Long-Term Care and Lazy Rotten Kids

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1 DISCUSSION PAPER SERIES IZA DP No Long-Term Care and Lazy Rotten Kds Helmuth Cremer Kerstn Roeder August 2013 Forschungsnsttut zur Zukunft der Arbet Insttute for the Study of Labor

2 Long-Term Care and Lazy Rotten Kds Helmuth Cremer Toulouse School of Economcs and IZA Kerstn Roeder LMU, Munch Dscusson Paper No August 2013 IZA P.O. Box Bonn Germany Phone: Fax: E-mal: Any opnons expressed here are those of the author(s) and not those of IZA. Research publshed n ths seres may nclude vews on polcy, but the nsttute tself takes no nsttutonal polcy postons. The IZA research network s commtted to the IZA Gudng Prncples of Research Integrty. The Insttute for the Study of Labor (IZA) n Bonn s a local and vrtual nternatonal research center and a place of communcaton between scence, poltcs and busness. IZA s an ndependent nonproft organzaton supported by Deutsche Post Foundaton. The center s assocated wth the Unversty of Bonn and offers a stmulatng research envronment through ts nternatonal network, workshops and conferences, data servce, project support, research vsts and doctoral program. IZA engages n () orgnal and nternatonally compettve research n all felds of labor economcs, () development of polcy concepts, and () dssemnaton of research results and concepts to the nterested publc. IZA Dscusson Papers often represent prelmnary work and are crculated to encourage dscusson. Ctaton of such a paper should account for ts provsonal character. A revsed verson may be avalable drectly from the author.

3 IZA Dscusson Paper No August 2013 ABSTRACT Long-Term Care and Lazy Rotten Kds * Ths paper studes the determnaton of nformal long-term care (famly ad) to dependent elderly n a worst case scenaro concernng the harmony of famly relatons. Chldren are purely selfsh, and nether sde can make credble commtments (whch rules out effcent barganng). The model s based on Becker s rotten kd specfcaton except that t explctly accounts for the sequence of decsons. In Becker s world, wth a sngle good, ths settng yelds effcency. We show that when famly ad (and long-term care servces n general) are ntroduced, the outcome s lkely to be neffcent. Stll, the rotten kd mechansm s at work and ensures that a postve level of ad s provded as long as the bequest motve s operatve. We dentfy the neffcences by comparng the lassez-fare (subgame perfect) equlbrum to the frst-best allocaton. We ntally assume that famles are dentcal ex ante. However, the case where dynastes dffer n wealth s also consdered. We study how the provson of long-term care (LTC) can be mproved by publc polces under varous nformatonal assumptons. Interestngly, crowdng out of prvate ad by publc LTC s not a problem n ths settng. Wth an operatve bequest motve, publc LTC wll have no mpact on prvate ad. More amazngly stll, when the bequest motve s (ntally) not operatve, publc nsurance may even enhance the provson of nformal ad. JEL Classfcaton: D13, H21, I13 Keywords: rotten kds, long-term care, famly ad, optmal taxaton Correspondng author: Helmuth Cremer Insttut d Econome Industrelle Unversté des Scences Socales 21 allée de Brenne Toulouse Cedex France E-mal: helmuth.cremer@tse-fr.eu * Fnancal support from the Chare Marché des rsques et creaton de valeur of the FdR/SCOR s gratefully acknowledged. Ths paper has been presented at the European Health Economcs Workshop, the Publc Economc Theory Conference, the Journées d Econome Publque Lous-André Gérard-Varet and n semnars at the Unverstes of Munch, and Constance. We thank all the partcpants for ther comments. We are partcularly grateful to Justna Klmavcute for her nsghtful remarks and suggestons.

4 1 Introducton Long-term care (LTC) concerns people who depend on help to carry out daly actvtes such as eatng, bathng, dressng, gong to bed, gettng up or usng the tolet (OECD, 2005). It s delvered nformally by famles manly spouses, daughters and stepdaughters and, to a lesser extent, formally by care assstants, who are pad under some form of employment contract. Formal care s gven at home or n an nsttuton (such as care centers and nursng homes). The governments of most ndustralzed countres are nvolved n ether the provson or nancng of LTC servces, or often both, although the extent and nature of ther nvolvement d ers wdely across countres. 1 In the future, the demand for formal LTC servces by the populaton s lkely to grow substantally. LTC needs start to rse exponentally from around the age of 80 years. The number of persons aged 80 years and above s growng faster than any other segment of the populaton. As a consequence, the number of dependent elderly at the European level (EU 27) s expected to grow from about 21 mllon people n 2007 to about 44 mllons n 2060 (European Commsson, 2009). We thus antcpate ncreasng pressure on resources demanded to provde LTC servces for the fral elderly, and ths pressure wll be on the three nsttutons currently nancng and provdng LTC servces: the state, the market and the famly. To assess the adequacy of LTC nancng and provson and to make projectons, t s mportant to assess he extent to whch countres wll be able n the future to rely on the nformal provson of care. Most senors wth mparments resde n ther home or that of ther relatves, and they rely largely on volunteer care from famly members. These nclude senors wth severe mparments (unable to perform at least four actvtes of daly lvng). An mportant feature that s often neglected s the real motvaton for famly soldarty. For long, we have adopted the fary tale vew of chldren or spouses helpng ther dependent parents wth joy and dedcaton, what we call pure altrusm. We now ncreasngly realze that famly soldarty s often based on forced altrusm (socal norm) or on strategc consderatons (e.g., Cox 1987; Cox and Rank, 1992; Canta and Pesteau, 2013). 2 1 For a more n-depth dscusson see the overvew by Cremer, Pesteau and Ponthère (2012). 2 Ether way famly care goes along wth dsutlty and forgone labor market opportuntes for the care-gvng person. See, for nstance, Boln, Lndgren and Lundborg, (2008); Hughes et al. (1999); and Schulz and Beach, (1999). 1

