Market selection with learning and catching up with the Joneses

0.33 MB·English

by  Roman Muraviev

MARKET SELECTION WITH LEARNING AND CATCHING UP
WITH THE JONESES
2
ROMANMURAVIEV
1
0 DEPARTMENTOFMATHEMATICSANDRISKLAB
2 ETHZURICH

n
a
J
Abstract. We study the market selection hypothesis in complete financial
5
markets, populatedbyheterogeneous agents. Weallowforarichstructureof
1
heterogeneity: individualsmaydiffer intheir beliefsconcerning the economy,

] information and learning mechanism, riskaversion, impatience and ’catching
M
up with Joneses’ preferences. We develop new techniques for studying the
P long-runbehaviorofsucheconomies,basedontheStrassen’sfunctionallawof
. iterated logarithm. In particular, we explicitly determine an agent’s survival
n
indexandshowhowthelatterdependsontheagent’scharacteristics. Weuse
i
f these results to study the long-run behavior of the equilibrium interest rate
-
q andthemarketpriceofrisk.
[

3
v
5 1. Introduction
2
0 Afundamentalquestioninthemoderntheoryoffinancialeconomicsisconcerned
3
with the so-called market selection hypothesis, dating back to the ideas of Fried-
.
6 man [16]. Motivated by the postulate that agents with inaccurate forecasts will
0
1 eventually be driven out of the economy, this hypothesis can be stated informally
1 as”Ifyouare sosmart,why aren’tyourich?”. Formally,marketselectionin finan-
:
v cialmarketsexamines the agents’long-runsurvival1capabilityand price impactin
i
X equilibrium models. There is a vast body of literature dealing with this topic; see
r e.g. Blume and Easley [7], Cvitani´c, Jouini, Malamud and Napp [8], Nishide and
a
Rogers [25], Sandroni [27], and Yan [32, 33].
Thispaperinvestigatesthemarketselectionhypothesis(or,naturalselection,for
short) and the long-run behavior of asset prices in a complete market setting with
highly heterogeneous investors. Individuals may differ in their beliefs concerning
Date:January9,2012.
2010 Mathematics Subject Classification. Primary: 91B69Secondary: 91B16.
Keywordsandphrases. NaturalSelection,HeterogeneousEquilibrium,DiverseBeliefs,Learn-
ing,SurvivalIndex, CatchingupwithTheJoneses.
Thispaperwaspreviouslycirculatedunderthetitle”Naturalselectionwithhabitsandlearning
inheterogeneous economies”.
1An agent is said to survive in the long-run if the ratio of his consumption to the aggregate
consumptionstayspositivewithpositiveprobabilityastimegoestoinfinity.
1
2 R.Muraviev
the economy, information and learning mechanism, risk aversion,impatience (time
preferencerate)anddegreeofhabit-formation. Eachindividualinourmodelisrep-
resented by a generalized versionof the catching up with the Joneses power utility
functionofChanandKogan[9]. Thismodelofpreferencesissometimesreferredto
inthe literatureasexogenous habit-formation, sinceit incorporatesthe impactofa
certaingivenstochasticprocessonthe individual’sconsumptionpolicy. Agentsare
assumed to possess only partial information regarding the events associated with
the evolution of the market. More precisely, the stochastic dynamics of the mean
growth-rateofthe economy2areunobservable, andthe agents’informationsetcon-
sists of the aggregate endowment and a publicly observable signal. Furthermore,
agents are allowed to have diverse beliefs concerning the values of the initial and
averagemean growth-rate. Individuals may be irrationalin the way they interpret
the public signal: some of them may be over- (or, under-)confident about the in-
formativeness of the public signal. We use the standard way of modeling over-(or,
under-)confidence, originatedin Dumas, Kurshev, and Uppal [14] and Scheinkman
andXiong [28]: we assume that agents’beliefs concerningthe instantaneouscorre-
lationofthepublicsignalwiththeeconomy’sgrowth-ratemaydifferfromitsactual
value.3 The agentsare rationalin the sense that they use a standardKalmanfilter
to update their expectations about the economy’s growth-rate. The heterogeneous
filteringrulesyieldhighlynon-trivialdynamicsfor the individualconsumptionand
theequilibriumstatepricedensity,determinedbythemarketclearingcondition. In
particular,subjectiveprobabilitydensitiesdescribingtheagents’beliefs giveriseto
multiple new state variables, which governthe dynamics of the economy. We refer
to Back [2] for a survey on filtering and incomplete information in asset pricing
theory.
