The Blanchard (1985) Model of Perpetual Youth

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1 c November 20, 2017, Chrisopher D. Carroll BlanchardFinieHorizon The Blanchard (1985) Model of Perpeual Youh This handou analyzes a way o relax he sandard assumpion of infinie lifeimes in he Ramsey/Cass-Koopmans growh model. The rick, inroduced by Blanchard (1985), is o assume ha he economy is populaed by agens who face a consan probabiliy of deah. Thus, an agen who has lived a housand years is no more likely o die in he nex year han an agen who was born yeserday. Time is measured coninuously. If here is an insananeous probabiliy d of dying, he probabiliy (as viewed by a person alive in period ) of sill being alive (no dead) in period τ is D τ = e d(τ ). (1) 1. Assuming he insananeous uiliy funcion is u(c) = log c, and ha he pure rae of ime preference is ϑ, explain why he objecive funcion a ime of an individual household in his model will be o maximize (log c τ )e (ϑ+d)(τ ) dτ. where c τ is he consumer s consumpion a ime τ. For convenience, you may wish o define ˆϑ = ϑ + d. (2) A consumer wih a pure ime preference rae of zero will downweigh he uiliy he receives condiional on being alive by he probabiliy ha he is sill alive, yielding a discouned uiliy of where ˆϑ = ϑ + d. (log c τ ) D τ e ϑ(τ ) dτ = = = (log c τ )e ϑ(τ ) e d(τ ) dτ (log c τ )e (ϑ+d)(τ ) dτ (3) (log c τ )e ˆϑ(τ ) dτ (4) A similar poin holds in he discree ime model. A sensible hing o assume is ha if you have died before + 1, you ge zero uiliy in + 1 and afer. Thus, in he wo-period conex, if he probabiliy of

2 deah beween T 1 and T was zero we would have ( ) 1 V T 1 = max u(c T 1 ) + u(c T ) 1 + ϑ while if here is a probabiliy d of dying beween T and T + 1 value would be: ( ) 1 V T 1 = max u(c T 1 ) + (1 d) u(c T ) + d β ϑ ( ) 1 d = max u(c T 1 ) + u(c T ) 1 + ϑ ( ) 1 max u(c T 1 ) + u(c T ) (5) 1 + ϑ + d Bu behavior in his case is virually indisinguishable from he behavior ha would be induced if he consumer had a ime preference rae of ϑ + d. In coninuous ime, he approximaion in (5) becomes exac. If he probabiliy of deah is consan, he expeced remaining life for an agen of any age is given by 0 dτe dτ dτ = d 1. which we will call he agen s horizon. For example, if he chances of dying per year are d = 1/50, hen he agen s horizon is 50 years. We will assume ha a every insan of ime, a large cohor, whose size is normalized o be d, is born. 2. For an economy ha has exised forever, explain why he formula for he aggregae populaion a ime, P, will be P = d D sds (6) and show ha his formula implies ha he populaion is P = 1. The aggregae populaion will be he sum of he sill-alive persons from all pas generaions. The proporion of a populaion born a ime s ha is sill living a ime τ is D τ s by equaion (1). Thus, he absolue size a ime τ of a cohor of size d born a ime s is d D τ s = de d(τ s). The economy s oal populaion will be he sum of he populaions of all he cohors 2

3 ha are currenly living. Since he economy has exised forever, here will be remaining members of every cohor back o s =. Thus, indexing each cohor by a ime index s, (6) is simply he sum of he populaions of all currenly living members of every generaion. Subsiuing he formula for D, he inegral becomes P = d = d = d 0 e d( s) ds (7) e d(s ) ds (8) e dτ dτ (9) = dd 1 (10) = 1 (11) where (9) comes from a change of variables τ = s and (10) follows from he hin. Now some noaion. We will define variable x(s, ) as he value a dae of he variable x for a consumer who was born a dae s. Thus, c(s, ) is consumpion a of a consumer born a s. Suppose ha he consumers in his economy do no have a beques moive. If hey have posiive asses a he insan when hey die, hey are no happier han if hey had zero asses. This means ha if someone were willing o pay hem somehing while hey are sill alive for he righ o inheri heir asses whenever hey die, hese consumers would happily ake ha deal. For a consumer wih wealh w(s, ) who has probabiliy of dying d, he flow value of he righ o inheri ha wealh is dw(s, ). We will herefore assume ha insurance companies exis ha pay a consumer wih wealh w(s, ) an amoun dw(s, ) in exchange for he righ o receive ha consumer s wealh when he dies. (The insurance company will make zero profis). Bu noice ha from he sandpoin of he consumer, his is equivalen o saying ha he ineres rae received on wealh is higher by amoun d. Now suppose he marginal produc of capial in his perfecly-compeiive economy is consan a r and suppose an agen born in s receives exogenous labor income in period of y(s, ). This plus he insurance scheme implies ha he agen s dynamic budge consrain is given by ẇ(s, ) = (r + d)w(s, ) + y(s, ) c(s, ). (12) Define he effecive ineres rae as viewed by a consumer as ˆr = r + d. 3