5 Knowng the foundaton of altrusm s very mportant to see how famly assstance wll react to the emergence of prvate or publc schemes of LTC nsurance. For example, the ntroducton of LTC socal nsurance s expected to crowd out famly soldarty based on pure altrusm (e.g., Cremer, Gahvar and Pesteau, 2013). On the other hand, where soldarty s based on strategc exchanges crowdng out s expected to be less sgn cant. Its precse extent, s then lkely to depend on the spec c way these nterfamly exchanges are determned. The exstng lterature (e.g., Bernhem, Shlefer, and Summers, 1985) concentrates on strategc bequest type models wth full commtment leadng to e cent barganng. In realty ths appears to be a rather strong assumpton. In ths paper, we study the determnaton of famly ad n what can be consdered a worst case scenaro as to the harmony of famly relatons. Chldren act n a purely sel sh way and nether sde can make credble commtments (whch would open the possblty of e cent barganng as n the strategc bequest settng). The model we use s based on Becker s (1974; 1991) rotten kd spec caton (see also Bergstrom, 1989; 1996) except that we explctly account for the sequence of decsons (lke Bruce and Waldman, 1990). In Becker s world, wth a sngle good, ths settng yelds an e cent outcome, even n the absence of commtment and when a chld s purely sel sh. We show that when famly ad (and LTC servces n general) are ntroduced the outcome s lkely to be ne cent. Ths s partcularly true when the parents value ther chldren s care more than the market substtutes. 3 Stll, the rotten kd mechansm s at work and ensures that a postve level of ad s provded as long as the bequest motve s operatve. We study how the ne cency can be corrected by publc polces under varous nformatonal assumptons. For most of the paper, we assume that famles are dentcal ex ante. However, the case where dynastes d er n wealth s also consdered. The desgn of publc polcy has to account for the (n)e cences of nformal ad. The conventonal wsdom s that publc polcy often creates or at least enhances such ne cences through crowdng out. In our settng, however, the relatonshp between publc LTC and famly ad s more complex. As long as the bequest motve s operatve, the chldren do provde some nformal ad to ther parents, but ts level s too low, except when the full mpact of ad s captured by ts monetary valuaton n the parent s 3 It was already ponted out by Bergstrom (1989) that Becker s rotten kd theorem holds f there s one commodty (money), the parent s an e ectve altrust and chooses after the chld (for ths see also Hrshlefer, 1977), and the model s statc. For a summary of cases when the rotten kd theorem fals see also Laferrère and Wol (2006). 2

6 utlty. Chldren s labor supply s then also ne cent, but ths problem s not drectly connected to the potental need for LTC. When the bequest motve s not operatve, no famly ad wll be provded and the case for publc nterventon wll be even stronger. Interestngly, ths falure of famly ad may e ectvely be related to prvate market ne cences. Partcularly, an ndvdual who cannot a ord to buy nsurance coverage to cover the potental monetary cost of LTC may be subject to a double punshment. In case of dependency the ndvdual wll not only run out of resources, but he can also not count on any famly ad (snce he has no resources to leave a bequest). Publc ad may then even result n a postve bequest and thus brng about a postve level of ad. To sum up, crowdng out of prvate ad by publc LTC s not a problem n ths settng. Wth an operatve bequest motve, publc LTC wll have no mpact on prvate ad. More amazngly stll, when the bequest motve s (ntally) not operatve, publc nsurance may even enhance the provson of nformal ad. Publc nterventon n LTC s often advocated because t mght help to overcome ne cences n the prvate nsurance market. 4 When, as s typcally observed n realty, prvate LTC nsurance nvolves sgn cant loadng costs (Brown and Fnkelsten, 2007), the lassez-fare soluton mples nsu cent nsurance. Publc nsurance may mtgate ths problem but once agan ts e ectveness wll hnge on the extent of crowdng out of famly ad. For the sake of realsm we consder the possblty that prvate nsurance may nvolve loadng costs, but ths s not the drvng force behnd our results. Ether way, the e ectveness and the desgn of publc LTC depend on the avalable nstruments whch s ultmately of course a queston of nformaton. We rst study the mplementaton of the rst-best (FB) under full nformaton. Though of lmted realsm ths s an nterestng benchmark to show whch nstruments are necessary wthn ths settng of mult-stage strategc nteracton to acheve the e cent soluton. We show that the FB can be decentralzed by a lump-sum transfer from the dependent to the healthy elderly supplemented by lnear subsdes on labor ncomes (of the young) and ad. Lump sum transfers are determned to mmc far prvate nsurance. Next, we look at a second-best soluton whch s acheved when ad s not observable (and thus cannot not be subsdzed). The set of nstruments now conssts of a lump-sum tax on the healthy old and lnear taxes on chld s ncome that both nance publc LTC provson 4 See e.g. Sclan (2013) for a dscusson of the relatve merts of publc and prvate nsurance. 3

7 to dependent parents. We show that transfers are used to acheve full nsurance of the old. The tax on labor, on the other hand, s not used to rase revenue, but because t ncreases nformal ad (whch becomes more attractve when market labor s taxed). If possble the tax on chldren s labor supply should be d erentated accordng to the dependency status of ther parents. The level of the tax on the chldren of dependent parents s then set to strke a balance between the deadweght loss of the labor tax and the bene ts assocated wth ts e ect on ad (a tax on labor e ectvely acts lke a subsdy on ad). Chldren of healthy parents, on the other hand, bene t from the rst-best subsdy on labor. Fnally, we turn to a settng where ndvduals are heterogenous and parents d er n wealth. Ths adds an extra potental just caton for publc nterventon, namely redstrbuton. It also makes a case where some ndvduals cannot a ord prvate LTC coverage more plausble and we can have an ntal equlbrum n whch the bequest motve s operatve for some ndvduals and not for others. To concentrate on dstrbutonal ssues, we assume that the government does not observe wealth, whle all other varables are observable. We consder a two-types settng and study the second-best allocaton acheved under ths nformaton structure. We show that bequests n rch famles are not dstorted (nether taxed nor subsdzed at the margn), but there s a downward dstorton on bequests n poor famles. In other words, bequests left by low wealth parents are subject to a postve margnal tax. Ths result s n lne wth standard ndngs n optmal tax models (Mrrlees, 1971). More surprsngly, the soluton mples a rst-best tradeo for labor supply and nformal care. Interestngly, ths rst-best tradeo does not mply the same margnal tax rates on labor or ad as n the decentralzaton of the rst-best under full nformaton. Ths s because the bequest tax a ects also labor supply and ad decsons and the tax or subsdy rates have to be adjusted accordngly. Fnally, hgh wealth ndvduals are fully nsured; the nsurance provson to the low wealth parents, on the other hand s dstorted (and the sgn of ths dstorton does not appear to be unambguous). Prvate nsurance s su cent when markets are far, but t wll be replaced by publc nsurance (publc LTC bene ts) when there are loadng costs. The paper s organzed as follows. In Secton 2 we ntroduce the economc setup. The rst-best allocaton s descrbed n Secton 3 followed by the analyss of the lassez- 4