Letusdescribethecontributionofthisworktotheliteratureonequilibriumand
naturalselection. Firstly,asdescribedabove,weanalyzeaverygeneralparadigmof
heterogeneous economies including diverse beliefs, Kalman filtering and exogenous
state-dependenthabitformationpreferences. We providea comprehensivedescrip-
tionoftheequilibriumcharacteristics,thatcanbeusedforfurtherresearchinother
possible directions. Secondly, this complex setting in turn allows detecting which
traits (both behavioral-preferentialand information-related) are beneficial for sur-
vival. That is, as in Yan [32], we reveal that there is a unique surviving agent in
thelong-run. Moreover,weshowthatthe interestrateandthe marketpriceofrisk
behave asymptotically as those of an economy, populated solely by this surviving
agent. Lastly, to derive our results, we develop new techniques based mainly on
the Strassen’s functional law of iterated logarithm. To the best of our knowledge,
these methods have never been used in the general equilibrium literature before.
2Weassumethatthemeangrowth-ratefollowsanOrnstein-Uhlenbeckprocess.
3Thisisarealisticassumptionascorrelationsareextremelydifficulttoestimateempirically.
MarketselectionwithlearningandcatchingupwiththeJoneses 3
The conclusions and implications on natural selection are as follows. Most im-
portantly,our findings indeed confirm, to a large extent, the validity of the market
selection hypothesis.
Ina growingeconomy,the less effectivelyrisk-averse4agentis the one to survive
in the long-run. This result is consistent with previous studies (see e.g. Cvitani´c,
Jouini, Malamud, and Napp [8]). However, the impact of habit-formation on the
effective risk aversion, and thus in particular on survival, is quite novel. As it
turnsout,ifthe (standardnon-effective)riskaversioncoefficientisaboveone,then
the individual with the strongest habit will survive. Intuitively, this makes sense,
as aggressiveness in a growing economy among somewhat moderate individuals is
supposed to be a plus. On the other hand, if the (standard non-effective) level
of risk aversion is below one (i.e., individuals are relatively risk-seeking in the
classicalsense), the agent with the lowest degree of habit-formationwill dominate.
This is not surprising at all, as excess aggressiveness can cause bubbles leading to
extinction.
Someofourconclusionsconcerningtheinteractionofdiversebeliefsandsurvival
arequiteintriguing,andseemtobequitehardtopredictwithoutadelicateanalysis.
Whenagentsdifferonlyintheirbeliefsconcerningtheaveragemeangrowth-rate,
theonewiththemostaccurateforecastwilldominatethemarket,asexpected. Ifall
agentsareover-confident(or under-confident),then,again, the agentwiththe best
guess will beat the others. However, if some agents are over-confident and others
are under-confident, the situation is more complex. For instance, it may happen
thatinasituationwherethepublicsignalprovidessomerelevantinformationabout
the market, the surviving agent will be the one who (wrongly) believes that this
signal is a pure noise, whereas the agent who is significantly overconfident in the
informativenessofthe signalwillbe eliminatedfromthe economy. Furthermore,in
some cases, agents that believe in a negative correlation of the signal will survive,
while individuals who believe in a (too high) positive correlation will be extinct,
despiteanactualpositivecorrelation. SeeFigure1foranexampledescribingthese
phenomena. Eventhoughitissomewhatdebatablewhichpropertyofthepreceding
two can be considered a more rational one, we still learn that theoretically, the
market selection hypothesis is valid, at least in some modified form.
