Exercise 2. Consumption with state-contingent good

Transcription

1 Poblem Set - Econ 0A (Second Half) Pof David Rome GSI: Victoia Vanaco Execie Conumption unde Uncetainty The baic model of conumption unde uncetainty (with quadatic utility, and uncetainty only about labo income) pedict that: A The change in income will not be pedictable on the bai of pat change in conumption B The change in conumption will not be pedictable on the bai of pat change in income C The change in conumption will not be coelated with the cuent change in income D (A) and (B) E (A) and (C) F (B) and (C) Anwe Note that option (A) i fale Ou baic model of conumption unde uncetainty pedict that change in conumption occu when new about change in income ae eceived In thi ene, change in conumption due to an expected futue inceae in income would help u pedict the change in income In othe wod, if we ee conumption jumping, with no change in cuent income, we hould expect an inceae in futue income Thu, change in income can be pedicted on the bae of pat change in conumption Thi ule out anwe (A), (D), and (E) Option (B) i tue When β(+ =, ou baic model of conumption with quadatic utility tell u that conumption follow a andom walk, which mean that the expected change in conumption i zeo Theefoe, any change in conumption hould be unpedictable, and thu we hould not be able to pedict them on the bai of pat change in income If β(+, we might be able to pedict that conumption i gowing o deceaing, but note that thi expected change in conumption ae unelated to change in income and have moe to do with the elative pice of conumption today v tomoow Option (C) i ticky On the one hand, if the change in cuent income wee unpedictable, we hould obeve a poitive coelation between the change in conumption today and the change in cuent income On the othe hand, if the change in cuent income wee pedictable, conumption aleady adjuted when new about thi change aived, and thu hould not eact to the actual change in cuent income; in thi cae, the coelation hould be zeo Since both cae ae poible, I would ay we hould obeve ome coelation between change in conumption and cuent income, and that thi coelation eect the unanticipated hock to income So I would ay (C) i fale Finally, (B) i the coect anwe Execie Conumption with tate-contingent good Conume maximize the expected peent value utility: ( ) t E U (c t)],u > 0 and U < 0 +δ t=0

2 Conume can puchae tate-contingent good, whee the pice of one conumption good at time t tate i given by p t Let S be the pace of all poible tate Given that the pobability of tate at time t i given by π t, and auming agent' income i alo contingent on the tate, we can wite the agent' poblem a: max {c t} t=0, S t=0 S a The conume' maximization poblem i: L = max {c t} t=0, S Taking FOC wt (c t ) we get: t=0 S t=0 S ( ) t π t U (c t) +δ t p t c t p t Y t t=0 S ( ) t π t U (c t) λ +δ ] p t c t p t Y t t=0 S ( ) t π t U (c t) = p tλ, t, +δ b Conumption at time t will be the ame fo tate and if: U (c t ) = U (c t) ( ) t ( π t +δ π t = λp t λp t π t = π t p t p t +δ ) t t=0 S The atio between the pobability of the tate occuing and the pice of conumption in that tate i equalized aco tate c In thi cae, let t > t WLOG: U (c t ) = U (c t ) ( ) t ( π t +δ π t = λp t λp t π t p t = π t p t +δ ) t ( +δ In thi cae we need to alo conide the dicount ate, but the condition i the ame a befoe (afte contolling fo dicount) d In conume ONLY die in the Y t, the tatement of the quetion i tue The poof goe a follow: Since conume do not die in anything ele, thei FOC with epect to c t ae the ame, but they might have dieent λ: ) t t

