INFINIE-HORIZON CONSUMPION-SAVINGS MODEL SEPEMBER, Inroducion BASICS Quaniaive macro models feaure an infinie number of periods A more realisic (?) view of ime Infinie number of periods A meaphor for many periods A meaphor for alruisic linkages beween generaions Mahemaically convenien because recursive represenaion Limi of a -period economy as One-period subjecive discoun facor β (a scalar in [,]) Wih infiniy of periods, can inerpre as simple way of modeling finie planning horizon Impaience aggregaes geomerically over ime Subjecive discoun for S periods ahead is β S ypically begin analysis from beginning of period zero Index ime wih arbirary indexes, +, +, Sepember,
Inroducion BASICS Recas/resae (some) consumpion heory resuls Recas/resae (some) financial heory resuls New echnical issues Sae-coningen decisions (sill) he key period- consumpion c = H c probabiliy q M c probabiliy p L c probabiliy -p-q corresponding o y = H y probabiliy q y probabiliy p L y probabiliy -p-q Describe as an opimal decision rule c (y ) A mapping from realized income sae o opimal choice in ha sae Infinie-horizon analog c (y ) Bu a more complicaed objec because branching of even ree Sepember, 3 Model Srucure UILIY Preferences v(c, c, c, ) wih all he usual properies Separable across ime Sricly increasing in each c individually Diminishing marginal uiliy in each of c individually Expeced lifeime discouned uiliy E vc (, c, c, c,...) = uc ( ) + βeuc ( ) + β Eu( c ) + β E u( c ) +... 3 3 3 = E β u( c ) = Planning ime aligned wih calendar ime by saring a period zero Condiional on informaion se of period zero Laer consider alernaive formulaions (wihin he class of expeced uiliy and raional expecaions) ime-non-separable uiliy Non-geomeric subjecive discouning Sepember, 4
Model Srucure UILIY Preferences v(c, c, c, ) wih all he usual properies Separable across ime Sricly increasing in each c individually Diminishing marginal uiliy in each of c individually Expeced lifeime discouned uiliy E vc (, c, c, c,...) = uc ( ) + βeuc ( ) + β Eu( c ) + β E u( c ) +... 3 3 3 = E β u( c ) = Planning ime aligned wih calendar ime by saring a period zero Condiional on informaion se of period zero Firs consider deerminisic β uc ( ) and hen consider limi = lim β uc ( ) before urning o sochasic = Sepember, 5 Model Analysis DEERMINISIC SEQUENIAL ANALYSIS Planning horizon max β uc ( ) subjec o { c, a} = = FOCs c a c a c,,, + a = y + ( + r ) a =,... Euler equaions u'( c ) = β u ( c )( + r ) u'( c ) = β u ( c )( + r) c. c a Do we need any condiions a all abou a? u'( c ) = βu ( c )( + r ) Sepember, 6 3
Model Soluion DEERMINISIC SEQUENIAL ANALYSIS Planning horizon max β uc ( ) subjec o { c, a} = = c,,, + a = y + ( + r ) a =,... Need o solve for (+) unknowns c, c, c,, c + unknowns a, a, a,, a + unknowns Need (+) condiions Flow budge consrains for periods,,,, + condiions Euler equaions for periods,,,, - condiions Require one more condiion a (second) boundary condiion Sepember, 7 Macro Fundamenals BOUNDARY CONDIIONS a - provides iniial condiion on problem Requiring second boundary condiion suggess problem is a (sequence of) second-order difference (differenial) equaions Consider Euler equaion for period u'( c) = β u'( c+ )(+ r) Second-order difference equaion in a Need wo boundary condiions o solve Subsiue c and c + using and + budge consrains ( + ( + ) a a ) β ( + ( + ) a a ) u' y r = u y r ( + r) + + Second boundary condiion o require? Candidae a s = consan for s Arbirary Does one arise naurally from he economics of he problem? Sepember, 8 4