We now reviewsomerelatedworks. The mostcloselyrelatedto oursarethe pa-
pers by Yan [32] and Cvitani´c, Jouini, Malamud and Napp [8].5 Specifically, these
authors consider a special case of our model corresponding to the case when there
is no learning and agents having standard CRRA preferences without any habit
formation. In terms of modeling heterogeneous beliefs and learning, our model
4Inourmodel,theeffective riskaversiondepends onthelevelofhabit-formation(see(4.2))
5BhamraandUppal[5],Dumas[13],andWang[29]consideredthesamemodel,butwithonly
twoagenttypes andheterogeneity comingonlyfromriskaversion.
4 R.Muraviev
closely follows the one of Dumas, Kurshev, and Uppal [14] and Scheinkman and
Xiong [28], who considered a special case of our model: a two-agenteconomy with
standard CRRA utility functions, and the public signal being a pure noise, uncor-
related with the economy’s growth-rate. Chan and Kogan [9] consider a special
case of our model with homogeneous ’catching up with the Joneses’ habit levels
and a continuum of agents with heterogeneous risk aversions. Xiouros and Zapa-
tero [31] derive a closed form expression for the equilibrium state price density in
the Chan and Kogan [9] model. Cvitani´c and Malamud [10] study how long-run
risk sharing depends on the presence of multiple agents with different levels of risk
aversion. Kogan,Ross, Wang and Westerfield [21] and Cvitani´c and Malamud [11]
studytheinteractionofsurvivalandpriceimpactineconomieswhereagentsderive
utility only fromterminal consumption. Fedyk, Heyerdahl-LarsenandWalden [15]
extendthe modelof Yan[32] by allowingfor manyassets. Kogan,Ross,Wang and
Westerfield[22]study thelink betweensurvivalandpriceimpactinthepresenceof
intermediate consumption, and allow for general utilities with unbounded relative
risk aversionand a generaldividend process. Another quite significant direction of
the complete market risk sharing literature concentrates on the equilibrium effects
of heterogeneous beliefs. Bhamra and Uppal [6] derive a characterization of the
equilibriumstatepricedensitybymeansofinfinite seriesthatadmitsaclosedform
solution for specific coefficients, in a two-agent economy with diverse beliefs and
heterogenousCRRApreferences. WithCRRAagentsdifferingonlyintheirbeliefs,
the equilibrium state price density can be derived in a closed form, and thus many
equilibrium properties can be analyzedin detail. See, e.g., Basak [3, 4], Jouini and
Napp [19, 20], Jouini, Martin and Napp [18] and Xiong and Yan [30].
The paper is organized as follows. In Section 2 we introduce the model and
provide some preliminary results. Section 3 is devoted to a brief description of
the equilibrium state price density in homogeneous and heterogeneous settings. In
Section 4, we present the main result of the paper and discuss some implications.
Section5dealswithsomeauxiliaryresultsthatarecrucialfortheproofofthemain
result. In Section 6 we prove the main result. Finally, in Section 7 we establish
long-run results for the interest rate and the market price of risk. Some of the
results appearing in sections 5 and 7 are of an independent mathematical interest.