8 fare allocaton (subgame perfect equlbrum of the ad for bequest game) n Secton 4. Secton 5 compares the two outcomes and shows how the rst-best allocaton can be decentralzed. Second-best polces under d erent nformatonal assumptons are presented n Sectons 6 and 7. The former determnes the optmal polcy nstruments when ad s not observable, whle the latter for the case where famles d er n (unobservable) wealth. Secton 8 o ers some concludng remarks. More techncal materal, ncludng all proofs, s relegated to the Appendx. 2 The model Consder a populaton (whch sze s normalzed to one) consstng of one parent (subscrpt p ) and one chld (subscrpt c ) famles. Whle the chld s sel sh, the parent s a pure altrust. The parent s retred and has accumulated wealth!. He faces the probablty of becomng dependent and needng long-term care. The need of LTC requres expendtures of amount L and comes along wth a utlty loss due to the deprvaton of autonomy captured by q 0. LTC nsurance coverage can be bought on the prvate market at a prce p. For p = LTC nsurance s actuarally far. The parent decdes how much LTC nsurance coverage I to buy and how much he wants to leave as a bequest to hs chld. The chld decdes how much labor to supply and how much nformal care to provde. On the one hand, care provded by the chld reduces the monetary loss from LTC by h(a) L (wth h 0 > 0; h 00 < 0) snce then the parent requres less professonal care servces. On the other hand, t reduces the (utlty) loss the parent su ers from LTC mplyng q 0 (a) 0 (wth q 00 < 0). The latter re ects the fact that the parent prefers care by hs chld to care provded by a stranger or to enterng a nursng home. The chld, earns ncome w`, where w denotes the chld s wage rate and ` labor supply. Labor supply as well as nformal care provson come along wth dsutlty captured by v, wth v 0 > 0; v 00 > 0. The altrustc parent maxmzes the followng welfare functon W p = U p + U c : Indvdual utlty of the parent U p s gven by U p = [u(! pi + I L + h(a) b) + q(a)] + (1 )u(! pi b b); 5

9 where a b ndcates the state of stayng healthy. Utlty of the chld s gven by h U c = [u(w` + b) v(` + a)] + (1 ) u(w b` + b b) The utlty functon sats es u 0 > 0, u 00 < 0 and u v( b`) : The tmng of the model s as follows: rst the government announces ts polcy. Then, the parent and the chld play the followng three stage game. In stage 1, the parent decdes how much LTC nsurance coverage, I, to buy. In stage 2, the state of nature s revealed, that s, the parent s ether dsabled or not. Then, the chld decdes how much labor to supply, ` and b`, and how much nformal care, a, to provde f the parent s dependent. Fnally, n stage 3 the parent decdes the level of bequests, b; b b 0, n each state of nature. To determne the subgame perfect Nash equlbrum we solve ths game by backward nducton. But, before we turn our attenton to the lassez-fare we study the rst-best allocaton whch provdes a benchmark aganst whch we can compare the lassez-fare allocaton. 3 Frst-best soluton Wth ex ante dentcal famles, we can de ne the optmal allocaton as the one maxmzng the expected utlty of a representatve dynasty. Ths problem can be wrtten as max W fb = [u(m p ) + q(a) + u(m c ) v(` + a)] `; b`;a;m p;m c; bm p; bm c h + (1 ) u( bm p ) + u( bm c ) h v( b`) s.t.! + (1 )w` + w b` + h(a) = [m p + m c + L] + (1 ) [ bm p + bm c ] ; where the decson varables are labor supples, ` and b`, nformal care, a, and consumpton levels of the parent and the chld n both states of nature. We denote the latter by m p, m c, bm p and bm c. In the rst-best all varables are drectly set, assumng full nformaton and dsregardng the mult-stage structure of the game. However, the spec caton of the game wll of course be relevant below when we study the decentralzaton of the rst-best optmum. Denotng the Lagrange multpler assocated wth the resource constrant (1) by, the rst order condtons (FOCs) characterzng the optmal soluton 5 The assumpton on the thrd dervatve s not essental for our analyss. It s used for a sngle result; see Subsecton 7.2. (1) 6

10 can be wrtten as follows u 0 (m fb p ) = u 0 (m fb c ) = u 0 ( bm fb p ) = u 0 ( bm fb c ) = ; (2) v 0 ( b`fb ) u 0 ( bm fb = w; c ) (3) v 0 (`fb + a fb ) u 0 (m fb = w; c ) (4) h 0 (a fb ) + q0 (a fb ) u 0 (m fb = w: (5) c ) These expressons are pretty much self-explanatory. Equaton (2) states the equalty of margnal utltes of ncomes across ndvduals and states of nature (full nsurance). Equatons (3) and (4) are the usual condtons descrbng the e cent choce of labor supply. For nformal care we obtan a smlar condton, except that ths varable nvolves both monetary bene ts, h 0 (a), and utlty gans, q 0 (a), whch translates nto the margnal rate of substtuton term on the LHS of equaton (5). 4 Lassez-fare allocaton 4.1 Stage 3: optmal bequests The parent s ether healthy or dependent. In both states of nature he observes the chld s labor ncome and nformal care provson (n case he requres LTC). The parent chooses hs optmal bequests by maxmzng welfare; equaton (6) when he s healthy and equaton (7) when he requres LTC: max b b c Wp =u(! pi b b) + u(w b` + b b) v( b`); (6) max b W p =u(! + (1 p)i L + h(a) b) + q(a) + u(w` + b) v(` + a): (7) Assumng an nteror soluton, the optmal bequests n each state of nature are mplctly gven c W = u 0 ( bm p ) + u 0 ( bm c ) = 0; (8) = u 0 (m p ) + u 0 (m c ) = 0: (9) That s, bequests are chosen so that consumpton levels between the parent and the chld are equal n both states of nature. Recall that bequests are restrcted to be nonnegatve, 7