3 ( π t +δ π t ( +δ ) t U ( c ) t ) t U ( c ) t = p t λ = p t λ Dividing one FOC by the othe we get (and note that it i key fo thi that pice, pobabilitie and dicount ate ae the ame fo both agent): U ( ) ( ) c t λ = U c t λ, t, So, if in one peiod c t > c t U ( c t) < U ( c t) (uing the fact that U < 0) Uing ou lat condition, we can ee that thi implie: λ < λ Since thi condition hold, t, we can then conclude that if c t > c t in time t, then: U ( c ) t < U ( c t), t, c t > c t, t, fo one paticula and at ome point QED The intuition fo thi eult i the following Agent ell all claim to thei futue income at the beginning of thei live and poceed to buy conumption fo all the futue peiod and tate with thi income So they tat thei live with lifetime income o wealth: p t Yt and t=0 S t=0 S p t Y t The fact that thei Y t die imply implie that one might be wealthie than the othe Since in eveything ele they ae identical, they would both behave the ame way if thei wealth wa the ame If in any peiod we obeve agent conuming moe than the othe, thi mut mean that agent had highe wealth initially (and thu fo thi conume the hadow value of wealth i le λ < λ ) and wa able to buy moe conumption fo all peiod In othe wod, both conume ubtitute intetempoally and intatempoally following the ame ule, o we hould expect dieence only to come fom dieence in the budget containt (lifetime wealth) e Let' take log of FOC fo agent : log(π t ) tlog(+δ)+logu ( c ) t = logpt +logλ And ee how thi change with time (no deivative ince we'e woking with dicete time, o let x t = x t+ x t ): log(+δ)+ U ( ) c t c t c t U (c t) c t RRA c t c t = p t p t = δ p t p t 3

4 Uing the fact that ince δ < log(+δ) δ Idem fo agent : Dividing one by the othe we get: RRA c t c t c t c t = δ p t p t = c t c t Thi tell u that if conume have the ame contant elative ik aveion, conumption gowth ate hould be pefectly coelated between them Thi i, if conume inceae he conumption between peiod t and t, then o will conume, and by the ame popotion Thi implie that the atio of conumption between conume and conume aco time hould be contant and given by c 0 (ince incement will be the ame c 0 aco conume, what detemine thi atio ae initial conumption) f In pactice, we often ee conumption eveal, one conume might initially have conumption highe than anothe, but late lowe conumption How do we explain thi with thi model? We can't Let' ee which aumption of the model we can elax to explain thi fact: Complete maket: agent can buy contingent claim to conumption in the futue Thi i one of the tonget aumption of thi model Let' change thi aumption, and go to the othe exteme: not only conume cannot buy contingent good, but they cannot paticipate in nancial maket of any kind Thi i, conume cannot ave no boow In thi cae, all conume can do i conume thei income: c t = Y t Clealy in thi ituation, conumption would uctuate a income doe, and we could obeve eveal wheneve thee i a eveal of thei income Again complete maket! But now let' focu on the uncetainty pat By having complete maket, we'e making uncetainty ielevant in the model, ince by tading conumption good, agent ae not expoed to idioyncatic ik Howeve, we do think uncetainty play a ole in obeved conumption patten So let' analyze a dieent way in which thi aumption fail to explain eality Let' aume agent do have acce to nancial maket, they can ave and boow Howeve, they face uncetainty about thei futue income To implify even futhe the analyi, cetainty equivalence hold in thi model (we have quadatic utility fo example) In thi cae, agent' deciion to lend and boow will be baed on expected futue income Now, if in one peiod one of the conume eceive a vey high poitive hock to income (eceive much moe income than expected), then he will ave and adjut he conumption upwad If peviouly to the hock thi conume wa conuming le than the othe (becaue he expected lowe income than the othe, then we would obeve a eveal Thi i, in a wold without complete maket, but with acce to nancial maket like the one we know, unexpected income change can explain eveal a the one peviouly decibed 3 I would continue talking about the poblem of auming complete maketbut let' think about omething ele: unique tate ditibution In the model we aumed that the pobability of a tate happening at a given time i given by π t, which i known by eveyone, eveyone agee Maybe thi i not ealitic, we don't ee people ageeing about the pobabilitie of futue outcome, if anything, we ee people dicuing about futue outcome a lot! So you can ee that if we emove thi aumption, and conide πt πt, nothing guaantee that conumption gow at the ame ate, and we might obeve eveal in peiod and in tate whee thee wa a lage initial diageement g Thi will have no impact on the conume' conumption in late peiod (o any peiod) Remembe that conume old the claim to thei own output in the t peiod With thi money, they decided how much to conume in evey futue peiod, at any given tate, and bought the epective contingent good to do o So conumption c t i pinned down initially fo all,t Conume decided in t = 0 how much they wanted conume at 4

