Empirical implications of limited commitment. Evidence from Mexican villages

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1 Emprcal mplcaon of lmed commmen. Edence from Mexcan llage Pedro Albarran and Orazo P. Aanao Fr Draf July 00 Th Draf February 003 A prelmnary draf of h paper wa preened a he SIEPR Conference "Effec of Cred Marke Frcon on he Macroeconomy" a Sanford, Augu 00 and wa prepared whle Aanao wa ng he Deparmen of Economc a he Unery of Chcago, whoe hopaly graefully acknowledged. The auhor hank Fernando Alarez, Marcel Fafchamp, Narayana Kocherlakoa, Vcor Ro-Rull and Rob Townend for ueful dcuon, a well a emnar audence a UCL, Cambrdge, Wharon, Kellogg, MIT, Prnceon, Mnneoa, Sanford, Chcago, Oxford Nuffeld and CSAE, Souhampon, LSE for ueful commen. Joep Pjoan-Ma helped wh he mulaon, whle Ana Sanago helped wh he Progrea daa. Seeral people n he Progrea aff, ncludng Danel Hernandez, Parca Munz and Suan Parker helped u wh nformaon abou he daa and he program n he early par of he projec. We alo wh o acknowledge our deb o he lae Joe Gomez de Leon who fr mulaed our nere no Progrea and whou whoe on none of h would hae been poble. CEMFI & Unerdad Carlo III Madrd Unery College London, IFS, and NBER

2 Inroducon In h paper we udy he oberable mplcaon of a cla of model of conumpon moohng characerzed by le han full rk harng. In parcular, we focu on model where he fr be allocaon of reource no acheed becaue of he preence of mperfec enforceably. Tha, whle he nddual whn he economy we conder hae full nformaon abou he endowmen and he acon aken by he oher nddual wh whom hey nerac, hey canno coney h nformaon o he exernal world. A a conequence, full enforceably of conrac mgh be exceedngly coly. We are herefore led o conder model where nddual only ener conrac ha are elf-enforceable. The model we conder are no new. They conder conraned-effcen allocaon, where he conran are gen by he nably of enforcng conrac agreemen. They hae been uded by eeral auhor, ncludng Thoma and Worrall 988, 990, Kocherlakoa 996, Alarez and Jerman 000a,b, Lgon, Thoma and Worrall 000, Kehoe and Perr 999, Krueger and Perr 00, Aanao and Ro-Rull 000, among oher. In he cla of model we conder, conrac are enforced by he hrea o reerng o he wor ub-game perfec equlbrum: auarky. Th equlbrum concep, propoed by Abreu e al. 988 ha proed ery ueful n h leraure. Thee model can generae ery complex dynamc and allocaon of reource ha dffer ubanally from hoe mpled by complee marke. In parcular, hey can replcae feaure of ner-houehold agreemen ha are reporedly prealen n many llage econome. Plaeau 987, for nance, clam ha he rk harng agreemen ofen obered n fherman commune are half way beween cred and nurance, whle Udry 994 repor ha nere rae and maure on loan n Norhern Ngera eem o be ae conngen and ary no only wh hock affecng he borrower bu alo wh hoe affecng he lender. Moreoer, he aumpon ypcally ued n hee model complee nformaon, lack of rong enforcemen mechanm, repeaed neracon, large cope for rk harng eem o be approprae o characerze poor and olaed rural llage n many par of he world. A we dcu below, he crucal ae arable of hee model, he rao of margnal ule, unlke n model wh perfec rk harng, moe oer me o guaranee ha he parcpaon conran are alway afed. One can nerpre moemen n h ae arable a moemen n he relae Pareo wegh ha a ocal planner gen o an

3 nddual: when he parcpaon conran of an nddual bndng, he ocal planner wll ncreae he wegh gen o ha nddual. I he paral nera creaed by h mechanm ha ge o he effcen conrac ome of he feaure of a deb conrac. Howeer, when he conran of anoher nddual become bndng, pa hory become rrelean and conrac reemble more an nurance one. No much edence ex on he emprcal releance of model wh mperfec enforceably. The only paper we are aware of are hoe by Foer and Roenzweg 00, Lgon, Thoma and Worrall 00, Krueger and Perr 00 and Albarran and Aanao 00. In her magnae and nereng paper, Foer and Roenzweg 00 conder he mplcaon of he model for ranfer. In parcular, hey noce ha, condonng on he curren hock, he ne ranfer an nddual recee from her parner n a elf enforcng rk harng agreemen negaely relaed o he cumulae of pa ranfer receed. Afer howng wh ome mulaon ha uch a relaonhp well approxmaed by a lnear one, hey conder fr dfference o relae change n ranfer o he change n hock and he lagged leel of ranfer. Such an equaon can be ued on panel daa ha conan a lea wo oberaon per houehold on ranfer o e ha he coeffcen on lagged ranfer negae. Foer and Roenzweg 00 alo exend he model o conder he preence of alrum. Applyng hee e o daa from Pakan and Bangladeh hey fnd a paern of coeffcen ha n accordance wh he predcon of he model. Lgon, Thoma and Worrall 00 ake a more rucural approach and emae a eron of he model wh mperfec enforceably by maxmum lkelhood ung he ICRSTAT daa ued, among oher, by Townend 994 o e full nurance. Ther emaon mehod nole, for each ealuaon of he lkelhood funcon, he numercal oluon of he opmal conrac. The numercal complexy of he problem force hem o ome mporan mplfyng aumpon. In parcular, hey aume he lack of a orage echnology o ha aggregae ncome and conumpon are alo he ame and, nead of conderng he conrac among he N houehold n he llage hey conder N conrac beween each houehold and he remanng N- houehold. Whle hey conncngly jufy he laer aumpon on he ba of an aggregaon argumen, he former lkely o be more crcal and hard o jufy. Howeer, een wh hee counerfacual aumpon, he model wh