11 and one obtans from (8) and (9) b b > 0 ()! pi > w b`; b > 0 ()! + (1 p)i L + h(a) > w`: In words, the net resources of the parents (ncludng LTC cost and the monetary value of nformal ad, f any) must be larger than that of the chldren, otherwse the bequest motve s not operatve. Let b b b b(i; b`) and b b(i; `; a) denote the optmal bequest levels. When the soluton s nteror, the dervatves wth respect to LTC nsurance coverage, labor supples and nformal care are @` = u00 (m p )h 0 (a) u 00 (m p ) + u 00 (m c ) = h0 (a) 2 ; (10) u 00 (m c )w u 00 (m p ) + u 00 (m c ) = w b ; b` = (1 p)u00 (m p ) u 00 (m p ) + u 00 (m c ) = 1 p 2 = p 2 : (12) When the chld ncreases hs LTC provson the parent ncreases hs bequest by half of the addtonal return. On the other hand, f the chld ncreases hs labor supply and thereby hs ncome, then half of ths addtonal ncome s taxed away by a reducton n the parent s bequest. The net return (when dependent) and the costs (when healthy) of addtonal LTC nsurance coverage are equally dvded between the parent and the chld. 4.2 Stage 2: optmal labor supply and nformal care provson The chld takes nto account the bequest he gets from the parent and chooses labor supply and nformal care by maxmzng max b` max `;a bu c =u(w b` + b b(i; b`)) v( b`); (13) U c =u(w` + b(i; `; a)) v(` + a): (14) 8

12 The FOCs wth respect to b`, ` and a are gven U b! b` = u0 ( bm c ) b` v 0 ( b`) = 0; = u0 (m c ) w v 0 (` + a) = = u0 (m c v 0 (` + a) = 0: Equatons (15) and (16) show that the antcpaton of a postve bequest reduces the chld s margnal bene ts of labor supply and the rotten kd becomes the lazy rotten kd. Ths s because the parent lowers the bequest as chld s ncome ncreases. In other words, part of the chld s extra revenue s taxed away by the parents. Equaton (17) mples that wthout a bequest, t s never optmal for the sel sh chld to provde LTC. If the bequest motve s operatve (b s determned by an nteror soluton) then the amount bequeathed ncreases wth nformal care provson snce nformal care reduces the monetary costs of LTC; see equaton (10). For an operatve bequest motve equatons (10) to (12) mply h 0 (a) =w; (18) u 0 (m c ) w 2 =v0 (` + a); (19) u 0 ( bm c ) w 2 =v0 ( b`): (20) These expressons mplctly determne labor supply n both states of nature, as functons of I (set n the prevous stage of the game): ` `(I) and b` b`(i). Equaton (18) spec es nformal care a whch s ndependent of LTC nsurance coverage. Snce the chld acts n a completely sel sh way when determnng hs nformal care provson, he only takes nto consderaton ts e ect on bequests. Both nformal care and labor cause the same dsutlty. So n the optmum the chld equalzes ther returns. Recall =@a = h 0 (a)=2 whch means that the chld wll receve half of the monetary value hs ad represents to the parent. However, snce wage ncome s also taxed away at 50% the tradeo represented by equaton (18) s e ectvely the e cent one as far the monetary value of ad s concerned. The non-monetary value of ad (that arses when q 0 (a) > 0) s not taken nto account because t does not translate nto a hgher bequest. Consequently, nformal ad s e cent when q 0 (a) = 0; otherwse t s too low because the utlty bene ts, q 0 (a)=u 0 (m p ), valued by parents are not accounted for. 9

13 The comparatve statcs of labor supply wth respect to I (under an operatve bequest motve) are @I = u 00 (m c ) @I u 00 2 (m c ) v 00 (` + a) = u 00 (m c )(1 p)w u 00 (m c )w 2 4v 00 < 0; (21) (` + b` u00 ( bm c ) u 00 ( bm c )pw 2 u 00 ( bm c ) b` v 00 ( b`) = > 0: (22) u 00 ( bm c )w 2 4v 00 ( b`) Snce LTC nsurance s a net bene t for the parent n the state of dependency t ncreases the parent s bequest and thereby reduces labor supply ncentves of the chld. In the state of beng healthy, by contrast, LTC nsurance s only a cost whch n turn reduces the bequest and the chld becomes less lazy. Fnally, when the bequest motve s not operatve, we obtan a =0; (23) u 0 (m c )w =v 0 (` + a); (24) u 0 ( bm c )w =v 0 ( b`): (25) In words, wth no bequests there s no ad, but labor supply decsons are e cent (snce there s no mplct tax anymore). Interestngly, once the bequest motve becomes operatve, the actual level of the bequest s of relevance for the level of labor supply but not for the level of famly ad; see equaton (18). 4.3 Stage 1: optmal LTC nsurance coverage We now turn to the rst stage n whch the parent chooses LTC nsurance coverage I to maxmze the followng welfare functon max I W p = [u(! + (1 p)i L + h(a ) b ) + q(a ) + u(w` + b ) v(` + a )] h + (1 ) u! pi b + u w b` + b v b` : (26) In ths functon a, b, `, b b and b` are determned at the equlbrum of the subsequent stages whch, as descrbed n the prevous subsectons, s contngent on the level of I 10

14 set n the rst stage. The FOC of (26) wth respect to I s gven = u 0 (m p ) 1 p + u 0 (m c ) " + w + + u 0 ( bm c + (1 )u 0 ( bm p b` b v 0 (` + # v ( b`) : Wth the envelope theorem, ths expresson reduces p = (m p ) 1 p (1 )u 0 ( bm p ) b` u0 (m p ) (1 ) b` u 0 ( bm p ) : @I Interestngly, equaton (28) mples full nsurance (m p = bm p ) when prvate nsurance s far ( = p). To see ths observe that under full nsurance (19) (20) yeld ` + a = b`. It then follows mmedately from (11) (12) together wth (21) (22) that full nsurance s a soluton to equaton (28). Ths result s not partcularly surprsng; obtanng full nsurance n a far market s a rather common result (see Mossn, 1968). On the other hand, n our settng t s not obvous at rst glance because of the mult-stage nature of the game. Full nsurance s only optmal (and for that matter feasble) when the symmetry between states of nature sought n the rst stage s not destroyed by the subsequent strategc nteractons. Ths happens to be the case n our settng n partcular because total labor supply (market labor plus ad) wll be the same n both states of nature. When < p, we obtan m p < bm p mplyng less than full nsurance. To see ths we take equaton (28) and evaluate t at full nsurance, that s, I s chosen so that u 0 ( bm p ) = u 0 (m p = ( u 0 (m p)=u 0 ( bm p)u0 (m p ) 1 p) u 00 (m c )w 2 2u 00 (m c )w 2 8v 00 < 0; (` + a) whch s negatve snce the rst term n brackets s negatve for < p and the second term n brackets s postve. In other words, a margnal reducton n I, from ts full nsurance level ncreases welfare. lower than full nsurance. 6 Consequently, the parents optmal choce nvolves 6 As long as W p s a concave functon of I (and gven that n stage 1, the parent s problem s sngle dmensonal). 11