5 time t if the good tate happened, o when it actually happen, they will eceive the c t they had bought initially and conume, but thi will not aect the conumption they decided initially fo the peiod ahead The intuition i that when we have complete maket, uncetainty i killed, o hock to income do not have any impact on conumption deciion Execie 4 Conumption Choice and Peent Value Hamiltonian The Houehold olve the following poblem: ˆ T max e D(t) u(c(t))dt {c t} T 0 0 t A(0) given Ȧ(t) = (t)a(t)+y (t) c(t) A(T) 0 whee D(t) = t τ=0 ρ(τ)dτ a) Setup the Peent Value Hamiltonian (PVH): b) Condition fo optimality: Optimality of Contol H t (c(t),a(t)) = e D(t) u(c(t))+µ(t)(t)a(t)+y (t) c(t)]] H t c(t) = 0, t u (c(t)) µ(t) = 0 Motion fo the cotate ] e D(t) µ(t) t = H t A(t) e D(t) µ(t) δ(t)µ(t)] = e D(t) µ(t)(t) µ(t) = δ(t) (t) µ(t) 3 Combining +, we get an expeion fo the gowth ate of conumption: ċ(t) c(t) = (t) δ(t) RRA whee emembe RRA = u (c(t))c(t) u (c(t)), and thi come fom howing that µ(t) µ(t) = (c(t))c(t) ċ(t) u u (c(t)) c(t) 5

6 Exta-Execie Pemanent Income Hypothei In epone to an unexpected, pemanent fall in he labo income, the conume will: Anwe B The conume eceived an unexpected negative hock to it' peent labo income To make thing eaie, aume that +δ (+ = In thi cae, the optimal epone i to adjut conumption by the amount of the fall in labo income, ince the change i pemanent Remembe that in the cae with cetainty and innite hoizon (and the intuition i the ame if we have T nite): c t = A t + + T =t ( ) ] t Y + If Ỹ = Y +ε fo all t and the conume nd thi out at time t, he will adjut he conumption afte the new to c t, given by: c t = c t = c t = c t = c t = A t + + A t T ( ) ] tỹ + =t T ( ) t (Y +ε)] + =t T ( ) t T ( ) ] t A t +ε + Y + + =t =t A t +ε + T ( ) ] t + Y + =t T ( ) ] t A t + Y +ε = c t +ε + =t We can ee that anwe A i not poible A a epone to a fall in pemanent income, the conume hould not be able to boow againt the futue ince it will violate the no-ponzi condition (it will neve eceive enough labo income to epay the inceaed debt) Thi alo tell u that the conume will have to omehow educe it conumption a a epone to a negative income hock A fo C and D, they both have the wod gadually, thi hould hint that thee cannot be ight anwe Remembe that unde ou aumption of concave utility function and + +δ thi conume pefe to mooth conumption ove time (thi doe not mean gadual, it mean contant ove time!) So, gadual adjutment would imply deceaing conumption ove time until ome point Given the concavity of the utility function, it i tictly pefeed to do the cut in conumption at once 6

7 SOLUTIONS FROM ROMER S BOOK Poblem 85 (a) Conide the uual expeiment of a deceae in conumption by a mall (fomally, infiniteimal) amount dc in peiod t With the CRRA utility function given by () u(c t ) = C -θ t /( - θ), the maginal utility of conumption in peiod t i C -θ t Thu the change ha a utility cot of () utility cot = C -θ t dc The maginal utility of conumption in peiod t + i C -θ t+ With a eal inteet ate of, the individual get to conume an additional ( + dc in peiod t + Thi ha a dicounted expected utility benefit of (3) expected utility benefit = E + θ t Ct ( + dc] + ρ If the individual i optimizing, a maginal change of thi type doe not affect expected utility Thi mean that the utility cot mut equal the expected utility benefit o (4) C θ t = + E t C t+ θ ], + ρ whee we have (athe infomally) canceled the dc' Equation (4) i the Eule equation (b) Fo any vaiable x, e lnx = x, and o we can wite C (5) E t Ct ] E t e t + θ θln = + ] Uing the hint in the quetion -- if x ~ N(µ, V) then Ee x ] = e µ e V/ -- then ince the log of conumption i ditibuted nomally, we have E E C ] E e t Ct t t+ θ θ ln = + θ σ / t e (6) θe C = e t ln t+ θ σ / e In the fit line, we have ued the fact that conditional on time t infomation, the vaiance of log conumption i σ In addition, we have witten the mean of log conumption in peiod t +, conditional θe C on time t infomation, a E t lnc t+ Finally, in the lat line we have ued the fact that e t ln t+ θ σ / e i imply a contant Subtituting equation (6) back into equation () and taking the log of both ide yield (7) - θlnc t = ln( + - ln( + ρ) - θe t lnc t+ + θ σ / Dividing both ide of equation (7) by (- θ) leave u with (8) lnc t = E t lnc t+ + ln( + ρ) - ln( + ]/θ - θσ / (c) Reaanging equation (8) to olve fo E t lnc t+ give u (9) E t lnc t+ = lnc t + ln( + - ln( + ρ)]/θ + θσ / Equation (9) implie that conumption i expected to change by the contant amount ln( + - ln( + ρ)]/θ + θσ / fom one peiod to the next Change in conumption othe than thi deteminitic amount ae unpedictable By the definition of expectation we can wite (0) E t lnc t+ = lnc t + ln( + - ln( + ρ)]/θ + θσ / + u t+, whee the u' have mean zeo and ae eially uncoelated Thu log conumption follow a andom walk with dift whee ln( + - ln( + ρ)]/θ + θσ / i the dift paamete (d) Fom equation (9), expected conumption gowth i () E t lnc t+ - lnc t ] = ln( + - ln( + ρ)]/θ + θσ /