4 mperfec enforceably eem o be able o f he obered daa beer han a model ha aume perfec rk harng. In a ery recen paper, Krueger and Perr 00 conder he mplcaon of he model for he relaonhp beween he eoluon of he cro econal arance of ncome and conumpon and preen ome edence ha no nconen wh he model. In Aanao and Albarran 00, we propoe a mple e ha conder how he nroducon of a publc ranfer program affec, under mperfec enforceably, prae ranfer. In ha paper, we explo he randomzed adopon of he welfare program mplemened n he llage ha form our daa bae. We ar by nocng ha a welfare program ha nole publc ranfer o ome or all of he parner of an nurance agreemen wh mperfec enforceably lkely o reduce or crowd ou, under many preference pecfcaon, prae ranfer. Th mplcaon, howeer, no unque o he model wh mperfec enforceably: mgh alo occur n model wh perfec rk harng and ceranly n model wh alrum. Howeer, whn he e of model we conder, he amoun by whch prae ranfer are reduced deermned by oher feaure of he economy, uch a he arance of ncome and perence. I h e of mplcaon, ha are unque o he model wh mperfec enforceably ha we conder. The man cope of h paper o propoe ome new e ha can be nformae abou he emprcal releance of uch model. We propoe wo dfferen approache ha focu on dfferen feaure of he model and of he aalable daa. Our approache dffer conderably from hoe menoned aboe. The man dfference of boh our e relae o hoe menoned aboe ha our approach focue on he mplcaon of he model for neremporal conumpon allocaon. We can herefore afford o be len abou he parcular mechanm ha are ued o achee ha parcular allocaon of reource. They are herefore n he pr of Townend 994 e. Th parcularly rue for our fr e, baed on llage leel ac. We apply our e o a unque daa e from rural Mexco. The daa, a 4 wae panel colleced beween 997 and 999, wa gahered o ealuae a large welfare program ared by he Zedllo admnraon n 998, called PROGRESA, whoe am o foer he accumulaon of human capal n rural commune by prodng fnancal ncene o he nemen n healh and educaon. To ealuae uch a program he admnraon ared 3

5 collecng a large daabae ha gahered nformaon on all he houehold lng n 506 locale n 7 ae ha qualfed for he program. The fr pece of edence we conder can be nerpreed a a e of full nurance and fr be allocaon. A we dcu below, and a wa noced by Deaon and Paxon 994, a raghforward mplcaon of model wh complee nurance marke ha he cro econal arance of he margnal uly of conumpon conan oer me. In wha follow, we ar from h mplcaon and conruc a meaure of he deaon from fr be allocaon gen by he magnude of he change n he cro econal arance of margnal uly. We hen relae h meaure wh arou feaure of he econome we udy, uch a he arance and perence of ncome, whch would affec under he mperfec enforceably of conrac. Our e predc ha hee quane relae n a parcular way wh he leel of rk harng: eeryhng ha ncreae he enforceably of conrac hould reul n a greaer amoun of rk harng. The econd e we propoe deelop ome of he dea propoed by Kocherlakoa 996. I ue he fr order condon mpled by he conraned opmzaon ha characerze he neremporal allocaon of reource under mperfec enforceably. In parcular, explo he fac ha cro econal dfference n he rae of growh of margnal uly of conumpon can be nformae abou whch houehold hae a bndng enforceably conran. Moreoer, he conraned effcency naure of hee conrac ha pecfc mplcaon for boh he e of conraned and unconraned houehold. We generalze he dea n Kocherlakoa o conder orage and meauremen error n conumpon. The re of he paper organzed a follow. In econ, we kech he heorecal framework ha we wll be negang emprcally. Whle mo of h maeral no new, we need o eablh noaon and ae ome reul ha can be ranlaed no eable mplcaon. In econ 3, we propoe our hree e of he model. In econ 4 we dcu A he program ery large and wa phaed n oer a perod of wo year, 86 randomly choen locale of he ealuaon ample were placed a he end of he queue o ha he program n hoe locale wa ared n December 999 raher han July 998. Th randomzaon cheme wa explcly mplemened wh he purpoe of faclang he ealuaon of he effecene of he program. In wha follow we do no ue he araon nduced by he program o denfy our model. We followed ha raegy n Albarran and Aanao 00. 4

6 he daa e we ue, whle n Secon 5, we preen our reul. Secon 6 conclude he paper wh ome hough on fuure reearch. The heorecal framework In h econ, we kech a mple model of rk harng wh mperfec enforceably. Whle mo of he maeral we preen no parcularly new, we need o eablh noaon and ae formally ome of he reul we wll be ung n our emprcal work. Moreoer, we could no fnd he characerzaon of ome of he propere of he equlbra we udy n he exng leraure. Th parly becaue he propere we udy are argeed oward our emprcal applcaon. In oher word, we are nereed n oberable propere of he equlbra wh mperfec enforceably. For expoonal mplcy we ar our preenaon conderng he cae of wo houehold whou orage opporune. We hen dcu how our reul exend o he cae of many houehold and orage.. The bac model Le u conder wo nfnely led agen who, n each perod recee an endowmen e, =,, whch a funcon of an aggregae hock z and an doyncrac hock υ : e = e υ, z. Bohυ and z hae dcree uppor. The wo doyncrac hock are ndependen of he aggregae hock and of each oher. A we ar by aumng ha here are no orage opporune, he ae of he world fully decrbed by he aggregae hock and he wo doyncrac hock. We denoe he ecor ha conan hee hree arable wh and aume ha Marko wh a ranon probably marx Γ. We hall denoe wh { } =,,..., he hory of he yem up o me. The aumpon of no orage make he alue of auarky, whch we wll denoe by Ω e, ery eay o compue: wll be equal o he preen dcouned alue of he uly of conumng he nddual endowmen mnu ome penaly P ha he communy could mpoe on he nddual who doe no comply wh he erm of a conrac: 5

7 j j. Ω e = u e + E + β u e e P j= where u a well-behaed uly funcon connuou, concae, and afyng he Inada condon and β he dcoun facor aumed o be beween zero and one. A he penaly wll no play any mporan role n wha we are gong o do, we wll e o zero n wha follow. We aume ha here complee nformaon beween he wo nddual: each obere compleely he ecor and hory. A he wo doyncrac hock are uncorrelaed here wll be cope for rk harng. A conrac beween he wo nddual wll pecfy he ne ranfer from nddual o nddual a a funcon of he curren hory of he world, κ. The alue of beng n uch a conrac a me, gen he curren hory of he world, for nddual wll herefore be:. U c + k = u c + k = e + E + k j= κ j β u c + k + j + j e, k = 0,,... Noce ha he conrac κ, n h framework, he only way he wo nddual hae o hare rk. The edence repored by Townend 994, Udry 994 and many oher, lead o belee ha n llage econome doyncrac hock are mporan enough and orage echnologe and acce o cred lmed enough o ha een n a more general framework, hee conrac conue an mporan way o hare rk. In he abence of enforceably problem, he fr be allocaon of reource can be acheed and he wo nddual wll hare doyncrac rk fully. Of coure full rk harng allocaon are no unque and wll depend on he reource conrolled by each nddual, or, f one chooe o characerze hem by conderng a ocal planner problem, by he Pareo wegh ued by he planner. The emprcal mplcaon of full rk harng hae long been recognzed and uded. In parcular, Townend 994 eed on daa from ome Indan llage he hypohe ha change n he nddual margnal uly of conumpon are no affeced by change n nddual ncome, whch, a we wll ee below, one of he mplcaon of complee nurance. 6