15 5 Lassez-fare versus rst-best allocaton The followng proposton summarzes the man results of Sectons 3 and 4 and spec cally compares the lassez-fare equlbrum to the rst-best allocaton. 7 Proposton 1 The lassez-fare soluton (subgame perfect equlbrum) of the three stage game wth altrustc parents and sel sh chldren has the followng propertes: () when prvate nsurance s far, = p, there s full nsurance, m p = bm p ; otherwse nsurance s less than full, m p < bm p. () when the bequest motve s operatve (so that b; b > 0): (a) we have a > 0 and de ned by h 0 (a) = w. Consequently, nformal ad s e cent when q 0 (a) = 0; otherwse t s too low because the utlty bene ts, q 0 (a)=u 0 (m p ), valued by parents are not accounted for. (b) chldren s market labor supply n both states of nature s ne cent; there s a downward dstorton because chldren face an mplct tax of 50% on ther labor ncome (va a reducton n bequests). () when the bequest motve s not operatve (so that b = b = 0): (a) we have a = 0; no nformal ad s provded. Consequently, ad provson s always ne cent. (b) chldren s market labor supply decson s e cent n both states of nature; they no longer face any mplct tax on ther labor ncomes. 5.1 The case for publc LTC polcy The results summarzed n Proposton 1 provde us wth a bass on whch we can buld to assess the opportunty and the desgn of publc LTC polcy. The rst tem suggests that publc nterventon s useful to overcome ne cences n the prvate nsurance market. When, as s typcally observed n realty, p > the lassez-fare soluton mples nsu cent nsurance. Though nterestng and realstc, ths s not at the heart of our analyss. Our focus wll be on the nteracton between publc polcy and famly ad. As long as the bequest motve s operatve the chldren do provde some nformal ad to ther parents, however, ts level s too low except when q 0 (a) = 0 (so that the full mpact of ad s captured 7 The comparson follows drectly from expresson (18) (20), (23) (25) and (2) (3). 12

16 by ts monetary valuaton n the parent s utlty). Chldren s labor supply s then also ne cent, but ths problem s not drectly connected to the potental need for LTC. When the bequest motve s not operatve no famly ad wll be provded and the case for publc nterventon wll be even stronger. Strkngly, ths falure of famly ad may e ectvely be related to prvate market ne cences. To see ths assume that pl >! > L. In that case the ndvdual cannot a ord to buy nsurance coverage to cover the potental monetary costs of LTC. In case of dependency, the ndvdual wll then not only run out of resources, but he can also not count on any famly ad (snce he has no resources to leave a bequest). However, as long as! > L the ndvdual can a ord nsurance coverage at a far rate. Interestngly, ths may even result n a postve bequest and thus brng about a postve level of ad. To sum up, crowdng out of prvate ad by publc LTC s not a problem n ths settng. Wth an operatve bequest motve, publc LTC wll have no mpact on prvate ad. More amazngly, when the bequest motve s (ntally) not operatve, publc LTC nsurance may even enhance the provson of nformal ad. Ether way, the e ectveness and the desgn of publc LTC depend on the avalable nstruments whch are ultmately of course a queston of nformaton. In the next secton we rst study the mplementaton of the FB under full nformaton. Though of lmted realsm ths s an nterestng benchmark to show whch nstruments are necessary wthn ths settng of mult-stage strategc nteractons to obtan the e cent soluton. Next, we look at a second-best soluton whch s acheved when ad s not observable (and thus cannot be subsdzed). Fnally, we turn to a settng where ndvduals are heterogenous and dynastes d er n wealth. Ths adds an extra potental just caton for publc nterventon, namely redstrbuton. It also makes the case where some ndvduals cannot a ord prvate LTC coverage more plausble and we can have an ntal equlbrum n whch the bequest motve s operatve for some ndvduals and not for others. 5.2 Decentralzaton of the rst-best allocaton Assume for the tme beng that there s no asymmetry of nformaton so that all relevant varables ncludng nformal ad are publcly observable. The followng proposton (whch s establshed n the Appendx 2) shows how ths FB allocaton wthn our mult-stage settng can be decentralzed by a lump-sum transfer from the healthy to the 13

17 dependent elderly ( b D; D) supplemented by lnear subsdes on labor ncomes (b y ; y ) and ad ( a ). Proposton 2 Under full nformaton, the FB allocaton can be decentralzed by a lump-sum transfer from the dependent to the healthy elderly supplemented by lnear subsdes on labor ncomes (of the young) and ad. To acheve ths, the nstruments are set at the followng levels: () the rates of subsdes on w` and w b` denoted y and b y are gven by y = b y = 1 (29) () nformal care s subsdzed at rate a gven by a = h 0 (a fb ) + 2 q0 (a fb ) u 0 (m fb c ) (30) () the lump-sum transfer to dependent elderly, D, and the lump-sum tax mposed on the healthy elderly, b D, are gven by h D =(1 ) L h(a fb ) a a fb w`fb y ; (31) h bd = L h(a fb ) + b y w b`fb : (32) The ntuton behnd these condtons s as follows. Condton (29) s the most straghtforward: snce the chldren face a 50% mplct tax on ther labor ncomes (va the reducton n bequests), we have to subsdze them at rate 1. In other words, the total ncome s multpled by 2 of whch the chldren receve half so that we get the correct tradeo. Expresson (30) s also qute ntutve, except that the factor 2 may appear to be surprsng at rst. The sole bene t chldren get from a s h 0 (a)=2; see equaton (10). Consequently, the remanng socal bene ts, namely h 0 (a fb ) 2 + q0 (a fb ) u 0 (m fb c ) ; are not taken nto account. Ths can be compensated by a subsdy. However snce half of the subsdy wll be lost due to the reducton n bequests, a must be set at a level of twce the unaccounted socal bene ts, whch yelds (30). Turnng to D and b D, the dependent old get the monetary loss of dependency (whch s exactly the net bene t a far prvate nsurance would gve) less the subsdes to ther chldren. In sum, snce parents nance the subsdes to ther own chldren, these 14