8 Clealy, a ie in aie expected conumption gowth We have Et ln Ct+ C ] () ln t = > 0 θ ( + Note that the malle i θ the the bigge i the elaticity of ubtitution, /θ the moe that conumption gowth inceae due to a given inceae in the eal inteet ate An inceae in σ alo inceae conumption gowth ince Et ln Ct+ ln Ct] θ (3) = > 0 σ It i taightfowad to veify that the CRRA utility function ha a poitive thid deivative Fom equation (), u '(C t ) = C -θ t and u ''(C t ) = -θc -θ- t Thu (4) u '''(C t ) = -θ(-θ - )C -θ- t = (θ + θ)c -θ- t > 0 So an individual with a CRRA utility function exhibit the pecautionay aving behavio explained in Section 76 A ie in uncetainty (a meaued by σ, the vaiance of log conumption) inceae aving and thu expected conumption gowth EXTRA EXERCISES Execie 83 (a) We need to find an expeion fo (C t+ + C t+3 )/] - (C t + C t+ )/] We can wite C t+, C t+ and C t+3 in tem of C t and the e' Specifically, we can wite () C t+ = C t + e t+, () C t+ = C t+ + e t+ = C t + e t+ + e t+, and (3) C t+3 = C t+ + e t+3 = C t + e t+ + e t+ + e t+3, whee we have ued equation () in deiving () and equation () in deiving (3) Uing equation () though (3), the change in meaued conumption fom one two-peiod inteval to the next i (4) C t+ + C t+ 3 C t + C t+ ( C t + e t+ + e t+ ) + ( C t + e t+ + e t+ + e t+ 3) C t + ( C t + e t+ ) =, which implifie to (5) C t+ + C t+ 3 C t + C t+ e t+ 3 + e t+ + e t+ = (b) Though imila manipulation a in pat (a), the peviou value of the change in meaued conumption would be (6) C t + C t+ C t + C t e t+ + e t + e t = Uing equation (5) and (6), the covaiance between ucceive change in meaued conumption i C t+ + C t+ 3 C t + C t+ C t + Ct+ C t + C t (7) cov, = et+ 3 + et+ + et+ et+ + et + e t cov, Since the e' ae uncoelated with each othe and ince e t+ i the only value of e that appea in both expeion, the covaiance educe to C t+ + C t+ 3 C t + C t+ C t + C t+ C t + C t σ e (8) cov, =, 4