8 7 If rk-harng conrac are no fully enforceable, one ha o rerc one aenon o conrac ha are elf-enforcng. I ha become pracce n h leraure o focu on conrac ha are conraned effcen and enforced by he hrea o reere o he wor ub-game perfec equlbrum, whch ealy proen o be he auarkc equlbrum. To characerze he elf-enforceable rk harng conrac we can herefore conder a modfed ocal planner problem where, n addon o he reource conran, he ocal planner face wo parcpaon conran. Th can be wren a follow:.3 : : :.. e U k e U k e e c c U U Max Ω Ω µ λ λ where λ and λ are he Pareo wegh agned by he ocal planner o he wo nddual, µ he mulpler aocaed o he reource conran, and k he mulpler aocaed o he enforceably conran of nddual. Wren a n.3, he program no mmedaely recure. Howeer, poble o rewre n a dfferen way ha make recure and allow one o dere ome ueful expreon. Th approach, propoed by Marce and Marmon 999 ha been ued, among oher, by Kehoe and Perr 00 and Aanao and Ro Rull For h purpoe, ueful o defne recurely a quany k K K + = wh K λ = 0. K, for each nddual, he cumulae of he mulpler aocaed o he enforceably conran for ha nddual, ung a an nal alue he Pareo wegh for ha nddual. Wh h defnon, raghforward o how ha one can re-wre he planner problem a:.4 { } :.. ] [ 0 e e c c e c u k c u K E Max + + Ω + = = µ β The equlbrum concep ued n h leraure are hoe adocaed by Abreu, Pearce and Sacche An alernae oluon mehod ued by Lgon e al. 00

9 Gen he repreenaon.4, a fr order condon for he effcen allocaon of reource gen by he followng equaon: u' c K K + k.5 = = x u' c K K + k The arable x, whch equal o he rao of margnal ule of conumpon of he wo nddual, fully repreen he eoluon of he yem. If he enforceably conran are neer bndng, x a conan and equal o he rao of Pareo wegh. Th he andard reul one ge under full rk-harng. When one of he enforceably conran bndng, he rao of margnal ule wll change. Noce ha, a no poble ha boh conran are bndng a he ame me, a each me a lea one of he wo mulpler k, =, ; gong o be equal o zero. Therefore, a each pon n me, wll be poble o eablh wheher he arable x ncreang or decreang. Moreoer, equaon.5 mple ha he margnal uly of he conraned nddual ncreae by le han he margnal uly of he unconraned nddual. Th fac can be ued o eablh, on he ba of oberable arable, whch of he wo conumer conraned. One can nerpre he rao of margnal ule a he relae wegh ha he ocal planner ge o he wo nddual. In oher word, he planner compenae an nddual whoe enforceably conran bndng wh an ncreae n her relae wegh. The model can generae ome nereng dynamc. I poble, for nance, no only ha he amoun of ne ranfer n abolue alue below he leel mpled by fr be, bu ha ha he oppoe gn. In oher word, we can hae uaon n whch he nddual who relaely le lucky make a ranfer o her lucker parner. Th happen f he relae wegh of an nddual n he ocal planner problem ha declned o much ha een f he recee a maller endowmen han ha of her parner, he aked o make a ranfer. 4 Th mgh happen when an nddual ha receed from her parner ranfer for ome me. Th example llurae well he ene n whch he conrac we are conderng are half-way beween deb and nurance. There are ae of he world n whch he opmal conrac keep rack of pa ranfer and make agen repay her deb. Howeer, when 4 Noce ha n a wo-peron conex, alway he peron makng a ne poe ranfer whoe enforceably conran mgh be bndng. A ery clear dcuon of hee ue can be found n Lgon e al

10 he ae of he world uch ha he deny of he conraned nddual change, he opmal conrac erae all pa deb ee he dcuon n Lgon e al. 00 on h.. Exenon There are eeral way n whch one can add orage o he model. We wll aume ha wo agen n our model hae acce o a orage echnology ha allow hem o ranform conumpon a me no conumpon a me + a a rae R, whch aumed conan. In prncple we could alo add a lqudy conran ang ha orage canno be negae and capurng he fac ha mo of he agen n llage econome hae only ery lmed acce o exernal cred. Th would complcae he noaon furher. A poned ou by Kehoe and Perr 00, equaon.5 wll hold een n he preence of orage. The exen o whch orage change he naure of he equlbrum depend on how orage affec he alue of auarky. In general, unle orage communally held, he amoun of prae orage wll affec he alue of auarky, whch wll now denoe wh Ω e, A. Addng orage o he model alo modfe he reource conran faced by he planner n equaon.3 and.4. In parcular, f we denoe by held by nddual, he aggregae reource conran wll be: A he amoun of orage.6 µ : c c + A + A e + e + + R A + A Gen h reource conran, wll be poble o dere an Euler equaon for each of he wo conumer: k, + + k+ Ω e+ A.7 u' c = β E + R + u' c + ; =, K K A Noce ha when he parcpaon conran no bndng for nddual, equaon.7 reduce o a andard Euler equaon of he knd analyzed n he conumpon leraure. When he mulpler k + poe, ha when he enforceably conran for nddual bndng, nroduce a doron n he neremporal allocaon of reource for uch nddual. The mulpler ener wce, reflecng wo dfferen effec. Mong reource o perod + make nddual beer off and herefore relaxe he enforceably conran. 9

