Family structure and long-run equilibrium distribution of wealth. Yue Xin 2018/03/12

Transcription

1 mily srucure nd long-run dribuion welh Yue 08/03/

2 Conen Review bsic model Assumpions Inrgenerion uiliy mximizion Inergenerion welh rnsformion elh divion welh union x Quesions: Suppose re endency o form mrrige prnerships wih persons ouside one s own welh clss. Assess liely implicions longrun impc herce x on dribuion welh. Consider policy encourgg n crese birh re. h would you expec o be effec on welh equliy long run?

3 Bsic model: Assumpions Assumpions: Number children (): Posiive eger xogenously deermed refore dependen welh Proporion fmily wih children p " sfyg: (Populion sionriy) Only generions re live ny given po ime.

4 Bsic model: Inrgenerionl uiliy mximizion mx s.. > > g: Per period growh ol welh γ: Tse for beques υ: Tse for leure when () posiive

5 : p b : [ + ] c : bove rgumen implies h h b populion : [ + ] c hs : welh rgumen proporion : p or given vlue h bove or mpions bou fmily: (i) h re sric ssorive mg rens follow policy equl ssumpions divion bequess mongs ir children. In dribuion mus sfy dribuion mus sfy bou fmily: (i) h re sric ssorive mg less period ( ) found s weighed sum vlues ( ) fmily ech prens heried welh perfecly correled nd (ii) h ll h model fmily chrcerized by jo welh level mrrige ech fmily prens heried welh pperfecly correled (ii)[h llfor wihgenerion. + ] n Recllg h welh proporion fmilies nd children n re dc levels previous quilibrium h requires ns follow policy equl divion bequess mongs ir children. In ( ) p. (6 rners nd by number children. children. prensrbirry follow policy equldivion bequess ir In dribuion subse :funcion rems mongs unchnged hrough ) so ( (bh + c ) generions model ech fmily chrcerized by jo welh level mrrige h model ech fmily by welh level mrrige chrcerized for ll. Cll hjo dribuion ( ). (bdefg c (7 + ) consns ners nd by number children. [ + ] prners nd by number children. : ( p) b : p[+ ] c : bove rgumen implies h. elh dynmics. (6) for ny vlue such h < h nd hs b welh + c < fo or given vlue proporion populion o dribuion mus sfy elh dynmics ih wo prens equl welh nd childrenless shrg herce period ( ) found s weighed sum vlues ( ) 3. elh dynmics cse for proporion ny vlue such h < 0 b +orc < for ll. The h lionship beween herce nd beques or given vlue populion hsnd welh dc welh levels previous generion. quilibrium requires h h wo prens equl welh nd children shrg herce ( )shrg sum (bherce + c) (7) ih wo equl welh nd schildren vlues less prens period ( ) found weighed ( ) In h cse ; hrough we re focusg on pr ionship beween herce ndrelionship beques beween herce dribuion funcion rems unchnged generions so popu h dc nd beques generion. welh levels previous quilibrium requires h I B. () solely idle rich or reniers. I priculr eres cse h 0 hrough Assumpions: funcion The for ll.unchnged Cll dribuion. Defg consn dribuion rems or generions so models h Recllg h re propor i close o herce welh-dribuion for ny vlue such h < nd b + c < for ll. I B. () welh for. Cll h dribuion consns. Defg (0) children shre eqully: ll : di erence bih c; we :re zero-erngs bove rgumen Bequion () : [.+ ]lso rbirry populion implies enbles subseh o :h ombg equions () nd() we fd followg becuse simplificion us es pin cse focusg on pr cons : p b : [ + ] c : resul. bove rgumen implies h r fmily wih () children: dribuion mus sfy bg equions (0) nd () we fd followg di erence The 0equion solely we fd idle reniers. I rich h priculr eres nowor jusn becu dribuion mus sfy Combg equions ()cse (0)nd rich followg di erence equion () Ifor people re so y choose no o fmily wih children: 8 for fmily wih children: i close o or herce bu ; we re focusg on models pr welh-dribuion populion consg Implicion: lierure In h cse (3) + ( ) esblh solely rich or reniers. I c ) priculr eres jus simplificion us obecuse sri lso ( )becuse zero-erngs (b + idle ( ) +enbles c(7) (7 (b )no + (3) 8 (3) + i close o or herce models welh-dribuion lierure bu here depends on (see equion ) nd n exogenous one-generion resul. The chrcerizion o condiion follows lso becuse zero-erngs simplificion enbles us esblh srig owh fcor given by e depends on (see equion ) for nd n exogenous one-generion given now where depends on (see ) n exogenous people re rich h choose o wor n cse. given 0 we hvepro b nyvlue suchequion hif < nd ndso b + cprevious < y one-generion for ll However.noor vlue resul. : [ + g]. (4) here h fcor given by growh fcor given forbyny vlue such h choose < nd b +less c for.becomes: equion (7) son condiion period ll ( ) foun < If people re so rich h y no o nd wor : [ + g]. (4) : comes [ + g].from (4) cse hs 0 summry if person curren The generion welh nd welh levels dc (8 fmily where rewere children prens mus ech hve hd welh mmry if person curren generion hs welh nd comes from In summry if person curren generion hs welh nd comes from ( ) (b ) > re focusg 0 >on pr dribuion funcion remi In h csethe cse ; we consg The idle rich: hen (8) populion mily children solely prens mus ech hve hd welh fmily where re were children prens mus ech hve hd welh [where ].re Thwere simple fc enbles us oderive evoluion welh idle rich or reniers. I priculr eres no jus becuse 8 for ll. Cll h equ The chrcerizion condiion sme les follows The chrcerizion condiion follows sme les s [ ]. Th simple fc enbles us o derive evoluion welh sribuion ]. Th simple fc enbles us o derive evoluion welh In h cse ; we re focusg on pr populion cons close o or herce welh-dribuion lierure bu from generion oi. e use equion (3) long modelswih given h where nd b for: ll. Observe h p b : [nd + ] P previous cse. However given 0 we now hve b : c c previous However given 0olong we b : nd cno 0jus hve so lsouse becuse zero-erngs enbles usi esblh srig dribuion generion idle cse. o e use equion (3) wih ( ) ibuion fromgenerion o e equion (3) long wih sumpions bsic model o.derive from s solely funcion.where rich or reniers. now priculr eres becus bsic model ; wih cresg simplificion dribuion mus resul. ssumpions bsic model o derive s funcion where equion (7) nd so condiion becomes: 8 s mpions bsic model o derive s funcion where equion condiion or (7) nd so < < which implies becomes: < <. lierure he dribuion funcion dividul welh iime. o close herce models welh-dribuion bu If people re so rich h y choose no o wor n dribuion funcion dividul welh ime. dribuion funcion dividul welh ime. Theorem ( Appendix) we fd h if (9) e ssume h suppor n ervl. There re zero-erngs wo impor- rom lso becuse simplificion enbles us o esblh sri e ssume h suppor n ervl. There re wo impore ssume h suppor n ervl. There re wo impor h implies h welh proprie dom Bsic model: Inergenerionl welh rnsformion elh divion