18 payments do not nvolve any transfers between famles (that s across states of nature). Such transfers are not necessary because the lump sum transfers between the elderly are already desgned to acheve full nsurance. Ths rst-best decentralzaton provdes an nterestng benchmark. However, n realty some of the relevant varables are lkely not to be publcly observable whch, n turn, wll restrct the avalable polcy nstruments. We shall now study the polcy desgn n second-best settngs where nformaton s no longer complete. We start by a settng n whch nformal ad s not observable so that t cannot be subsdzed. 6 Second-best: unobservable ad Assume the government employs a lump-sum tax on the healthy old, D b and taxes chld s ncome at a proportonal rate t when parents are dependent and at a rate bt when they are healthy to nance publc LTC provson, D to dependent parents. 8 The optmzaton problem s then characterzed by max t;bt;d; b D W (t; bt; D; b D) = [u(! + (1 p)i + D L + h(a ) b ) + q(a )] + [u((1 t)w` + b ) v(` + a )] h + (1 ) u(! pi D b b ) + u (1 bt)w b` + b v b` s.t. tw` + (1 )btw b` + (1 ) b D = D: (33) The FOCs of the above problem wth respect to t, bt, D and b D are =u0 (m p ) h 0 w` u 0 (m c + =u0 (m p ) u0 ( bm b` D b = u0 ( bm p ) + q 0 @t = 0; (34)! u 0 ( bm c )w b` + w b` + = 0; tw = D b + 1 D b = 0: (37) 8 A change n the dstrbuton of ncome between famles does not alter ther consumpton as long as the parent s an e ectve altrust. In other words, whether the lump sum transfer s payed by the chldren or by the parent s rrelevant. Ths neutralty property s the bass of the Rcardan equvalence (see Barro, 1974). 15

19 Appendx B shows that rearrangng and combnng these FOCs and de nng the compensated e ects as @ yelds the followng = ; D b ; (39) Proposton 3 Assume that nformal ad, a, s not observable and that polcy nstruments are restrcted to publc LTC provson to dependent parents, D, nanced by a lump-sum tax on the healthy old, b D, and a lnear tax on the chldren s labor ncome at rates t and bt. The optmal polcy s characterzed by: () u 0 ( bm p ) = u 0 (m p ): the transfers are used to acheve full nsurance whch (for an operatve bequest motve) also mples full nsurance for the chldren, u 0 ( bm c ) = u 0 (m c ). () the followng optmal tax rates n the dependent and healthy state h h 0 (a ) + q0 @`c u 0 t ; (40) bt = 1: (41) To explan these results let us rst nterpret the expresson for t; equaton (40). The numerator can be nterpreted as Pgouvan terms. The rst one s a drect e ect whch resembles the Pgouvan expresson derved n the prevous secton. The chld does not consder the postve externalty of nformal care provson on the parent s utlty. Ths calls for a postve ncome tax whch s e ectvely a subsdy on ad. However, the labor and ad varatons nduced by the tax also have ndrect e ects on the parent s utlty. These e ects are negatve and operate va the adjustment n bequests. Both more ad and lower labor supply ncrease bequests and thus reduce parent s utlty, whch counteracts the rst postve e ect of hgher ncome taxes. The denomnator represents the deadweght loss of ncome taxaton. Fnally, the optmal tax rate n the healthy state, bt, smply re ects the rst-best subsdy on labor whch s equal to one. To sum up, transfers are used to acheve full nsurance of the old. The tax on labor, on the other hand, s not used to rase revenue but because t ncreases nformal ad 9 The sgn of the last term changes because one s a transfer and the other a tax, so the compensaton (to reman at the same utlty level) goes n opposte drectons. We expect these compensated e ects to be negatve. 16

20 (whch becomes more attractve when market labor s taxed). The level of t s then set to strke a balance between the deadweght loss of the labor tax (the denomnator of (40)) aganst the net bene ts assocated wth ts e ect on ad (whch arse provded that the drect e ects n the numerator (40) domnate the negatve ones), whle bt s at ts rst-best level. 7 Heterogenous famles So far, famles were dentcal. In ths secton we ntroduce parents who d er n ther wealth! ( = l; h) where! l <! h. Ths brngs n another mportant just caton for government nterventon, namely redstrbuton. It also makes the case where some parents cannot a ord prvate LTC coverage more plausble, and we can have an ntal equlbrum n whch the bequest motve s operatve for some famles and not for others. The share of type- famles s gven by. We assume that! s unobservable to the government. To concentrate on the mplcatons of wealth heterogenety we assume that all other varables, that s, bequests (b ; b ), labor supples (`; b`), nformal care a and LTC care nsurance coverage I can be observed. The chldren s consumpton levels are thus e ectvely known by the government. Under the consdered nformaton structure nstruments nclude a (possbly nonlnear) transfer scheme for parents and chldren n each state of nature gven by D(I ; b ), D(I b ; b ) and T (w`; a ), T b (w b`) respectvely. In other words, long-term care nsurance coverage, bequests, labor supply and ad can be taxed or subsdzed and D ; D b and T ; T b can be postve or negatve. Wth wealth heterogenety, the (utltaran) rst-best allocaton contnues to be de ned by expressons (2) (5) whch apply for all types. The full nformaton mplementaton of the rst-best allocaton remans as descrbed n Proposton 2 except that the transfers D and D b are now type spec c and desgned to elmnate wealth d erences. 10 Consequently, they mply a transfer from hgh- to low-wealth famles. In a second-best world wth unobservable wealth and a utltaran welfare functon, the ncentve constrant from type-h to type-l famles wll then be bndng. We proceed as follows: rst, we reconsder the varous states of the ad for bequest game wth nonlnear taxes and transfers and n partcular parent s and chld s optmza- 10 All the other expressons n the proposton contnue to apply and the taxes and subsdes on `, b` and a are the same for all types. 17

