The Lucas Asset Pricing Model

Transcription

1 c January 0, 206, Chrisopher D. Carroll The Lucas Asse Pricing Model LucasAssePrice 0. Inroducion/Seup Lucas 978 considers an economy populaed by infiniely many idenical individual consumers, in which he only asses are some idenical infiniely-liverees. Aggregae oupu is he frui ha falls from he rees, and canno be sored i would ro!; because u c > 0 c, i is all eaen: c L = K where c is consumpion of frui per person, L is he populaion, is he exogenous oupu of frui ha drops from each ree and K is he sock of rees. In a given year, each ree produces exacly he same amoun of frui as every oher ree, bu varies from year o year depending on he weaher. An economy like his, in which oupu arrives wihou any deliberae acions on he par of residens, is called an endowmen economy or, someimes, an exchange economy. 0.2 The Marke for Trees In equilibrium, he price of rees mus be such ha, each period, each idenical consumer does no wan eiher o increase or o decrease his holding of rees if, a a hypohesized equilibrium price, every idenical consumer waned say o increase heir holdings, ha price could no be an equilibrium price!. Le P denoe he equilibrium price, and assume ha if he ree is sold, he sale occurs afer he exising owner receives ha period s frui P is he ex-dividend price. The oal resources available o consumer i in perio are he sum of he frui received from he rees owned, k i, plus he poenial proceeds if he consumer were o sell all his sock of rees, P k i. Toal resources are divided ino wo uses: Curren consumpion c i anhe purchase of rees for nex period k i + a price P, Uses of resources { }} { k i +P + c i = Toal resources { }} { k i + P k i 2 k i + = + /P k i c i /P. 3 As in Aggregaion.

2 0.3 The Problem of an Individual Consumer Consumer i maximizes vm i = max E i s.. ] β n uc i +n n=0 4 k i + = + /P k i c i /P m i + = P k i +. Rewriing in he form of Bellman s equaion, vm i = max {c i } uc i + β E i he firs order condiion ells us ha so 0 = u c i + β E i v m i + d dc i u c i = β E i v m i + vm i + ], m i + { }} { P /P k i c i /P } {{ } k+ i P P } {{ } R + = β E i R+ v m i + ] 5 where R + is he reurn facor ha measures he resources in perio + ha are he reward for owning a uni of rees a he end of. The Envelope heorem ells us ha v m i + = u c i +, so 5 becomes 0.4 Aggregaion ] u c i = β E i u c i P P ] u P = β E i c i + P u c i The assumpion ha all consumers are idenical says ha c i = c j i, j, so henceforh we jus call consumpion per capia c. Since aggregae consumpion mus equal aggregae producion because frui canno be sored, normalizing he populaion o 2

3 L = and sock of rees o K =, equaion becomes: c =. 7 Subsiuing c and c + for c i and c i + in 6 anhen subsiuing for c we ge ] u + P = β E P u We can rewrie his more simply if we define an objec u M,+n = β n +n u which is callehe sochasic discoun facor because a i is sochasic hanks o he shocks beween an + n ha deermine he value of +n ; and b i measures he rae a which all agens in his economy in perio will discoun a uni of value received in a fuure period, e.g. + : 9 P = E M,+ P ]. 0 A corresponding equaion will hold in perio+, and in perio+2, and beyond: P + = E + M +,+2 P ] P +2 = E +2 M +2,+3 P ], 2 so we can use repeaed subsiuion, e.g. of ino 0, and of 2 ino he resul, ec o ge P = E M,+ + ] + E M,+ E + M +,+2 +2 ]] The law of ieraed expecaions says ha E E + P +2 ]] = E P +2 ]; given his, and noing ha M,+2 = M,+ M +,+2, 3 becomes: P = E M,+ + + M, M, ]. 4 So, he price of he asse is he presen discouned value of he sream of fuure dividends, where he sochasic facor by which poenially sochasic dividends received in + n are discouned back o is M,+n. 0.5 Specializing he Model This is as far as we can go wihou making explici assumpions abou he srucure of uiliy. If uiliy is CRRA, uc = ρ c ρ, subsiuing u d = d ρ ino 8 yields P = βd ρ E d ρ +P ] 5 anhe paricularly special case of logarihmic uiliy which Lucas emphasizes corresponds o ρ =, which again using he law of ieraed expecaions allows 3