Macro Fundamenals BOUNDARY CONDIIONS No-Ponzi game condiion Would be opimal o end wih (infinie) deb Ruling ou sricly negaive asses a end of planning horizon seems naural So le s require a No-Ponzi a no-bankrupcies resricion Presen in mos models (wheher aware of i or no ) (NB Models of defaul do exis bu ofen are opimal defauls ) NO quie he complee second boundary condiion we need Sepember, 9 Macro Fundamenals BOUNDARY CONDIIONS β s λ s measures (presen-discouned) marginal value of resources in period s Saring from allocaion ha solves problem Suppose (c *, c *, c *,, c *; a *, a *, a *, a *) solves problem An inerior allocaion Le V(.) be maximized value (uiliy) * * * * V( a ;.) β u( c ) + λ y + ( + r ) a c a = { } Marginal injecion of resources in period s increases value of objecive funcion by V( a ;.) s = β λs ys Envelope heorem do no need o consider effec of marginal change in y s on opimal choices Suggess a consrucive proof of a second boundary condiion Sepember, 5
Macro Fundamenals CONSRUCING A BOUNDARY CONDIION Define period- objecive funcion ( a ) u ( a, a ) u y+ ( + r ) a Dynamic opimizaion problem β ua { a } = max (, a) = Firs-order necessary condiions u ( a a ), >, u ( a, a ) < u ( a, a ) + βu ( a, a ) =, =,,,..., * * * * + Define maximized value saring from dae ( s s ) * s * * ;.) β u s= V ( a a, a Sepember, Macro Fundamenals CONSRUCING A BOUNDARY CONDIION Assumpions Opimal soluion is inerior (i.e., c > ) (weak assumpion) u (,) (weak assumpion) Implies V () Lifeime uiliy is finie for all feasible pahs Implies * lim β V( a ;.) = Also noe V (.) Is increasing Is concave because u(.) is concave Sepember, 6
Macro Fundamenals CONSRUCING A BOUNDARY CONDIION Envelope condiion * V ( a ;.) * * u * a, a a Firs-order necessary condiions u ( a, a ) + βu ( a, a ) =, =,,,..., * * * * + ( ) V ( a ;.) u ( a ) u a a a a V a ;.) * * * * * * * * * a, a = β (, ) = β ( * β a Recall < By envelope condiion Because V() and V(.) concave Implies > By No-Ponzi condiion, a a end of planning horizon β V ( a ;.) β u ( a, a ) a + * * * * Sepember, 3 Macro Fundamenals CONSRUCING A BOUNDARY CONDIION β V ( a ;.) β u ( a, a ) a + * * * * Consider By No-Ponzi condiion, a a end of planning horizon lim β V ( a ;.) li m β u ( a, a ) a + * * * * = β u (c *)a * + * * * lim β V( a;.) lim β u'( c) a = because lifeime uiliy assumed finie for any feasible pah Sepember, 4 7
Macro Fundamenals BOUNDARY CONDIIONS No-Ponzi game condiion Would be opimal o end wih (infinie) deb Ruling ou sricly negaive asses a end of planning horizon seems naural So le s require a ransversaliy condiion * * lim β u'( c) a = VC he second boundary condiion (along wih iniial condiion) Subopimal o end planning horizon wih posiive asses ha would have sricly raised lifeime discouned uiliy Sufficiency sraighforward o esablish Necessiy much more difficul (Kamihigashi, Economerica) VC is sronger saemen han No-Ponzi condiion Sepember, 5 Model Soluion DEERMINISIC SEQUENIAL ANALYSIS Complee soluion of -period planning horizon? Soluion o consumer problem is a consumpion and asse sequence * * c, a ha saisfies { } = Sequence of flow budge consrains * * * + a + + r a c = y ( ), =,,,... Sequence of Euler equaions u' ( c ) = ( )( ), =,,,... * * β u c+ + r + condiions condiions VC for finie horizon aking as given { } * * * β u'( c) a = a = ( r,,, y a r = ) Exacly as in woperiod model! Sepember, 6 8
Model Soluion DEERMINISIC SEQUENIAL ANALYSIS Complee soluion of infinie-period horizon? Soluion o consumer problem is a consumpion and asse sequence { c * *, a } ha saisfies = Sequence of flow budge consrains * * * + a + + r a c = y (,,, ) =,... Sequence of Euler equaions u'( c ) = ( )( + ), =,,,... * * β u c+ r VC aking as given { } * * lim β u'( c ) a = ( r,,, y a r = ) Sepember, 7 9