9 whee σ e denote the vaiance of the e' So the change in meaued conumption i coelated with it peviou value Since the covaiance i poitive, thi mean that if meaued conumption in the twopeiod inteval (t, t + ) i geate than meaued conumption in the two-peiod inteval (t -, t - ), meaued conumption in (t +, t + 3) will tend to be geate than meaued conumption in (t, t + ) When a vaiable follow a andom walk, ucceive change in the vaiable ae uncoelated Fo example, with actual conumption in thi model, we have C t - C t- = e t and C t+ - C t = e t+ Since e t and e t+ ae uncoelated, ucceive change in actual conumption ae uncoelated Thu if C t wee bigge than C t-, it would not mean that C t+ would tend to be highe than C t Since ucceive change in meaued conumption ae coelated, meaued conumption i not a andom walk The change in meaued conumption today doe povide u with ome infomation a to what the change in meaued conumption i likely to be tomoow (c) Fom equation (5), the change in meaued conumption fom (t, t + ) to (t +, t + 3) depend on e t+, the innovation to conumption in peiod t + But thi i known a of t +, which i pat of the fit two-peiod inteval Thu the change in conumption fom one two-peiod inteval to the next i not uncoelated with eveything known a of the fit two-peiod inteval Howeve, it i uncoelated with eveything known in the two-peiod inteval immediately peceding (t, t + ) Fom equation (5), e t+3,e t+ and e t+ ae all unknown a of the two-peiod inteval (t -, t - ) (d) We can wite C t+3 a a function of C t+ and the e' Specifically, we can wite (9) C t+3 = C t+ + e t+3 = C t+ + e t+ + e t+3 Thu the change in meaued conumption fom one two-peiod inteval to the next i (0) C t+3 - C t+ = C t+ + e t+ + e t+3 - C t+ = e t+ + e t+3 The ame calculation would yield the peviou value of the change in meaued conumption, () C t+ - C t- = e t + e t+ And o the covaiance between ucceive change in meaued conumption i () cov(c t+3 - C t+ ), (C t+ - C t- )] = cov(e t+ + e t+3 ), (e t + e t+ )] Since the e' ae uncoelated with each othe, the covaiance i zeo Thu meaued conumption i a andom walk in thi cae The amount that C t+ diffe fom C t- doe not povide any infomation about what the diffeence between C t+ and C t+3 will be Execie 86 (a) Subtituting the expeion fo conumption in peiod t, which i () C A E t t+ ] t = t +, + = 0 ( + Y ) into the expeion fo wealth in peiod t +, which i () A t+ = ( + A t + Y t - C t ], give u (3) A A Y A Y E Y E Y t t+ t t+ t+ = ( + ) t + t t t Κ ( + Obtaining a common denominato of ( + and then canceling the ( + ' give u E Y E Y t t+ t t+ (4) A t+ = A t + Yt + + Κ + ( + Since equation () hold in all peiod, we can wite conumption in peiod t + a (5) C A E t+ t+ + ] t+ = t+ + + = 0 ( + Y )

10 Subtituting equation (4) into equation (5) yield (6) C A Y E Y E Y t t+ t t+ E t+ Yt+ t+ = t + t + + Κ + E t+ Yt Κ ( + + Taking the expectation, conditional on time t infomation, of both ide of equation (6) give u (7) E C A Y E Y E Y t t+ t t+ E tyt+ t t+ = t + t + + Κ + E tyt Κ, ( + + whee we have ued the law of iteated pojection o that fo any vaiable x, E t E t+ x t+ = E t x t+ If thi did not hold, individual would be expecting to evie thei etimate eithe upwad o downwad and thu thei oiginal expectation could not have been ational Collecting tem in equation (7) give u (8) E C A Y t t+ = t + t E tyt+ + E ty + + t+ + + Κ, ( + which implifie to (9) E C A Y E Y E Y t t+ t t+ t t+ = t + t Κ + + ( + Uing ummation notation, and noting that E t Y t = Y t, we have (0) E C A E t t+ t t+ = t + + = 0 ( + Y ) The ight-hand ide of equation () and (0) ae equal and thu () E t C t+ = C t Conumption follow a andom walk; change in conumption ae unpedictable Since conumption follow a andom walk, the bet etimate of conumption in any futue peiod i imply the value of conumption in thi peiod That i, fo any 0, we can wite () E t C t+ = C t Uing equation (), we can wite the peent value of the expected path of conumption a E t C t ] C + t (3) = = C t = 0 ( + = 0 ( + = 0 ( + Since /( + <, the infinite um on the ight-hand ide of (3) convege to / - /(+] = ( + / and thu E t C t+ ] + (4) = C t = 0 ( + Subtituting equation () fo C t into the ight-hand ide of equation (4) yield E t C t+ ] + E t Yt ] + E t Yt+ ] (5) = A A t + = t + = 0 ( + ) + = 0 ( + = 0 ( + Equation (5) tate that the peent value of the expected path of conumption equal initial wealth plu the peent value of the expected path of income (b) Taking the expected value, a of time t -, of both ide of equation () yield (6) E C A E Y t t+ ] t t = t +, + = 0 ( + whee we have ued the fact that A t = ( + A t- + Y t- - C t- ] i not uncetain a of t - In addition, we have ued the law of iteated pojection o that E t- E t Y t+ ] = E t- Y t+ ] Subtacting equation (6) fom equation () give u the innovation in conumption:

