Present and Future Value

Transcription

1 Pesen and Fuue Value Fuue value measues he value of a speifi sum o seies of sums on a given dae in he fuue. If an individual boows 00 unis of good oda wih he inees ae () =., she has o epa (.)(00) = 0 in he following peiod. If he individual boows fo hee peiods, she has o epa (.)(0) = (.)(.)(00) = in he hid peiod. This amoun is ()()(00) = ( )(00). If she boows fo T peiods (duaion of he loan), he amoun of epamen will be ( T )(00). The fuue value is disouned b he disoun fao T. We all 00 a pesen value of fuue pamen o seies of fuue pamens, 00 = (0)/(.) = (0)/. Likewise, 00 = ()/[(.)(.)] = ()/( ). We ould wie he elaionship as fuue value = T (pesen value), fuue value pesen value =. T

2 The Savings Deision We ae now poviding ou model a deailed look a individual deisions abou saving and onsumpion. Wih he wo-peiod-lived ovelapping geneaions models, assume ha people have endowmen goods when oung and goods when old. Young people fae a goss eal inees ae,, and hoose he numbe of goods he wish o save, s. Budge onsain in he fis peiod: + s., Budge onsain in he seond peiod: + s., +, + Lifeime budge onsain: + +., / and / ae pesen values of he old-age onsumpion and, + endowmen.

3 Wealh and Consumpion Inome epesens he goods an individual podues o eeives in a single peiod, while wealh is he pesen value of an individual s lifeime inome seam. A ombinaion of onsumpion is onsained b individual s wealh. Wealh is a measue of individual inome ove enie lifeime. w = +. An individual s onsumpion bundle will be eniel deemined b wealh w and he inees ae. When wealh ineases, he individual s lifeime onsumpion will also inease. Howeve, hange in eah peiod s onsumpion depends on eah peson s pefeenes fo onsumpion when oung and old.

4 If he onsumpion of a eain good deeases when wealh ineases, ha good is alled an infeio good. If he onsumpion of a eain good ineases when wealh ineases, ha good is alled a nomal good. In ou model, we assume ha and ae boh nomal goods. I is wealh, no inome, whih deemines onsumpion. People do no ae when o how muh inome is eeived. The will hoose he same onsumpion bundle as long as hei ombinaion of inome in eah peiod gives hem he same level of wealh.

5 Inome and Saving How an he individual aange he same onsumpion bundle when endowmen paens diffe? If a peson s inome is no enough o suppo he desied onsumpion in some peiod, she an adjus he saving so ha onsumpion will exeed he inome. Suppose =., wealh = 00, = 50, and = 55. The level of saving adjuss o he level of inome when oung in ode o ahieve he desied onsumpion bundle. s =. Negaive saving (dissaving) suggess ha individual an boow agains fuue inomes. In sum, onsumpion hanges aoding o wealh, while saving adjuss o inome.

6 The Effe of Taxes on Consumpion and Savings We will invesigae he effes of ax poliies on onsumpion and savings. Conside wo lump-sum axes; a ax of τ goods on eah oung peson and a ax of τ goods on eah old peson. The budge onsains beome:, + s,, + + s,, +, + + w. The igh-hand side of equaion is he pesen value of a axpae s afe-ax endowmens. I is he limi on he pesen value of he axpae s lifeime onsumpion.

7 Wealh Effes fom Tax Changes Conside hanges in axes ha do no ale an individual s wealh. Le τ * and τ * be he iniial axes on he oung and old. Suppose he axes of he oung is ineased b 0 ( τ * + 0) and he axes of he old is deeased b 0 ( τ * 0 ). The deease in axes paid when old exal offses he inease in axes paid when oung. Theefoe, he axpae s wealh has no hanged. The desied onsumpion ombinaion is also unaffeed b he hange beause wealh is no affeed and individual an adjus savings o mainain he onsumpion bundle. Consumpion is deemined b he pesen value of lifeime axes, no b he iming of hose axes. Suppose now axes ae ineased when oung wihou an offseing ax deease when old. Wha will happen?