4 us o simplify 5 o P = β E d d +P ] ] P+ = β + E d + ] P+2 = β + β + E +2 = β + β + β E lim { β = β + β E lim n βn n βn P+n +n ]}. P+n If he price is bounded i canno ever go, for example, o a value such ha i would cos more han he economy s enire oupu o buy a single ree, i is possible o show ha he limi erm in his equaion goes o zero. Using he usual definiion of he ime preference facor as β = / + ϑ where ϑ is he ime preference rae, he equilibrium price is: β P = β = /β = + ϑ = 6 ϑ or, equivalenly, he dividend-price raio is always /P = ϑ. 2 I may surprise you ha he equilibrium price of rees oday does no depend on he expeced level of frui oupu in he fuure. You migh reason ha higher expeced fuure frui producion increases he araciveness oday of buying he rees ha will produce ha abundan fuure frui, so demand for anhus price of rees should be higher. This logic is no wrong; bu i is exacly counerbalanced by anoher, and subler, fac: Since fuure consumpion will equal fuure frui oupu, higher expeced frui oupu means lower marginal uiliy of consumpion in ha fuure period of abundan frui basically, people ge weary of eaing frui, which reduces he araciveness of buying he rees. These wo forces are he manifesaion of he pure income effec and subsiuion effec in his model here is no human wealh, +n ]] 2 A derivaion parallel o he one above shows ha in he CRRA uiliy case he soluion is d ρ /P = ϑ. 4

5 anherefore no human wealh effec. In he special case of logarihmic uiliy considered here, income and subsiuion effecs are of he same size and opposie sign so he wo forces exacly offse each oher. 0.6 The Ineres Rae anhe Rae of Reurn in a Lucas Model We can decompose he reurn facor aribuable o ownership of a share of capial cf. 5 by adding and subracing P in he numeraor: P+ + P P + + R + = P = + P + + d + 7 P P so he rae of reurn is r + = P P P which is a useful decomposiion because he wo componens have naural inerpreaions: The firs is a capial gain or loss, anhe second can plausibly idenified as he ineres rae paid by he asse because i corresponds o income received regardless of wheher he asse is liquidaed. In models ha do no explicily discuss asse pricing, he implici assumpion is usually ha he price of capial is consan which migh be plausible if capial consiss mosly of reproducible iems like machines, 3 raher han Lucas rees. In his case R + = + d + P says ha he only risk in he rae of reurn is aribuable o unpredicable variaion in he size of dividend/ineres paymens. Indeed, if addiional assumpions are made e.g., perfec capial markes ha yielhe conclusion ha he ineres rae maches he marginal produc of capial, hen such models generally imply ha variaion in reurns a leas a high frequencies is very small, because aggregae capial ypically is very sable from one perioo he nex in such models, and, if he aggregae producion funcion is sable, his implies grea sabiliy in he marginal produc of capial. 3 The key insighs below remain rue even if here is a gradual rend in he real price of capial goods, as has in fac been rue. 5

6 0.7 Aggregae Reurns Versus Individual Reurns One of he subler enries in Arisole 350 BC s caalog of common human reasoning errors was he fallacy of composiion, in which he reasoner supposes ha if a proposiion is rue of each elemen of a whole, hen i mus be rue of he whole. The Lucas model provides a vivid counerexample. From he sandpoin of any individual aomisic agen, i is quie rue ha a decision o save one more uni will yield greaer fuure resources, in he amoun R +. Bu from he sandpoin of he sociey as a whole, if everyone decideo do he same hing save one more uni, here would be no effec on aggregae resources in perio +. Pu anoher way, for any individual agen, i appears ha he marginal produc of capial is r +, bu for he sociey as a whole he marginal produc of capial is zero. The proposiion ha he reurn for sociey as a whole mus be he same as he reurn ha is available o individials is an error because i implicily assumes ha here are no general equilibrium effecs of a generalized desire o save more or, more broadly, ha here is no ineracion beween he decisions one person makes anhe oucomes for anoher person. The Lucas model provides a counerexample in which, if everyone s preferences change e.g., ϑ goes down for everyone, he price of he fuure asse is affeced indeed, i is affeced in a way ha is sufficien o exacly counerac he increased desire for ownership of fuure dividends since here is a fixed supply of asses o be owned, he demand mus be reconciled wih ha preexising supply. 6