11 On he oher hand, an ncreae n he prae orage of nddual alo ncreae he alue of auarky and herefore make he enforceably conran wore. Howeer, gen ha equaon.5 hold, we are able o gn he dfference beween he wo effec: for he unconraned nddual he expeced rae of growh of margnal uly hould be hgher han for he conraned nddual. Moreoer, for he unconraned nddual he expeced rae of growh of margnal uly, mplyng ha β + R k ', + u c+ Ω e+ A E > 0 '. K u c A The cae wh wo nddual parcularly mple becaue a mo one of he wo enforceably conran can be bndng a any gen me. Howeer, een when we hae more han wo nddual, here mu be a lea one for whom he conran no bndng. In general, here wll be a group of nddual for whom he conran no bndng and whoe margnal ule of conumpon change a he ame rae, and anoher for whom he conran bndng and for whom f he fr effec dcued n he preou paragraph ronger he margnal uly grow le han ha of he unconraned group. Noce alo ha, a noed by Kocherlakoa 996, f an nddual gen by he ocal planner a ceran Pareo wegh mpled by he alue of curren margnal uly of conumpon relae o he oher nddual n he economy, and, on enerng perod + he ae of he world uch ha he conraned n ha her parcpaon conran bndng, he new allocaon of conumpon wll be deermned unquely by uch parcpaon conran. Tha, he cenral planner wll ge o her enough conumpon and connuaon uly o make her curren alue a lea a large a he alue of auarky. Therefore, he alue of conumpon for uch an nddual wll depend only on he curren ae of he world and on oher nddual lagged margnal uly. The forgene of he yem, whch crucal for he conraned effcency of he conrac, wll mply ha her curren conumpon wll be ndependen of her lagged margnal uly. In econ 3.3, we dcu how o explo h propery and ha dcued n he preou paragraph o conruc a e of he model. Mo reul obaned n he wo-nddual model exend o he cae n whch we hae many nddual, a long a he complee nformaon rucure preered. To conder many nddual, ueful o modfy he planner problem and wre a he maxmzaon of he 0

12 uly of an arbrarly choen nddual gen he wegh gen o he oher nddual ubjec o parcpaon and prome-keepng conran. Th he framework ued, for nance, by Lgon e al Characerzng propere of he equlbra Two naural benchmark o compare he allocaon of reource mpled by he equlbrum conrac are he auarkc and he fr be one. Whle he equlbrum conrac can concde wh one of hee wo, n general wll call for ome lmed amoun of rk harng. The effec of changng ome of he feaure of an economy on he lkelhood ha he equlbrum allocaon concde wh fr be or auarky reaonably nue and ha been dcued n he leraure. Propoon 4.9 n Alarez and Jermann 000 and Propoon n Lgon e al. 00 conan he man comparae ac reul. In parcular, eay o proe ha for alue of he dcoun facor ha are hgh enough, alway poble o mplemen he fr be allocaon. The ame rue for he coeffcen of relae rk aeron: enough curaure n he uly funcon wll deler fr be equlbra. The nuon behnd he wo reul reaonably obou: a he punhmen for deang from he opmal conrac n he fuure and con n he excluon from ome rk harng agreemen, more paen and more rk aere conumer wll fnd uch punhmen harher. I wll herefore be eaer o enforce rk-harng agreemen. A he oher exreme, here wll be alue of he dcoun facor and of rk aeron ha wll mply he exence of no rk-harng n equlbrum and no equlbrum oher han he auarkc one. Smlarly nue reul can be dered f one conder change o he enronmen nddual face. In parcular, a decreae ncreae n he arance of nddual and aggregae ncome wll make people beer wore off n auarky and ge more cope for rk harng. I herefore no urprng ha low enough alue of he ncome arance wll yeld only he auarkc equlbrum, whle hgh enough alue of he ame parameer wll yeld full rk harng. The nuon, once agan, n wha happen o he alue of auarky when one change he arance of he ncome proce. A hgher arance mple ronger ncene o ay n a rk-harng agreemen. I alo raghforward and nue o ee wha happen when one change he perence of doyncrac hock. More peren hock are harder o nurance a an nddual wll

13 re harng her permanen luck. In oher word, wll be poble o fnd a hgh enough leel of perence ha wll caue auarky o be he only equlbrum oucome. Whle mo reul n he leraure are aed n erm of a how aryng a gen parameer can ge re o eher full rk harng or on he oher exreme o auarkc equlbra, ueful, epecally f one wan o e he emprcal releance of he model, o conder meaure of how far a gen allocaon from fr be and how h change when changng ae and echnology parameer. In ome uaon, namely n model wh wo nddual and mple error rucure, a meaure of he amoun of rk harng que naural and he reul we hae menoned o far generalze n ha he conergence of he economy from auarky o fr be when ome parameer change monoonc. 5 ee Alarez and Jermann 00. Howeer, prong hee reul n more general uaon or derng he parcular funconal form beween ome parameer and he amoun of rk harng can be hard. For h reaon, afer conderng a meaure of he amoun of rk harng n an economy, n Appendx A, we ue mulaon o characerze hee relaonhp. In eablhng a meaure of he amoun of rk harng, we wan o eablh a merc ha allow u o meaure he dance of a gen obered neremporal allocaon from he benchmark of full rk harng. In oher word, we wan o meaure how much rk harng happen n equlbrum compared o full rk harng. One pobly would be o compare equlbrum ranfer wh he ranfer ha one would obere under fr be. Whle h meaure ha an nue appeal, alo ha ome mporan drawback. Fr, gnore oher mechanm uch a ang ha nddual mgh be ung a a gen pon n me, o ge cloe o fr be allocaon. The relaonhp beween nddual ang and borrowng and prae ranfer could be que complex, epecally n model where he effec of he ock of prae ang on he alue of auarky can be mporan. Second, a we dcu aboe, een n he abence of orage, n our model, enforceable ranfer mgh hae he oppoe gn of fr be ranfer, makng he rao meaure le meanngful. Becaue of hee wo problem, we focu on a dfferen meaure. In a ymmerc fr be equlbrum full rk harng mple complee equaly acro nddual. On he conrary, auarky mple ha a large fracon of hock o endowmen 5 Wha we mean by monoonc ha an ncreae n a gen parameer eher ncreae or decreae he amoun of rk harng up o he pon where he equlbrum conrac concde wh eher fr be or auarky.

14 or ncome are refleced no conumpon. In he cae n whch nddual hae no acce o elf nurance he cro econal arance of conumpon wll acually be equal o he alue of ncome. Followng Aanao and Ro Rull 000, we can herefore ake he rao of he cro econal arance of conumpon o he cro econal arance of ncome a our meaure of he amoun of rk-harng acheed by a gen economy. Under full rkharng and wh ymmerc endowmen, uch a meaure zero. Under auarky and whou orage uch a meaure. A we menoned aboe, he fr order condon ha one ge from he cenral planner problem, are len abou he deermnaon of he Pareo wegh. A compee equlbrum no necearly ymmerc. Howeer, gen a e of Pareo wegh, whch mgh generae a non-zero cro-econal arably of conumpon n equlbrum, under fr be, he cro econal arance of conumpon wll no change. Inead, under a elf-enforceable rk harng conrac, he cro econal arance of conumpon wll ary f he parcpaon conran become bndng. Therefore, o accommodae he pobly of aymmerc compee equlbra, we ue he change n he cro econal arance of conumpon a our meaure of rk-harng. A ceran change n he cro econal arance of log conumpon mgh mean ery dfferen hng n erm of he amoun of rk harng ha acheed n a gen economy. In a gen economy llage he cro econal drbuon of conumpon mgh change ery lle een f here no rk-harng f he cro econal drbuon of ncome doe no change. On he oher hand, one could obere a llage where a ubanal proporon of doyncrac rk derfed and ye he cro econal arance of conumpon change ubanally. In oher word, o conruc a proper meaure of he degree of rk harng of a gen economy, change n he cro econal arance of conumpon hould be normalzed by omehng ha ake no accoun he need for rk harng of a gen economy. We propoe o normalze by he alue of he arance of ncome. Smple mulaon, whoe reul we repor n Appendx A, how ha uch a rao behae exremely correlaed wh he rao of arance ha we dcued n econ. 6 6 We hae alo expermened wh normalzng he change n he arance of conumpon wh he change n he arance of ncome. The reul we obaned boh n he mulaon and n he emprcal exerce were ery mlar. 3