11 ] E t Y E Y t t t E t Yt E Y + ] + ] + ] t t+ (7) C t E t C t = = + = 0 ( + ) = 0 ( + ) + = 0 ( + The innovation in conumption will be faction /( + of the peent value of the change in expected lifetime income The next tep i to detemine the peent value of the change in expected lifetime income That i, we need to detemine E t Yt+ ] E t Yt+ ] E tyt+ E t Yt+ E tyt+ E t Yt+ (8) = Y E Y t t t] + = ( Κ 0 ( + In what follow, "expected to be highe" mean "expected, a of peiod t, to be highe than it wa, a of peiod t - " We ae told that u t = and thu (9) Y t - E t- Y t = In peiod t +, ince Y t+ = φ Y t + u t+, the change in Y t+ i expected to be φ Y t = φ highe Thu the level of Y t+ i expected to be highe by + φ Thu (0) E t Y t+ E t Y t+ = + φ + + In peiod t +, ince Y t+ = φ Y t+ + u t+, the change in Y t+ i expected to be highe by φ Y t+ = φ Thu the level of Y t+ i expected to be highe by + φ + φ Theefoe, we have () E t Y t+ E t Y t+ = + φ + φ ( + ( + The patten hould be clea We have E t Yt+ ] E t Yt+ ] () = + + = ( φ φ φ φ φ φ + Κ 3 0 ( + ( + Note that thi infinite eie can be ewitten a E t Yt+ ] E Y ] t t+ φ (3) = = ( + + ( φ φ Κ ( + ( + φ φ + + Κ 3 ( + ( + + Κ Fo eae of notation, define γ /( + Then the fit um on the ight-hand ide of (3) convege to /( - γ) The econd um convege to φγ/( - γ) The thid um convege to φ γ /( - γ) And o on Thu equation (3) can be ewitten a ] φγ φ γ ] E t Yt+ ] E t Yt+ (4) = = = 0 ( + γ Κ ( γ) ( φγ) Uing the definition of γ to ewite equation (4) yield ] E t Yt+ ] E t Yt+ ( + ( + (5) = = = 0 ( + ( + ] φ ( + ] ( + φ ) Subtituting equation (5) into equation (4) give u the following change in conumption: ( + ( + ( + (6) Ct E t C t = = ( + ( + φ) ( + φ )

12 (c) The vaiance of the innovation in conumption i ( + ( + (7) va( Ct E t C t) = va u t = va( u t ) > va( u t ) ( + φ) ( + φ) Since ( + /( + - φ) >, the vaiance of the innovation in conumption i geate than the vaiance of the innovation in income Intuitively, an innovation to income mean that, on aveage, the conume will expeience futhe change in income in the ame diection in futue peiod It i not clea whethe conume ue aving and boowing to mooth conumption elative to income Income i not tationay, o it i not obviou what it mean to mooth it Execie 8 Let E t denote X t fom the book (a) The change in puchae in peiod t, de t, mut leave the peent value of pending unchanged, o that () de t + de t+ + de t+ = 0 In addition, it mut leave conumption in peiod t + unchanged, o () ( - δ) de t + ( - δ)de t+ + de t+ = 0 To ee why equation () mut hold, note that we can wite the change in C t a dc t = de t The change in C t+ i dc t+ = ( - ρ)dc t + de t+ o ubtituting fo dc t, we have dc t+ = ( - ρ)de t + de t+ The change in C t+ i dc t+ = ( - δ)dc t+ + de t+ o ubtituting fo dc t+, we have dc t+ = ( - δ) de t + ( - δ)de t+ + de t+ Thu if C t+ i not to change, equation () mut hold Thu we have two equation in two unknown Solving equation () fo de t+ give u (3) de t+ = -de t - de t+ Subtituting equation (3) into equation () yield (4) ( - δ) de t + ( - δ)de t+ - de t - de t+ = 0 Expanding and collecting tem give u (5) de t - δ + δ - ] + - δ - ]de t+ = 0, and thu (6) de t+ = (δ - )de t Subtituting equation (6) into equation (3) yield (7) de t+ = -de t - (δ - )de t, and thu (8) de t+ = ( - δ)de t (b) Since C t = ( - δ)c t- + E t, then (9) dc t = de t Since C t+ = ( - δ)c t + E t+, then (0) dc t+ = ( - δ)dc t + de t+ Subtituting equation (9) and (6) into equation (0) give u () dc t+ = ( - δ)de t + (δ - )de t = -de t Since only C t and C t+ ae changed C t+ i unchanged by contuction we only need to look at expected utility in peiod t and t + Since intantaneou utility i quadatic, the maginal utility of conumption in peiod t i - ac t Thu the change in utility in peiod t i ( - ac t )(de t ) The maginal utility of conumption in peiod t + i given by - ac t+ Since dc t+ = -de t, the change in expected utility in peiod t + i the expected value of ( - ac t+ )(-de t )