7 Appendix: Analyical Soluions in CRRA Uiliy Case 0. When Dividends are IID Suppose +n is idenically individually disribued in every fuure period, so ha is expecaion as of is he same for any dae n > 0: Now noe ha 5 can be rewrien as ] P P+ d ρ = β `d + E d ρ + ] = β `d + β `d P+2 + β E d ρ +2 `d E d ρ +n]. 9 = β `d ]] + β `d + β `d 2 P+n E lim n βn d ρ +n } {{ } assume goes o zero β = `d β `d = β `d To make furher progress, suppose ha he iid process for dividends is a mean-one lognormal: log +n N σ 2 /2, σ 2 n so ha E +n ] = n see ELogNormMeanOne], in which case ELogNormTimes] can be useo show ha `d = e ρρ /2σ2 23 and if we approximae β e ϑ hen β `d e ϑ ρρ /2σ2 and so 22 becomes P d ρ e ϑ ρρ σ2 /2 d ρ P ϑ ρρ σ 2 /2 where we used ExpEps] o ge from he firs o he second equaion. So he log of 22 is log P ρ log logϑ /2ρρ σ

8 anhus he variances obey varlog P = ρ 2 varlog d. 25 Given ha ρ >, his derivaion yields some ineresing insighs:. he log of asse prices will be more volaile han he log of dividends 2. An increase in risk aversion ρ increases P because ρρ σ 2 /2 > 0 and an increase in ρ increases is size The second poin is surprising, so le me say i again: an increase in risk aversion increases asse prices. In a sense, his is an implicaion of he proposiion ha risk aversion increases he volailiy of asse prices when hey are high, hey mus be very high; when low, very low. Bu, i does no correspond very well o he common narraive in which marke analyss ofen aribue a decline in asse prices o increased risk aversion. 0.2 When Dividends Follow a Random Walk The polar alernaive o IID shocks would be for dividends o follow a random walk: log+ / N σ 2 /2, σ 2. Now divide boh sides of 5 by, and rewrie he objec inside he expecaions operaor by muliplying he firs erm by + and dividing he seconerm by +, yielding ] P = βd ρ E d ρ P d + d+ ρ ] P+ = β E Now noe ha our assumpion here abou he disribuion of + / is idenical o he assumpion abou + above, so he expecaion will be he same `d; and 2 hypohesize ha here will be a soluion under which he price-dividend raio is a consan; call i r : ] r = β `dr + 28 = β `d + r 29 β `d β `d = r 30 β `d = r 3 8

9 so ha remarkably we obain a formula for r = P / ha is idenical o he formula for P /d ρ in 22; corresponding derivaions leao log P log logϑ /2ρρ σ 2 32 The difference wih 44 is only he absence of he ρ muliplying log. The main subsanive difference is herefore ha he variance of log prices anhe variance of log dividends is now he same. The surprising resul ha he price-dividend raio increases when risk aversion increases coninues o hold. 0.3 When Dividends Follow an AR Process Sar wih 34: P = βd ρ E d ρ P+ + = β E d+ and subsiue for + = α + z + : P = β E αd + z + ] ]. 34 ρ P+ + + ρ ] P We canno make furher analyical progress so long as he z + erm is presen. Numerical soluions eno work bes when i is possible o define he limis as he sae variables approach heir maximum possible values, so he nex sep is o ry o compue such limis As d In he limi as approaches, he z + erm becomes arbirarily small relaive o. Thus, ] P lim = β α ρ P βα ρ = 37 βα ρ = 38 β α ρ 9

10 0.3.2 As d 0 Suppose ha log z + N σ 2 /2, σ 2. Then ELogNormTimes] says: 2 ρ log E z + d ] = ρd σ 2 /2 + d σ 2 /2 39 ρ = d ρσ 2 2 /2 + σ 2 /2 40 whose limi is so lim d 0 lim log E d 0 z + d ] = P 2 ρ σ /2 2 2 ρ = β σ 2 /2 P+ so since P + /+ is a finie number we should have ha 2 d 2 P ρ lim = β σ 2 /2 d 0 4 ] P+ ] which should imply ha P is a finie number even as 0. To have boh limis be finie, we migh be able o use a rick like he ones proposed by Boyd 990. This would involve muliplying by some fd ha approaches d 2 as approaches zero bu approaches as approaches infiniy. Like, fd = d 2 +d? The idea is ha 2 fdp / migh be finie in boh limis and everywhere in beween even if P / is no. Think more abou his laer]. Alernaive. The soluion o he AR case is surely somewhere beween he soluions o he IID and RW cases. Tha means ha i is beween and log P ρ log logϑ /2ρρ σ 2 44 log P log logϑ /2ρρ σ 2 45 which can surely somehow be useo produce a reasonable limi. Acually, i seems prey clear ha he relevan comparison is o he IID case. 0

11 References Arisole 350 BC: On Sophisical Refuaions. The Wikipedia Foundaion. Boyd, John H. 990: A Weighed Conracion Mapping Theorem, Journal of Economic Theory, 6, Lucas, Rober E. 978: Asse Prices in an Exchange Economy, Economerica, 46, , Available a hp://