15 We can herefore confdenly ake h rao a a meaure of he dance of a gen economy allocaon from ha mpled by perfec rk harng. To oban a relaonhp beween our meaure of rk harng and he parameer of he model, we ar from a baelne economy and hock randomly he parameer ha characerze he endowmen proce o generae many dfferen econome. We can hen ole for he equlbrum conrac and mulae each of hee econome o udy he relaonhp beween effcen allocaon and he endowmen parameer. In parcular, we focu on he meaure of rk harng we ue n he emprcal analy ee he dcuon n Secon 3.. In Appendx A, for power uly, we how he followng reul: an ncreae n he arance of ncome eher doyncrac or aggregae ncreae monooncally rk harng; an ncreae n he perence of doyncrac ncome decreae monooncally rk harng; 7 an ncreae n he mean of he proce keepng he arance conan decreae rk harng; In econ 3., we dcu how o make our meaure of rk harng operaonal and we how how o ue hee reul o propoe a e of he model wh mperfec enforceably. 3 The propoed e The man am of h paper o propoe wo dfferen e of he emprcal releance of model wh mperfec enforceably. Our e conder ome explc mplcaon of he model we keched aboe and dffer n ha he fr baed on llage leel ac, whle he econd rele on he explc emaon of he Euler equaon mpled by he model. 3. Change n he cro econal arance of he margnal uly of conumpon and llage characerc The arng pon of our fr approach a e of he perfec nurance model and fr be allocaon. A well known, f one conder he opmzaon problem of a ocal planner 7 Aanao and Ro-Rull 000b, ung mulaon, how ha an ncreae n he perence of aggregae hock ha leae he mean and he arance of he proce unaffeced, wll caue le rk harng n equlbrum. 4

16 ha maxmze expeced uly of a e of nddual wh a ge e of Pareo wegh, one can dere he followng fr order condon: 3. U c, z λ β = µ c where U c he margnal uly of conumpon for nddual, whch aumed o depend on non-durable conumpon and, pobly, on a ecor of oher arable z, ome of whch mgh be unoberable. λ he Pareo-wegh gen by he ocal planner o nddual n he maxmzaon problem. Dfferen e of wegh wll correpond o dfferen compee equlbra wh full rk harng. The heory len abou wha deermne hee wegh, excep n ayng ha hey are conan oer me. β he dcoun facor for nddual, and µ he Lagrange mulpler aocaed wh he reource conran a me. Snce he work of Townend 994, Mace 99 and Cochrane 99, equaon 4. ha been ued exenely o e he emprcal mplcaon of a model wh full rk harng. If one ake he log of equaon 3. and conder a wo dfferen me perod, one can elmnae boh he dcoun facor and he unobered Pareo wegh. In parcular, one ge: 3. log U c c, z log U c c, z = log µ log τ τ τ τ µ τ Noce ha me and τ need no be adjacen perod. 8 If hey are k perod apar, one can wre equaon 3. a: k 3.3 log U c, z = ν, c k The man mplcaon of equaon 3.3 ha change oer me n he margnal uly of dfferen nddual hould be he ame. Dfferencng ha elmnaed he Pareo wegh. The change n margnal uly hould be unaffeced by he doyncrac hock receed by nddual. Full nurance mean ha change n he amoun of reource aalable o an nddual oer and aboe he aggregae change hould no be refleced n change n margnal uly. The reource conran can be aken no accoun by conderng eher me 8 Moreoer, f one conder many par of oberaon, he dance beween he wo me perod need no be he ame. 5

17 dumme, or, a n Mace 99, aggregae conumpon. Any oher arable, uch a nddual ncome, hould herefore no ener equaon 3.3. In he abence of panel daa ha follow he ame nddual oer me, one can ll e he mplcaon of perfec rk harng by aggregang equaon 3.3 oer nddual belongng o a gen group whoe memberhp aumed o be fxed oer me. Inurance acro group mple ha he aerage margnal uly of conumpon for dfferen group hould change n he ame way and hould be unaffeced by group leel hock. Aanao and Da 996 ued ynhec panel o e equaon 3.3 by formng year of brh and educaon group and followng he aerage for hee group oer me. The ynhec panel approach ha wo bg adanage and one dadanage. The adanage are he pobly of eng equaon 3.3 een n he abence of longudnal daa and he gan n power ha mgh be obaned aeragng meauremen error n wage, ncome or whaeer meaure of nddual reource are ued oer he member of a group. The dadanage he fac ha one focue only on he nurance acro group. By akng aerage oer he member of a group, one canno ay anyhng abou he exen of rk harng whn a group. An alernae e of he perfec nurance model can be conruced by conderng, nead of he cro econal aerage of he arable n equaon 3.3, her cro econal arance. Conder once agan he log of equaon 3. aboe, and re-wre a follow: 3.4 log U c, z = log µ log λ β c Hang defned group wh fxed memberhp one can compue he cro econal arance of boh de of equaon 3.4. A he reource conran mulpler common acro nddual doe no conrbue o he cro econal arance. Under perfec rk harng, he Pareo wegh and he dcoun facor are conan oer me: herefore an mplcaon of he heory ha he cro econal arance of he margnal uly of conumpon conan oer me. If we aume ha he uly funcon oelac, equaon 3.4 can be wren a: 3.5 Var log c = d 6