4 3. Wrie he curren-value Hamilonian for he individual s maximizaion problem and use i o show ha he growh rae of consumpion in period for a consumer born a ime s is given by (ċ(s, ) ) = ˆr c(s, ) ˆϑ (13) = r ϑ. (14) The curren-value Hamilonian is wrien H(c, w, λ) = log c(s, ) + λ(ˆrw(s, ) + y(s, ) c(s, )) so he firs opimaliy condiion H/ c(s, ) = 0 implies and he second opimizaion condion implies ( ) λ λ ( ) ċ(s, )c(s, ) 1/c(s, ) = λ (15) ċ(s, )/c(s, ) 2 = λ (16) c(s, ) (ċ(s, 2 ) ) c(s, ) λ = ˆϑλ H/ w(s, ) (17) = ˆϑλ ˆrλ (18) = ˆϑ ˆr (19) = ˆϑ ˆr (20) = ˆr ˆϑ (21) = r ϑ. (22) 4. Use he firs order condiion for consumpion and he ineremporal budge consrain implied by (12) o show ha he level of consumpion in ime for an individual born a ime s is c(s, ) = ˆϑ(w(s, ) + h(s, )) where recall ha ˆϑ = ϑ + d and human wealh is ( h(s, ) = y(s, τ)/ ˆR ) τ dτ where ˆR τ = e τ (rµ+d)dµ (23) (his is simply he compound discoun facor necessary o ake accoun of ime- 4

5 varying ineres raes; if ineres raes are consan a r i reduces o he usual erm e r(τ ) ). The IBC says ha he PDV of consumpion mus equal wealh plus he PDV of fuure labor income: (c(s, τ)/ ˆR τ )dτ = w(s, ) + y(τ, )/ ˆR τ dτ (24) (c(s, τ)/ ˆR τ )dτ = w(s, ) + h(s, ). (25) Bu if ċ(s, )/c(s, ) = ˆr ˆϑ hen implying (c(s, τ)/ ˆR τ )dτ = so ha he IBC becomes: c(s, τ) = c(s, )e τ s ˆrµdµ e ˆϑ(τ ) = c(s, ) c(s, )e τ s ˆrµdµ e ˆϑ(τ ) e τ s ˆrµdµ dτ (26) e ˆϑ(τ ) dτ (27) = c(s, )/ˆϑ (28) c(s, )/ˆϑ = w(s, ) + h(s, ) (29) c(s, ) = ˆϑ(w(s, ) + h(s, )). (30) Suppose we define upper-case variables as he aggregae value across all generaions currenly living of he corresponding lower-case value, e.g. aggregae consumpion is C() = dc(τ, ) D τdτ. (31) 5. (Easy) Show ha he aggregae level of consumpion in his economy is C() = C() = ˆϑ(W () + H()). (32) = dc(τ, ) D τdτ (33) d ˆϑ(w(τ, ) + h(τ, )) D τdτ (34) 5

6 from he definiion of W () and H(). = ˆϑ(W () + H()) (35) Suppose all living agens in his economy receive he same noncapial income, y(s, ) = Y (). Since every member of he populaion has he same income, and he size of he populaion is one, aggregae income will also be Y (). Aggregae human wealh a is herefore where H() = (Y (τ)/ ˆR τ )dτ (36) Ḣ() = ˆr H() Y (). (37) The differenial equaion for aggregae wealh can be shown o be Ẇ () = w(, ) dw () + ẇ(τ, )de d( τ) dτ, (38) where w(, ) = 0 is he wealh of newly born generaions, dw () is he wealh of hose who are dying a he momen, and he las erm is he change in wealh for hose who neiher die nor are born in his period. Bu (12) implies ha so (38) becomes ẇ(τ, )de d( τ) dτ = (r + d)w () + Y () C() (39) Ẇ () = rw () + Y () C(). (40) Collecing, wriing ou ˆr = r + d and ˆϑ = ϑ + d, and dropping he () argumens gives us he following equaions for aggregae variables: 6. Use hese equaions o show ha in his economy C = (d + ϑ)(h + W ) (41) Ḣ = (r + d)h Y (42) Ẇ = rw + Y C (43) Ċ = (r ϑ)c d(d + ϑ)w (44) Hin: Differeniae (41) and subsiue ou for H by solving (41) for H. Time differeniae (41) and subsiue for Ḣ, Ẇ, and H o ge Ċ = (d + ϑ) [(r + d)h Y + rw + Y C] 6