21 ton. Then, we determne the second-best allocaton and show how the transfer scheme must be desgned to mplement ths allocaton. 7.1 Equlbrum wth nonlnear taxes and transfers Gven the above descrbed transfer scheme, parent s and chld s utltes are gven by 11 Wp = [u(! + (1 p)i L + h(a ) b + D(I ; b )) + q(a )] + (1 )u! pi b b + D(I b ; b ) + Uc; (42) Uc = [u(w` + b + T (w`; a )) v (` + a )] h + (1 ) u w b` + b + T b (w b`) v b` : (43) In stage 3 parents maxmze (42) wth respect to bequests n both states of nature, b and b b, whch are then mplctly gven by u 0 (m p) u 0 (m c) = 1 1 Db ; (44) u 0 ( bm p) u 0 ( bm c) = 1 : 1 D b bb (45) D b and D b b denote partal dervatves whch represent margnal tax (or subsdy) rates on bequests. When bequests are taxed (subsdzed) D b ; D b b < (>)0, parents have a hgher (lower) level of consumpton than ther chldren. calculate the followng comparatve statcs From the above equatons we = (1 D b )u00 (m p)h 0 (a ) u 00 (m u 00 (m c) + (1 Db )u00 (m ; @ b b` u 00 (m = c)(1 + T ì )w u 00 (m c) + (1 Db )u00 (m p) ; (47) u 00 ( bm c)(1 + T b )w b` = u 00 ( bm c) + (1 D b bb )u 00 ( bm p) : (48) In the second stage the chldren choose labor supples, ` and b`, and nformal care provson, a, agan takng nto consderaton ther e ects on bequests. The margnal 11 Type ndces are generally subscrpts but they become superscrpts when ether the subscrpt s already used to ndcate the famly member,.e. the chld or the parent, or when the subscrpt s already used for a partal dervatve. Ths s smply to avod multple subscrpts. 18

22 rates of substtuton are as follows v 0 (` + a ) u 0 (m c) v 0 (` + a ) u 0 (m c) = (1 + T ì = " 1 # u 00 (m c) u 00 (m c) + (1 Db )u00 (m (1 + T ì )w; (49) p) = Ta = Ta + (1 D b )u00 (m p)h 0 (a ) u 00 (m u 00 (m c) + (1 Db )u00 (m ; (50) p) " # u 00 ( bm = 1 c) (1 T b u 00 ( bm c) + (1 D b bb )u 00 ( bm b` )w: (51) p) v 0 ( b`) u 0 ( bm c) = (1 + b T b` )w b` In the rst stage, parents choose ther prvate nsurance protecton, I. However, snce we show below that prvate nsurance s not necessary to mplement the secondbest allocaton, we do not reconsder the prvate nsurance decson at ths pont. 7.2 Second-best soluton The problem Ths subsecton characterzes the optmal utltaran allocaton constraned by the nformaton structure just sketched. The optmzaton problem of the government s gven by 12 max W = X I ;D ;b ;a ;`;T ; D b ; b ; b`; T b 2fl;hg [u(w` + b + T ) v(` + a )] + [u(! + (1 p)i + D L + h(a ) b ) + q(a )] h + (1 ) u! + D b pi b b + u w b` + b + T b v( b`) subject to the resource constrant X 2fl;hg h n[d + T ] + (1 ) bd + T b o = 0 and subject to the followng ncentve constrants for ; j 2 fl; hg [u(! + (1 p)i + D L + h(a ) b ) + q(a ) + u(w` + b + T ) v(` + a )] h + (1 ) u! + D b pi b b + u w b` + b + T b v( b`) [u(! + (1 p)i j + D j L + h(a j ) b j ) + q(a j ) + u(w`j + b j + T j ) v (`j + a j )] h + (1 ) u! + D b j pi j b bj + u w b`j + b j + T b j v b`j 8 6= j: (52) Ths problem characterzes the best allocaton (n terms of utltaran welfare) that can be acheved gven the nformaton structure. Observe that snce all chldren have 12 We assume that the soluton mples a strctly postve level of ad and thus an operatve bequest motve for all types. 19

23 the same wage rate ther labor supples, ` and b`, are e ectvely observable so that the chldren of the mmckng parents must have the same labor supples (and level of a) as those of the mmcked parents; ths explans the wrtng of the ncentve constrant. As usual n models wth dscrete types (and partcularly wth only two types) the soluton can be mplemented n many ways. 13 In the remander of ths secton we shall look at the mplementaton whch nterferes as lttle as possble wth the structure of our mult-stage ad for bequest game. In other words, we wll set the margnal tax rates (whenever possble) so that both chldren and parents choose the optmal allocaton as nteror soluton of ther optmzaton problem n the relevant stage of the game. We shall rst examne the taxaton of bequest, and then turn to the labor supply and ad decsons and determne f and how they are dstorted and whether they are subject to taxaton or subsdzaton. Fnally, we consder prvate nsurance as well as the lumpsum transfers (or taxes) between the elderly (D and b D) whch can be seen as publc LTC nsurance (or bene ts) scheme. Denote j, ; j 2 fl; hg, the Lagrange multpler assocated wth the self-selecton constrant from type- to type-j famles and the one assocated wth the resource constrant. The FOCs of ths problem are stated n Appendx C. Whle our formal condtons make no assumpton as to the bndng ncentve constrant, our nterpretatons below wll concentrate on the case where only the downward ncentve constrant s bndng ( hl > 0 and lh = 0). Ths s the relevant case wth a utltaran socal welfare functon (whch mples redstrbuton from the hgh-wealth to the low-wealth ndvduals) Taxaton of bequests Rearrangng the FOCs wth respect to b and b b yelds the followng margnal rates of substtuton between parent s and chld s utlty u 0 (m + j p) u 0 (m c) = u 0 ( bm + j p) u 0 ( bm c) = j u 0 (m j c ) u 0 (m c) 1 u 0 (m j p ) u 0 (m j c ) 8 ; (53) + j u 0 ( bm j c ) u j 1 0 ( bm j p ) u 0 ( bm c ) u 0 ( bm j c ) 8 : (54) + j 13 And the nonlnear functons consdered above can smply be used to control quanttes by assgnng large penaltes to any choces d erent from the optmal allocaton. Though extreme, ths shows that mplementaton s always possble. 20