18 where he ubcrp ndcae he fac ha he arance compued whn a group llage, he d are group dummy arable and reflec he arance of Pareo wegh and dcoun facor. Takng fr dfference of equaon 3.5 one ge: 3.6 Var log = 0 c Equaon 3.5 can be eed by regreng he cro econal arance of he margnal uly of conumpon on group dumme and oher arable uch a ncome hock ndcaor or he arance of ncome and e ha he coeffcen on hee oher arable are no dfferen from zero. Analogouly, equaon 3.6 could be eed regreng he change n he cro econal arance of conumpon on ncome hock ndcaor or he change n he ncome arance and e he hypohe ha hee arable hae no yemac effec. Whle h mplcaon had been noed n pang by Deaon and Paxon 994, Aanao and Szekely 00 and Aanao 00 9 propoe and mplemen explc e of uch a hypohe. In h paper, we ue h approach a a arng pon o udy he mplcaon of he model wh mperfec enforceably. Before mong on, howeer, one pon ha hould be noced here ha one can readly hnk of uaon n whch he null of full nurance olaed bu n whch he e baed on arance change fal o pck up h olaon. Anoher mplcaon of he null ha nddual manan her relae poon n he drbuon of he margnal drbuon of conumpon. I poble ha nddual change her poon herefore olang he null, whle leang he cro econal arance unaffeced. In a recen paper, Jappell and Paferr 00 e he mplcaon of he perfec nurance model of no mobly n he cro econal drbuon of conumpon by conrucng he Shorrock mobly ndex n a panel of Ialan houehold. Whle hey preen conncng edence rejecng he mplcaon of full rk harng, hey do no conder he mplcaon of alernae model. The e baed on he eoluon of he cro econal arance of conumpon complemenary o hoe baed on mean uch a Townend 994 and Aanao and Da 996. Th boh becaue focue on he nurably of hock whn a group or llage raher han acro group and becaue n ome crcumance could be more 9 See alo Aanao and Jappell 00. 7

19 powerful han he e baed on mean. An example of he laer uaon when large meauremen n ncome reduce he power of he e baed on ncome, whle, f he arance of meauremen error doe no change oer me, doe no affec he e baed on arance. The expreon on he lef-hand de of equaon 3.6 mlar o our meauremen of he degree of rk-harng we dcued n Secon.3. There we poned ou ha he farer away he neremporal allocaon of conumpon from fr be, he larger are he change n he cro econal arance of he log of he margnal uly of conumpon. The nex ep hen o ue h meaure of he dance from perfec rk harng, 0 o relae o arable ha, accordng o he model wh mperfec enforceably are mporan deermnan of he deaon from perfec rk harng. In parcular, we conder he arance of ncome, auocorrelaon and mean. A menoned n Secon, econome wh hgher alue of ncome arance hould achee greaer degree of rk harng, whle he oppoe rue for econome where he ncome procee are hghly peren. The effec of he mean, keepng conan he arance of log ncome and auocorrelaon, depend on he form of he uly funcon. Gen hee conderaon, we conder he followng relaonhp: 3.7 Var log c = α 0 + α. de.log y Var log y α ρ log y 4 + α arlog y + α µ log y 5 + α ρlog y + α µ log y where ρ and µ are he auocorrelaon and mean of log ncome repecely. A he relaonhp beween our meaure of deaon from fr be and he feaure of he llage we conder no necearly lnear, we conder a flexble funconal form and emae equaon 3.7 by LAD. The model wh mperfec enforceably mple ha he oerall effec of ncome arance negae whle ha of he auocorrelaon poe. In econ 4 and n he Appendx, we decrbe n deal how we oban emae of he arou quane n equaon Such a meaure no perfec. A we menon aboe, here mgh be uaon where he hypohe of perfec rk harng olaed and ye our meaure equal o zero. 8

20 3. A e baed on Euler equaon The econd e we conder baed on he Euler equaon ha we dered n econ. for conumer who parcpae n a rk harng agreemen ubjec o enforceably conran. We fr decrbe he e conderng he model wh wo nddual and no meauremen error. We hen relax hee wo aumpon. Aumng a CRRA uly funcon, we can re-wre equaon.7 aboe a: γ k, + c+ k+ Ω e+ A 3.8 = β E + + R ; K c K A where k + he mulpler aocaed wh he parcpaon conran of nddual and K = k τ = 0 τ. A menoned n Secon, f n perod + he enforceably conran for nddual no bndng, 3.8 reduce o a andard Euler equaon for he neremporal allocaon of conumpon of an nddual wh a orage echnology ha pay an nere rae of R, and a dcoun facor of β. If here are only wo nddual, only one of he wo enforceably conran can be bndng a any pon n me and a any ae of he world. Therefore, a each pon n me, for a lea one of he wo conumer, here wll be an Euler equaon holdng. Moreoer, f we hae acce o perfec daa on conumpon, we can ealy eablh whch he conumer who poenally facng a bndng conran. If none of he wo conran bndng, he fr be allocaon correpondng o a gen e of Pareo wegh can be mplemened and he margnal uly of he wo conumer wll grow a he ame rae. Howeer, f he conran of conumer one bndng, her margnal uly wll be growng le and, herefore, her conumpon wll be growng faer. The oppoe rue f he conran of he econd conumer bndng. If a me + he conran for he fr conumer bndng, no only her conumpon a me + wll be growng faer, bu wll be fully deermned by he upgrade n he connuaon uly deermned by he ocal planner. A noed by Kocherlakoa 996, an mplcaon of h ha for he conraned nddual, conumpon a me + wll be a funcon of her hock, her parner hock and her parner 9

21 lagged margnal uly of conumpon, bu no of her own lagged margnal uly of conumpon. Th dcuon mple ha we can wre a wchng regreon yem for he conumpon growh of he wo nddual. Each conumer n one of wo regme. If we denoe wh z + he conumpon growh of conumer mnu ha of conumer, wh ε he ncome hock of nddual, we hae: log c log c + + = con. + r + u = g ε +, ε + +,log c ; ; f f z z > 0 where g a funcon and r=log+r. The rong eable mplcaon of equaon 3.9 ha log c + doe no depend, afer conrollng for curren hock, on log c. Th a rong mplcaon of conraned effcency reed by Kocherlakoa 996 and follow from he lack of memory of he yem once a conran become bndng. One problem n eng he wchng regreon yem n 3.9 ha we do no hae, n general, a cloed form oluon for he funcon g. A pobly, herefore, o approxmae g by a flexble funcon of argumen, uch a a polynomal. To make h dea operaonal and apply o our conex, we hae o exend he e o conder many nddual and he pobly of meauremen error. We ar wh he laer. In he preence of error n he meauremen of conumpon, no poble o exacly clafy he oberaon n he wo regme n equaon 3.9. Howeer, f one wre down an explc model of meauremen error, one can eablh, for each oberaon, he probably of each of he wo regme and herefore denfy he model. In parcular, le u aume a clacal mulplcae meauremen error n he leel of conumpon, o ha meaured conumpon C = c, where a lognormal meauremen error. We furher h h h aume ha he meauremen error proce..d. acro nddual and oer me wh a andard deaon of σ. Denong wh Z + he meaured equalen of z +, we hae: 3.0 log C log C + + = con. + log + r + u = g ε +, ε +,log c, wh wh prob. Φ Z prob. Φ Z + + / 4σ / 4σ 0

