary suh that O O'Z". roof of Lemma : Grahially it is lear that for any oints on a segment line JD of X a whih is the subset of solution oints for A. one onstraint '³ is not binding. So from the first order ondition we hae a following equality: A + b + r + r + A. Taking as a funtion of and differentiate A. with regard to we obtain: u"a + b+ r æ u" + r + d è d + ö ø A. 7

29 Considering u "< it follows diretly that d d < A.3. roof of Lemma : lugging the onstant remium ondition l ' k A.4 into.4a we obtain the first order ondition: æ æ k öö A + b + r ç + r ç è è øø A.5 Taking as a funtion of k and differentiate A.4 with regard to k we obtain; u" A d b + r + dk æ æ k ööæ d ö u" ç + r ç ç è è øøè dk ø A.6 d dk from whih it follows that < < or >. Sine from A.4; dk d d d dk A.7 d d we hae <. Now the roof is done. d roof of roosition 4: roof of i: arents assoiated indiret utility inluding default risk is gien in 5.3. Maximizing 5.3 subet to 5. with regard to and imliitly we hae the first order ondition for oint h E : A + h b + r + r / + hb + r + r A3. elaing with h as a funtion of h in A3. differentiating the equality with regard to h and imlementing the omaratie statis immediately rodues the following equation: 8

30 ' h{ ' A h + [ h b + r + hb + r ' + r h} b + r[ + r h / + r h] / ] ' + r h / A3. The oeffiient of ' h in L.H.S { ' A h + [ h b + r / ] ' + r h / + hb + r ' + r h} is learly negatie sine u '' <. The sign of.h.s is negatie zero and ositie orresonding to s > s and s < resetiely. elaing h with h E hae omleted the roof. roof of iii: The roof is almost the same as roof of i exet for maximizing 5.3 subet to 5. instead of 5.. The first order ondition for oint A h + b + r[ + h] + hb + r + r h We get a following equation for omaratie statis: ' h{ ' A h + b + r [ + h] + hb + r ' + r h} b + r + r h / h ' b + r [ + h] / h ' h E is: we + r h[ + h]/ h A3.3 + r h[ + h]/ h + r h[ + h]/ h A3.4 Considering u '> u '' < learly.h.s is negatie and the oeffiient of ' h in L.H.S { ' A h + b + r [ + h] + hb + r ' + r h} / h] ' is also negatie. So relaing h with h E + r h[ + h]/ h the roof is done. 9

31 roof of ii: Sine 5.3 must be onstant subet to 5. we hae: u A h + h bu + r h / + hbu + r h Const. oer h The first order ondition with regard to h rodues: ' h{ A h + h b + r + r h / + hb + r + r h} b[ u + r h / u + r h] > Ealuating A3.5 at h ' { A + b + r + r / } > A3.5 A3.6 So we need the sign of the oeffiient of ' in L.H.S { A + b + r + r / }. A3.7 h But this is exatly the first order ondition at E E whih should be at E. Sine h I I is loated in the leftdown side of E along l so A3.7 should hae a ositie alue. The roof is now done. roof of i: Sine 5.3 must be onstant subet to 5. we hae: u A h + h bu + hbu + r h Const. + r h[ + h]/ b h oer h. A3.8 Differentiating A3.8 with regard to h rodues: ' h{ A + b + r[ + h] + hb + r + r h} b[ u + b + h + r h[ + h]/ h + r h[ + h]/ h u + r h] r / h + r h[ + h]/ h h Ealuating A3.9 at h '{ A b[ u + r / u + r ] + b + r + r / > + b + r + r / } Now we hae only to examine the sign of the oeffiient of ' in L.H.S; A3.9 A3. 3

32 { A + b + r + r / } whih is exatly the same as A3.7. Sine l h l now done. we an aly the same argument as after A3.7 in the roof of ii. The roof is roof of roosition 5: roof of i: Children s assoiated indiret utility inluding default risk is gien in 5.6. Maximizing 5.6 subet to 5.4 with regard to and imliitly and we hae the first order ondition with regard to for some alue of suh that ˆ arg max ' b h r W s.t. 5.4: æ ' ö ç + r W ˆ ' è h ø A3. + r W + ˆ Sine u ' is a dereasing funtion u '' < for h > we hae: W ' ' ˆ ˆ > W + A3. So we get W W > and ' W W > A3.3. elaing and ' with h and h and onsidering ' W W and ' W W A3. is equialent with ' < ' <. The roof is now done. roof of ii: Quite similar to roof of i. Maximizing 5.6 subet to 5.5 in stead of subet to 5.4 with regard to and imliitly and we hae the first order ondition with regard to for some alue of suh that arg max ˆ ' b h r W s.t. 5.5: 3

33 Then we hae: æ ç + r W è ' h ö ˆ ' ø + r W + ˆ A3.3 ' W W h ' h + ' ' h and W W h ' h + ' A3.4. Considering ' W W and ' W W A3.4 imlies ' > ' <. The roof is now done. roof of roosition 6: roof of i: arents assoiated indiret utility is gien by.5a. Maximizing.5a subet to 5.8 with regard to and imliitly we hae the first order ondition: A / z + b + r[ + z ] + r [ + z ]/ z A3.5 elaing with z as a funtion of z in A3.5 differentiating the equality with regard to z and imlementing the omaratie statis immediately rodue the following equation: ' z { / z ' + b + r [ + z ] / z z ' + b + r b + r [ + A z / z / z ' A z / z + r z [ + z ]/ z + r z [ + z ]/ z z ] / z z ' + r z [ + z ]/ z A3.6 Considering u '> u '' < learly.h.s is ositie and the oeffiient of ' h in L.H.S is negatie. So relaing z with z the roof is done. } roof of ii: arents assoiated indiret utility is gien by A.5. Maximizing.5a subet to 5.8 with regard to and imliitly we hae the first order ondition: 3

34 A + b + r[ + z ] + r [ + z ]/ A3.7 elaing with z as a funtion of z in A3.7 differentiating the equality with regard to z and imlementing the omaratie statis immediately rodue the following equation: ' z { ' + b + r + b + r A z [ + b + r [ + z ] ' + r z [ + z ]/ } + r z [ + z ]/ z ] / z ' + r z [ + z ]/ A3.8 Considering u '' < learly the oeffiient of ' h in L.H.S is negatie. On the other hand.h.s an be rewritten as: C + C ' C. H. S b + r u ' A3.9 where C º + r[ + z]/z Considering the form of utility. the sign of.h.s is ositie zero and negatie orresonding to s < s and s > resetiely. So relaing z with z the roof is done. roof of roosition 7: roof of i: Children s assoiated indiret utility inluding default risk is gien in.5b. Maximizing.5b subet to 5.9 with regard to and imliitly and we hae the first order ondition with regard to for some alue of suh that ˆ arg max ' b r W s.t. 5.9: æ ' ö ç + r W ˆ ' è z z ø A3. + r W + ˆ 33

35 Sine u ' is a dereasing funtion u '' < for h > we hae: So we get W ' ' z ˆ ˆ > W + ' z W ' + ' z W A3. > A3.. L.H.S of A3. is dereasing with regard to z. elaing with z and onsidering ' W W A3. is equialent with ' <. The roof is now done. Unlike roosition 5 the sign of ' is still unertain. 34