24 Consder the top famly, that s, the hgh-wealth famly who s not mmcked mplyng lh = 0. For such a famly equatons (53) and (54) are equal to one. In other words, rch famles bequests are not taxed at the margn whch s the tradtonal no dstorton at the top result. Wth (44) and (45) we thus have D h b = b D h b b = 0. Low-wealth famles, by contrast, are those who are mmcked by hgh-wealth famles mplyng hl > 0. Snce the chld s consumpton level s e ectvely observed, we have u 0 (m j c ) = u 0 (m c) and u 0 ( bm j c ) = u 0 ( bm c). Rch famles who mmc poor famles, however, no longer equalze consumpton levels between parents and chldren, but u 0 (m j p ) < u 0 (m j c ) and u 0 ( bm j p ) < u 0 ( bm j c ) due to the parents hgher wealth. Wth (44) and (45), we have 1 1 D l b u 0 (m hl p ) c ) l hl h1 u = 0 (m hl < 1; (55) l u 0 ( bm hl p ) 1 l hl h1 u = 0 ( bm hl c ) < 1; (56) 1 D b l bb l mplyng D l b ; b D l b b < 0. In other words, poor famles face a downward dstorton on ther bequests. A tax on ther bequests relaxes a bndng ncentve constrant; snce type-h famles want to bequeath more to ther chldren due to ther hgher wealth, they are also hurt more by a tax on these transfers Taxaton of chldren s labor supply and ad Rearrangng the FOCs wth respect to ` and b`, we get the followng margnal rates of substtuton for labor supply n the the dependent and the healthy state respectvely u h + j 0 (m j c ) j u 0 (m c ) w v 0 (` + a ) u 0 (m c) = v 0 ( b`) u 0 ( bm c) = + j j = w 8 ; (57) h + j j u 0 ( bm j c ) u 0 ( bm c ) w + j j = w 8 : (58) Snce consumpton levels of the chldren are e ectvely observed n both states of nature, the tradeo that both rch and poor famles face n the second-best s the same as n the rst-best; see equatons (3) and (4). Gven the mult-stage nature of our problem ths however does not mply a margnal tax rate equal to zero. Combnng equaton 21

25 (49) wth (57) and equaton (51) wth (58), we obtan " # (1 + T ì )w u 00 (m = 1 u 00 (m c) + (1 Db )u00 (m (1 + T ì )w = w 8 ; (59) p) " # 1 + T b b` w u 00 ( bm = 1 c) (1 T b` u 00 ( bm c) + (1 D b bb )u 00 ( bm b` )w = w 8 : (60) p) That s, the tax on labor s chosen to o set the downward dstorton of bequests on labor supply. Solvng equatons (59) and (60) for T ì and b T b` yelds T ì u 00 (m = c) (1 Db )u00 (m p) Tb` u 00 ( bm = c) (1 Db )u 00 ( bm p) 8 ; (61) 8 : (62) Snce the rch face no dstorton on bequests (D h b = Dh b b = 0 so that u 00 (m h c ) = u 00 (m h p) and u 00 ( bm h c ) = u 00 ( bm h p)) these expressons mply the rst-best margnal tax (subsdy) rates on labor supply: T h` = b T h b` = 1 (see Proposton 2). For the poor, by contrast, we have a postve tax rate on bequests, and wth our assumptons on utlty ths mples u 00 (m l p) u 00 (m l c) and u 00 ( bm l p) u 00 ( bm l c). 14 In other words, the optmal subsdes on labor supples for the poor are smaller than the rst-best levels. Ths s qute ntutve. Recall that poor famles face a tax on ther bequest, whch mtgates the negatve mpact bequests have on labor supply. Because of the bequest tax, an ncrease n the chld s labor ncome nduces a smaller reducton n net bequests than n the absence of (bequest) taxaton. Put d erently, t allevates the lazy rotten kd phenomenon. Consequently, labor supples need to be subsdzed at a lower rate to reach the rst-best tradeo. Let us consder the margnal rate of substtuton for nformal care provson whch can be wrtten as v 0 (` + a ) u 0 (m c) = u h + j 0 (m j c ) j h 0 (a u 0 (m ) c) + q0 (a ) + j j u 0 (m c) = h0 (a ) + q0 (a ) u 0 (m c) As for labor supply, the tradeo for nformal care provson n the second-best s the same as n the rst-best. But, agan ths does not mply zero margnal tax rates. From equaton (50), we get 8 : T = T a + (1 D b )u00 (m p)h 0 (a ) u 00 (m c)t a u 00 (m c) + (1 D b )u00 (m p) = h 0 (a ) + q0 (a ) u 0 (m c) 8 : (63) 14 Ths result makes use of our assumpton that u

26 Informal care provson n the lassez-fare s ne cently low and to acheve the rstbest tradeo, t must be subsdzed at the margn (see Proposton 2). Solvng equaton (63) for T a yelds T a = u 00 (m c) (1 D b )u00 (m p) h0 (a ) + " u 00 (m c) (1 D b )u00 (m p) + 1 # q 0 (a ) u 0 (m c) > 0 8 : Snce rch famles face no dstorton on bequests, the subsdy on ad s agan the rstbest one, T h a = h 0 (a h ) + 2 q0 (a h ) u 0 (m h c ) : Poor famles, by contrast, face a dstorton on ther bequests (Db l > 0) and to acheve the rst-best tradeo for nformal care, ther subsdy on ad s lower. The reason behnd ths result s that the chld equalzes the return of labor wth the return of ad. Snce the poor s labor supply s already subsdzed at a lower than rst-best rate, nformal care must also be subsdzed at a lower rate to obtan the rst-best tradeo for both varables Prvate nsurance and lump-sum transfers We now turn to prvate nsurance and publc LTC bene ts. Before worryng about a possble taxaton or subsdzaton of prvate nsurance, we have to examne whether t s used at all n the second-best. To study ths we use the FOCs gven n Appendx C. Substtutng equatons (84) and (85) nto equaton =(1 p) j u 0 (m j p ) + j (1 (1 )p j u 0 ( bm j p ) + p)u 0 (m j p ) (1 )pu 0 ( bm j p ) = (1 p) (1 )p 0: (64) That s, when prvate nsurance s o ered at hgher than far rates ( < p), t s not used n the second-best soluton mplyng I = 0 for = l; h. In other words, prvate nsurance s domnated by publc nsurance provded through transfers and taxes D and b D. Wth our nonlnear tax scheme, there s no cost of publc funds (or deadweght loss) so that the publc sector can always o er full nsurance at far rates. As we shall now show, ths s not necessarly the optmal polcy, but the argument shows that prvate nsurance s a domnated nstrument, unless t s far ( = p) n whch case t does no harm but cannot perform better than publc coverage ether. Lets turn our attenton to the optmal lump-sum transfers whch provde nsurance and redstrbute from hgh- to low-wealth famles. Note that D can also be nterpreted 23