6 Bsic model: Inergenerionl welh rnsformion () mus belong o exended ype I Preo dribuion fmily: pple ( ) A where p. quilibrium welh equliy (α) depends on growh fcor (β) nd dribuion fmily size (p). Tryg ou differen fmily size dribuion (p): p p p 3 p 4 p 5 p 6 cse cse cse d h proporion sgle child fmilies domes mgniude welh equliy.

7 Quesion Consider policy encourgg n crese birh re. h would you expec o be effec on welh equliy long run? Due o domn effec proporion fmily wih sgle child i policy significnly decrese proporion sgle child fmily n re will be significn effec on welh equliy long run. If proporion sgle child fmily didn chnge much from policy n re will be mild effecs on welh equliy long run.

8 Bsic model: Inergenerionl welh rnsformion elh union: ocus only on upper il where () 0 i.e. ll welh heried. ch fmily chrcerized by ( δ ) where welh poorer couple (+δ) welh richer couple number children exogenously deermed elh received by ech child such fmily If Φ(δ) dribuion funcion δ n [+ ] s n ss ( ) Z /p [ + ] d ( ).

9 Bsic model: Inergenerionl welh rnsformion Assumg δ fixed number Lower correlion beween prner s welh (δ) leds o higher equliy (α) ech growh fcor (β). Omig quie few seps o ob: igure 6: The ( )-relionship nd mrrige p +[+ ] [ + ].

10 Quesion Suppose re endency o form mrrige prnerships wih persons ouside one s own welh clss. Assess liely implicions long-run impc herce x on dribuion welh. p +[+ ] [ + ]. > p +[+ ] [ + ]. ~ here β β τ < β Holdg ll else consn α negively correled wih β so herce x (lower β) will be liely o promoe welh equliy. Also seen from grph previous slide lrger mgniude when δ lrger. (* Therefore wih higher endency o mrry ouside one s clss will imply lrger pril effec herce x on promog welh equliy. ()

11 Reference rn Cowell & Dir Vn de ger 07. "Condorce ws rong Preo ws Righ: milies Inherce nd Inequliy" STICRD - Public conomics Progrmme Dcussion Ppers 34 Sunory nd Toyo Inernionl Cenres for conomics nd Reled Dciples LS. <hps://ides.repec.org/p/cep/sippp/34.hml>

12 Thn you.