7 = (d + ϑ) [(r + d)h + rw ] C(d + ϑ) [ ( ) ] C = (d + ϑ) (r + d) d + ϑ W + rw C(d + ϑ) = (r + d)c + (d + ϑ) [rw (r + d)w ] C(d + ϑ) = (r ϑ)c (d + ϑ)dw. Now assume here is a sandard producion funcion F(K) = K α δk and assume perfec compeiion so ha he ne ineres rae r is equal o he ne marginal produc of capial, r = F (K) = αk α 1 δ (45) and he aggregae capial sock a ime is he same as aggregae nonhuman wealh, K() = W (). The aggregae accumulaion equaion is jus he usual 7. Show ha he following equaion K = K α δk C. (46) d(d + ϑ)k = C(αK α 1 δ ϑ) (47) describes he Ċ = 0 locus. Use his equaion o show ha lim C 0 = 0 (48) lim C = ((ϑ + δ)/α)1/α 1. (49) Rewriing he Ċ equaion as a funcion of K yields Ċ = (αk α 1 ϑ δ)c (d + ϑ)dk. (50) The Ċ = 0 locus is herefore given by 0 = (αk α 1 ϑ δ) d(d + ϑ)k/c (51) d(d + ϑ)k/c = (αk α 1 δ ϑ) (52) d(d + ϑ)k = C(αK α 1 δ ϑ) (53) In he limi as C 0 his expression approaches which can be rue only if lim {C 0} K = 0. d(d + ϑ)k = 0 (54) On he oher hand, as C we know ha K mus remain finie (because he DBC does no allow infinie accumulaion of K infinie K would imply infinie depreciaion which could never be paid for by a 7

8 producion funcion wih diminishing marginal reurns), which means ha lim {C } αkα 1 = (ϑ + δ) (55) K = ((ϑ + δ)/α) 1/α 1. (56) 8. Using your resuls from he previous quesion and he aggregae dynamic budge consrain, draw he phase diagram for his model and discuss how and why i differs from he phase diagram for he sandard Ramsey-Cass/Koopmans model. (please use K o designae he RCK seady-sae capial sock and K o designae his model s seady-sae K.) Using (46), he K = 0 locus is C = K α δk (57) which yields he usual hump-shaped K = 0 locus. Rewriing he Ċ equaion as a funcion of K yields Ċ = (αk α 1 ϑ δ)c (d + ϑ)dk. (58) The Ċ = 0 locus is herefore given by 0 = (αk α 1 ϑ δ) d(d + ϑ)k/c (59) d(d + ϑ)k/c = (αk α 1 δ ϑ) (60) d(d + ϑ)k = C(αK α 1 δ ϑ) (61) In he limi as C 0 his expression approaches which can be rue only if lim {C 0} K = 0. d(d + ϑ)k = 0 (62) On he oher hand, as C we know ha K mus remain finie (because he DBC does no allow infinie accumulaion of K), which means ha lim {C } αkα 1 = (ϑ + δ) (63) K = ((ϑ + δ)/α) 1/(α 1). (64) In he infinie horizon economy we have d = 0 and so he seady-sae ineres rae would be he K where αk α 1 δ = ϑ. Bu since K/C and d(d + ϑ) are sricly posiive, in his finie-horizon economy we would have Ċ/C < 0 a K = K. I is clear herefore ha in order 8

9 for (51) o hold we will need αk α 1 o be larger han i is a K, which is o say we need a higher seady-sae ineres rae, and hus we need a lower seady-sae capial sock, which is depiced in he figure as K. This makes sense because he finie-horizon consumers in his economy discoun he fuure more han he represenaive agen does, because hey die bu a represenaive agen does no. These resuls are combined in he figure, which shows ha he inersecion of he Ċ = 0 locus inersecs he K = 0 locus a poin A which corresponds o a lower level of he capial sock han in he infinie horizon model. Now consider he inroducion of a governmen ha finances spending eiher by lump-sum axes or by deb. Is dynamic budge consrain is Ḋ = rd + G T (65) where D is governmen deb, G is governmen spending, and T is a lump-sum per capia ax. Defining R s = e s rvdv (66) as he compound ineres facor beween ime and ime s, he governmen is also required o saisfy he ransversaliy condiion lim D /R s = 0. (67) Consider he following fiscal policy experimen. Unil ime here has been no governmen (G s = D s = T s = 0 s < ). A dae he governmen issues a quaniy D of deb and announces ha fuure lump sum axes will be imposed in 9

10 amouns exacly large enough o pay he ineres on his deb (so subsequenly, Ḋ = 0 forever). The governmen rebaes he proceeds of is sale of deb o he public as a per-capia lump sum of D per person. The governmen will never engage in any spending (aside from paying ineres on he deb). Define he new variables W = K + D (68) Y = Y T (69) H = Y/ ˆR s ds (70) 9. Explain why he effec of his policy is o modify he aggregae specificaion of he economy o (see op of nex page) 10