13 (c) Fo thi change in expected utility to be zeo a it mut be, if the individual i optimizing we equie () ( - ac t )(de t ) + expected value of ( - ac t+ )(-de t )] = 0 Canceling the de t ' (which i omewhat infomal), ubtacting one fom both ide and then dividing both ide by (- a) yield (3) expected value of C t+ = C t Thu conumption follow a andom walk ince change in conumption ae unpedictable The bet etimate of conumption in peiod t + i imply what conumption equal thi peiod (d) Reaanging C t = ( - δ)c t- + E t to olve fo E t give u (4) E t = C t - ( - δ)c t- Equation (4) hold fo all peiod and o we can wite (5) E t- = C t- - ( - δ)c t- Subtacting equation (5) fom equation (4) give u (6) E t - E t - = C t - ( - δ)c t- - C t- + ( - δ)c t-, which implie (7) E t - E t- = (C t - C t- ) - ( - δ)(c t- - C t- ) Since conumption i a andom walk, we can wite (8) C t = C t- + u t, whee u t i a vaiable whoe expectation a of t - i zeo Uing equation (8), and the fact that (8) hold in all peiod, equation (7) can be ewitten a (9) E t - E t- = u t - ( - δ)u t- Equation (9) tate that the change in puchae fom t - to t ha a pedictable component a component that i known a of t - which i u t-, the innovation to conumption in peiod t - Thu puchae of duable good will not follow a andom walk A explained in Section 7, any change in expected lifetime eouce i pead out equally among conumption in each emaining peiod of the individual' life Although we ae implifying by uing a dicount ate of zeo, the baic idea ae geneal Now uppoe that in peiod t -, the individual' etimate of lifetime eouce change in uch a way that C t- i one unit highe than C t-, that i, u t- = Thi alo mean that expected conumption in all futue peiod i one unit highe than it ued to be In ode to get C t- up by one, puchae in peiod t - mut be one highe than they wee expected to be But now look at the change in puchae fom t - to t Fom equation (9), the expectation (a of peiod t - ) of the change in puchae fom t - to t i -( - δ), ince u t- i aumed to equal one Intuitively, ome of the new good puchaed in peiod t - will till be aound in peiod t Thu to keep expected conumption in peiod t at the new highe path one highe than it wa befoe it i not expected to be neceay to buy one unit of good all ove again The individual only ha to puchae enough to eplace the faction of the exta t - puchae that depeciated, which i faction δ Thu puchae in peiod t ae expected to be le than puchae in t - Specifically, they ae expected to be lowe by the amount that doe not depeciate, which i ( - δ) Thu, a of peiod t -, pat of the change in puchae between t - and t i pedictable and thu puchae do not follow a andom walk Now conide what happen if δ = 0, the cae of no depeciation Then fom equation (9), the expectation (a of peiod t - ) of the change in puchae fom t - to t i - Now all of the new good puchaed in peiod t - will till be aound in peiod t Thu to keep expected conumption at it new highe path one highe than it wa befoe it i not expected to be neceay to puchae anything new

14 in peiod t Thu puchae ae expected to fall by the whole amount of the innovation in puchae the peviou peiod

15