22 where Φ he normal cdf. Nex, we can generalze h approach o he cae n whch we hae many conumer. In h cae, a lea one conumer wll no be conraned. More generally, we wll hae a e of unconraned conumer, whoe margnal uly of conumpon wll be growng a he ame rae and he remanng conumer, who wll hae bndng parcpaon conran and margnal uly of conumpon wll be growng a a rae lower han ha of he unconraned conumer bu no conan. Once agan, wh perfec nformaon abou conumpon wll be poble o clafy each conumer n one of wo e and o wre down equaon mlar o 3.9. Wh meauremen error, we hae o generalze equaon 3.0. We hae an addonal parameer o emae ha he unobered rae of growh of conumpon for he unconraned conumer. For a generc conumer h n llage, we hae he followng yem: 3. log C log C h, + h, + = con. + r + u = g ε h, +, ε + +,log c + + w. prob. Φlog C w. prob. Φlog C h + h + µ µ,, / σ / σ where µ, he unobered mean conumpon growh for he unconraned houehold n llage a me, and ε +, and log c are he ecor of me + hock for all he oher houehold and he ecor of me log conumpon for he unconraned houehold n llage. Once agan noce ha he eable rercon mpled by equaon 3. he fac ha + conumpon for he conraned houehold doe no depend on he lagged alue of her own conumpon. The emprcal mplemenaon of 3. mple wo addonal problem. Fr, ε +, and log c are poenally hghly dmenonal objec, a llage can hae a few hundred houehold. Second, he ecor of parameer µ, can alo be of ery hgh dmenon, a we hae eeral hundred llage and eeral me perod. To ackle he fr problem, we ummarze he drbuon of hock n he houehold, and lagged margnal ule of he currenly unconraned houehold by a few drbuonal ac. Thee ac, he mean, arance, kewne and kuro of he drbuon, can be emaed along wh he oher parameer of he yem. Moreoer, we can chooe o mplemen h exerce n each llage eparaely or poolng all he llage and make he

23 drbuonal ac funcon of oberable llage characerc n order o keep he dmenon of parameer manageable. To ole he econd problem, we noce ha, a a pon n me and for each llage, he aerage conumpon growh n he llage conue an upper bound o µ, whch he rae of conumpon growh of unconraned nddual. By how much he llage aggregae growh of conumpon exceed µ, depend on how many conumer are conraned and how bndng are he conran. Th n urn depend on he drbuon of ncome. We herefore make he followng aumpon:, 3. µ = ϕ y gc,, c where φ a funcon ha ake alue beween 0 and and gc he aggregae growh of conumpon n llage a me. y and c are he ncome hock and lagged margnal uly drbuon n llage. We approxmae he funcon φ a a logc funcon n he momen of he ncome and margnal ule drbuon. An alernae way o wre equaon 3., whch ncorporae he aumpon and approxmaon dcued o far, herefore he followng: 3.3 log C h, + con. + r + u = con. + r + u f ε h, +, ε +,log c w. prob. w. prob. p p,, where = +, m a ecor of h p, Φlog C µ, / σ, µ, = gc, + exp θm, momen d gc, he obered growh of conumpon n llage a me. Noce ha he econd lne of equaon 3.3 ha he ame lef-hand de of he fr. The f funcon mply he mulpler aocaed o he parcpaon conran and rcly poe for conraned nddual. For equaon 3.3 o reduce o equaon 3. for he conraned nddual, he f funcon ha o be of he form: 3.4 h, h, f ε +, ε+,log c = g ε+, ε+,log c con r + πlog c

24 wh he coeffcen π=. Th rercon, whch can ealy be eed, requred by he fac ha lagged conumpon margnal uly ha o cancel ou from equaon 3.3 o ge 3.. A we menoned aboe, he fac ha lagged own margnal uly doe no deermne curren conumpon for conraned nddual a ery rong rercon ha a he hear of he lmed enforceably model, a reflec he conraned effcency of he elfenforcng mechanm. An addonal rercon ha can be eed ha curren ncome doe no ener he equaon for he unconraned nddual. Th rercon analogou o he exce eny e n he Conumpon Euler equaon leraure. 4 Daa The e propoed n he preou econ are carred ou n h paper ung a unque daa e: he ealuaon ample of a large welfare program n rural Mexco, called PROGRESA. A we dcu below, PROGRESA a program amed a foerng he accumulaon of human capal by ncreang chool enrolmen, and mprong nuron and healh pracce. In order o ealuae h program, he admnraon colleced nformaon on all he houehold lng n 506 locale abou 5,000 houehold were ureyed before and afer he mplemenaon of he program. Furhermore, n a ube of 86 randomly choen locale he program mplemenaon wa delayed for almo wo year. Th randomzaon cheme wa explcly mplemened wh he purpoe of faclang he ealuaon of he effecene of he program. 4. The PROGRESA program and he ealuaon daa e In 997 he Mexcan goernmen decded o ar he Educaon, Healh and Nuron Program, called PROGRESA for Spanh acronym. PROGRESA a new and large welfare program argeed o rural commune. I am o rae he lng andard of ery poor famle by mean of hoe hree cloely lnked componen healh, food and educaon; he underlyng dea ha ner-acon among hem enhance he effecene of an negraed program. 3