11 C = (d + ϑ)(h + W) (71) Ẇ = rw + Y C (72) Ḣ = (r + d)h Y (73) The effec of he governmen policy is wofold. On he one hand, he disribuion of governmen bonds increases he consumers wealh W by an amoun equal o he value of he bonds received, resuling in a new definiion of wealh W which includes he bonds. On he oher hand, he higher value of axes off o infiniy reduces he consumers human wealh by an amoun equal o he presen discouned value of he axes. The change in (redefined) wealh is now ne income Y ϑ minus consumpion. 10. Use hese equaions o show ha he new dynamic equaions for he economy are Ċ = (r ϑ)c d(d + ϑ)(k + D) (74) K = K α δk C (75) Show how he policy change affecs he economy over ime, using a phase diagram and a diagram showing he dynamics of aggregae consumpion afer he policy is inroduced. Explain he impac of he policy on differen generaions in he economy. Time differeniaing (71) yields Now solve (71) for H, Ċ = (d + ϑ)(ḣ + Ẇ) (76) = (d + ϑ)(rw + (r + d)h C) (77) H = and subsiue ino (77) o obain ( ) C W (78) d + ϑ Ċ = (d + ϑ)(rw + (r + d)(c(d + ϑ) 1 W) C) (79) = (r + d)c (d + ϑ)c d(d + ϑ)w (80) = (r ϑ)c d(d + ϑ)(k + D). (81) 11

12 Since we are assuming ha D is a consan (afer he fiscal experimen), any combinaion of C and W ha would have been on he Ċ = 0 locus before he policy shif now has a value Ċ = d(d + ϑ)d. This means ha he C ha would resore Ċ = 0 mus be a larger C, which says ha he Ċ = 0 locus shifs up (or, equivalenly, o he lef). Thus, he new equilibrium will be a a lower value of K and a higher ineres rae. The phase diagram shows ha he new equilibrium poin A is o he lef of he original equilibrium. This is because a a given level of he aggregae capial sock, consumers spend more because W > W. Thus, Ricardian equivalence does no hold in his model, because a ax cu oday financed by a fuure perpeual ax is a ransfer of resources from fuure consumers o oday s consumers, and here are no alruisic links ha make curren consumers offse his by saving more on behalf of fuure generaions. The nex figure shows he pah of consumpion per capia in his economy. Prior o ime 0, he economy was in is seady-sae equilibrium a he level of consumpion C 0 corresponding o he equilibrium labeled A in he phase diagram. A ime 0, he fiscal policy is carried ou. The fiscal policy immediaely increases consumpion because i amouns o a ransfer of resources from fuure o curren generaions. However, he higher level of consumpion runs down he capial sock per capia, and so over ime consumpion asypmoically approaches a new, lower equilibrium level of consumpion C. 11. Blanchard shows ha if he model is changed so ha each consumer s income declines exponenially a rae γ afer birh, he resul is equivalen o assuming ha fuure labor income is discouned a an ineres rae ha is higher by γ. 12

13 C 0 C' 0 Time He furher shows ha he equaions of moion of he model change o Ċ = (αk α 1 δ + γ ϑ)c (d + γ)(d + ϑ)k (82) K = K α δk C. (83) Noe ha i is possible o rewrie he Ċ = 0 locus as (αk α 1 δ + γ ϑ)c = (d + γ)(d + ϑ)k (84) Use his equaion o show ha as C, he Ċ = 0 locus asympoes o r = αk α 1 δ = ϑ γ (85) Thus, for a large enough value of γ i is possible ha he ne ineres rae in his economy could be negaive. Draw a phase diagram corresponding o an equilibrium wih a negaive ne ineres rae, and commen on why his is an ineresing case o hink abou. In paricular, wha new ligh does i shed on he fiscal policy experimen examined above? As C goes o infiniy on he LHS of (84), he only way he equaion can coninue o hold is if which implies (85). lim C (αkα 1 δ + γ ϑ) = 0 (86) 13

14 The new phase diagram shows he Ċ = 0 locus inersecing he K = 0 locus o he righ of he maximum of he K = 0 locus. This is implied by he fac ha he ne ineres rae is negaive, which means ha he ne ineres rae could be increased by reducing he capial sock. The reason his is an ineresing case is ha in his case i is possible for he economy o be in a condiion of dynamic inefficiency, jus as in he 2-period OLG models discussed early in he class. The idea is o hink of declining labor income as a way o generae a life cycle saving moive. In such a case he fiscal experimen examined above is ineresing because i could rescue an economy wih oo much capial from a sae of dynamic inefficiency. 14

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16 References Blanchard, Olivier J. (1985): Deb, Deficis, and Finie Horizons, Journal of Poliical Economy, 93(2),