25 The healh componen con of a number of nae orened o prode beer nformaon abou accnaon, nuron, conracepon and hygene and of a program of for chldren and women o healh cenre. Parcpaon no he healh componen a pre-condon for parcpang no he nuron componen; h ge, n addon o a bac moneary ubdy receed by all benefcary houehold, ome n knd ranfer nuron upplemen o houehold wh ery young nfan and pregnan women. The large componen of he program he educaon one. Benefcary houehold wh chool age chldren recee gran condonal on chool aendance. The ze of he gran ncreae wh he grade and, for econdary educaon, lghly hgher for grl han for boy. In addon o he paymen, benefcare wh chldren n chool age recee a mall annual gran for chool upple. Fnally, he moher n he houehold recee eery wo monh all he ranfer. The benef of he program conue a ubanal help: oer 0% of he benefcare aerage ncome. The elecon of elgble houehold a mul-age proce. The Program fr argeed he poore commune n rural Mexco. Roughly peakng, he wo crera commune had o afy o qualfy for he program were a ceran degree of poery a meaured by wha called an ndex of margnalzaon and acce o ceran bac rucure chool and healh cener. Once a localy qualfe, followng a cenu n he relean locale n 997, nddual houehold could qualfy or no for he program, dependng on a ngle ndcaor ha affeced by a number of poery arable ncome, houe ype and o on. The program wa phaed n lowly and currenly ery large: a he end of 999 budge wa US$777m and wa mplemened n more han 50,000 locale. A ha me, abou.6 mllon houehold, or 40% of all rural famle and one nnh of all houehold n Mexco, were ncluded n he program. The co of he program abou 0.% of Mexcan GDP. The program ha receed a conderable amoun of aenon and publcy and mlar program are currenly beng mplemened n Hondura, Ncaragua and Argenna. See IFPRI 000 for addonal deal on he program and ealuaon. The agency runnng he program ued he fac ha, for logc reaon, he program could no be ared eerywhere mulaneouly, o ar an ealuaon ample. Among he benefcare locale, 506 where choen randomly and ncluded n he ealuaon ample. Among hee, 30 randomly choen were agned o he commune where he program 4

26 ared early, whle 86 were agned o he commune where he program ared almo n December 999 raher han May 998. The randomzaon eem well execued: Behrman and Todd 999 preen edence n h repec. In parcular, mo arable eem o be no acally dfferen beween he reamen and conrol llage. 4. The daa. The daa we ue come from he Surey of Houehold Soco-Economc Characerc ENCASEH and from he Ealuaon Surey ENCEL of PROGRESA. The Surey of Houehold Soco-Economc Characerc wa carred ou before he program began a he end of 997. In fac, h cenu wa ued o elec whch houehold n he elgble commune would parcpae n PROGRESA. On he oher hand, he Ealuaon Surey wa pecfcally degned for he ealuaon. Fr, he abou 5,000 houehold of he ealuaon ample from 506 llage were nerewed n May 998 for a baelne urey whch complemened he ENCASEH cenu. Then, follow-up urey were carred ou eery x monh: Noember 998, March 999, Noember 999 and Aprl 000. Whn each llage n he ealuaon ample, he urey coer all he houehold and collec exene nformaon on conumpon, ncome, labour upply, chool enrolmen, ranfer, and a arey of oher ue. Whle each nrumen conaned a core queonnare, ome of hem alo conaned ome addonal module. For nance, he queon on nerhouehold ranfer were only aked n he Noember urey. The houehold urey are upplemened by a localy queonnare ha prode nformaon on eeral llage pecfc arable, ncludng, among oher hng, nformaon on prce of arou commode and llage leel hock. We ue daa on houehold conumpon from he wae of Noember 998, March 999 and Noember 999. We hae added up acro he dfferen group of commode on whch he urey prode nformaon: food, ranporaon, hygene and healh, educaon, clohng and durable good. A, dependng on he caegory, houehold are aked abou expendure made durng he preou week, monh, emeer or year, we fr conered all Of coure one would expec a 5% of rejecon. I a b worryng ha one of he few arable for whch a rejecon wa obaned wa pre-program chool enrollmen. 5

27 no weekly ba. For food, we hae ncluded n our meaure an emae of he conumpon of good produced by he houehold alued a marke prce. In he urey, each houehold member aked o repor eparaely her/h ncome from eeral ource. We buld our meaure of houehold ncome by addng-up he followng ource for each houehold member: wage boh from a prmary and from a econdary occupaon, ne prof reenue mnu expendure from bunee or ncome generang ace, communary earnng, and ncome from penon, from nere, and from renal of land, anmal or machnery. 3 A he reponden can repor daly, weekly, bweekly, monhly or annual ncome, all repored ncome fr conered no weekly ncome; we ued nformaon on he number of day worked n he preou week o compue h weekly ncome when daly earnng were repored. For reaon ha wll come clear laer, we focu on houehold prodng nformaon on ncome and conumpon for all he round aalable; h leae u wh a b more han en houand houehold. In Table, we preen ome ummary ac relang ncome and conumpon: Table. Summary Sac for Weekly Houehold Income and Conumpon Sandard Wae Mean Medan Deaon Noember 998 Conumpon Income Conumpon March 999 Income Number of Oberaon 038 To mplemen ome of he econd of he e we dcued n he preou econ, we need o compue eeral momen of he drbuon of conumpon and ncome n each of he llage. Fr, we need o compue he cro econal arance of log conumpon and ncome, a hey wll form our meaure of he amoun of rk harng. Second, we need o compue he me ere arance and auocorrelaon of doyncrac ncome n each llage n addon o mean, a hey wll conue he explanaory arable of our regreon. Neerhele, alo rue ha ome queon are lghly dfferen from one wae o anoher. 6

28 In wha follow we ue he wo cenral wae o compue he change n he arance of conumpon and ncome, whle we ue all four wae o compue he arance, mean and auocorrelaon of ncome. The cro econal arance of ncome nformae abou he rk faced by an nddual ncome only under pecal crcumance. Fr, heerogeney acro nddual no necearly an ndcaon of rk. Second, moemen oer me mgh be predcable and herefore could no be defned a rk. In an aemp o keep hee ue no accoun, we fr regre ncome on a e of oberable and predcable facor. We hen compue he me ere arance and auocorrelaon of he emaed redual for each nddual. Fnally we compue he aerage of hee nddual arance and auocorrelaon n each llage. Thee are he meaure we ue n our regreon. Appendx B prode addonal deal on h procedure. 5 Reul 5. Change n he cro econal drbuon of margnal ule and mperfec enforceably In h ubecon, we mplemen he fr e decrbed n Secon 3. In parcular, we emae a flexble relaonhp beween a meaure of he amoun of rk harng obered n a llage compared o he amoun of rk harng ha would occur under perfec rk harng and ome feaure of he llage ha can explan h deparure under mperfec enforceably: he arance of ncome, perence and mean ee equaon 3.7. Followng he dcuon n ubecon 3., our meaure of how far a llage from perfec rk harng gen by he abolue alue n he change of he cro econal arance of conumpon normaled by he cro econal arance of ncome. We call h rao r. We hae calculaed he rao r ung hree dfferen defnon of conumpon: a food, b food and erce hygene, healh, educaon, and energy, and c food, erce, ranporaon, and durable good ncludng clohng. A commened n econ 4., we ued daa from Noember 998, March 999 and Noember 999. A we do no hae a cloed form oluon for he relaonhp beween our meaure of rk harng and he llage leel momen of doyncrac ncome, we regre he rao r on a 3 Our meaure of houehold ncome doe no nclude ranfer neher he prae ranfer nor he 7