2. Preliminaries
We consider a continuous-time Arrow-Debreu economy with an infinite hori-
zon,inwhichheterogeneousagentsmaximizetheir utilityfunctions fromconsump-
tion. The uncertainty in our model is captured by a (complete) probability space
(Ω, ,P) and a continuous filtration :=( ) , with = ,Ω . We fix
F∞ F Ft t∈[0,∞) F0 {∅ }
threestandardandindependentWienerprocesses(W(i)) ,i=1,2,3,adapted
t t [0, )
∈ ∞
to the filtration . There are N different types of agents in the economy, labeled
F
MarketselectionwithlearningandcatchingupwiththeJoneses 5
by i = 1,...,N. Each agent i is equipped with a non-negative endowment process
ǫi adapted to the filtration (see (2.7)). We denote by D := N ǫi the
t t [0, ) G t i=1 t
∈ ∞
aggregate endowment process and assume that (D ) satisfies
(cid:0) (cid:1) t t [0, ) P
∈ ∞
dD
(2.1) t =µDdt+σDdW(1) , D =1,
D t t 0
t
or equivalently
t 1
(2.2) D =exp µDds (σD)2t+σDW(1) ,
t s − 2 t
(cid:18)Z0 (cid:19)
wheretheconstantσD >0representsthevolatility. Themeangrowth-rate(µD)
t t [0, )
∈ ∞
is an Ornstein-Uhlenbeck process that solves uniquely the SDE
(2.3) dµD = ξ(µD µ)dt+σµdW(2) , µD =µ,
t − t − t 0
that is
t
(2.4) µD =µ+(µ µ)e ξt+σµe ξt eξsdW(2),
t 0− − − s
Z0
where µ,µ and σµ are some real numbers and ξ > 0. The numbers µ,µ will be
0 0
referred to as the average and initial mean growth-rate,respectively.
2.1. The Financial Market. We consider a financial market that consists of
at least two long-lived securities: a risky stock (S ) and a bank account
t t [0, )
∈ ∞
(S0) . Inadditiontothis,thereareother(notexplicitlymodeled)assetsguar-
t t [0, )
∈ ∞
anteeingthatthemarketisdynamicallycomplete6for adaptedclaims(wherethe
G
filtration :=( ) is defined in (2.7)). We emphasize that this filtrationco-
G Gt t [0, )
∈ ∞
incideswiththesymmetricinformationsharedbyallagents. Thebondisinzeronet
supply andthe stock is a claim to the totalendowmentofthe economy(D )
t t [0, )
and has a net supply of one share. The risk-less bond is given by St0 = eR∈0trs∞ds,
where (r ) is the risk-free rate process. We assume a unique positive state
t t [0, )
∈ ∞
price density denoted by (M ) , that is, a positive adapted process to that
t t [0, )
∈ ∞ G
satisfies
Mt =E eRtursdsMu t ,
G
for all u>t, and h (cid:12) i
(cid:12)
∞ Mu
S =E D du ,
t u t
M G
(cid:20)Zt t (cid:21)
(cid:12)
(cid:12)
6Inthissetting,themodelcanbeimplementedbyacompletesecuritiesmarketwithaunique
statepricedensityderivedinequilibrium(asforinstanceinDuffieandHuang[12]). Morespecif-
ically, the filtration G is generated by the Brownian motion st (which is interpreted as a public
signal)andthe aggregate endowment process Dt. Nevertheless, as explained inRemark2.2, the
filtration G is also generated by the Brownian motions st and Wt(0). Thereby, the market can
be completed by adding one additional security to St. However, since the price of this security
wouldbedeterminedendogenously, onewouldhavetoverifyendogenous completeness. Thiscan
bedonebyusingthe techniques of Hugonnier, Malamudand Trubowitz[17]. Otherwise, wecan
justassumethattherearesufficientlymany(derivative) assets,completingthemarket.
6 R.Muraviev
for all t > 0. Note that our assumption excludes arbitrage opportunities in the
model. The state price density, as well as all other parameters, are to be derived
endogenously in equilibrium.
2.2. Preferences and Equilibrium. Agentiismaximizinghisintertemporalvon
Neumann-Morgensternexpected utility
sup EPi ∞e−ρitUi(cit)dt ,
(cit)t∈[0,∞) (cid:20)Z0 (cid:21)
from consumption, under the constraints that the consumption stream (c )
it t [0, )
∈ ∞
is a positive process adapted to (which is defined in (2.7)) and lies in the budget
G
set
E ∞c M dt E ∞ǫiM dt .
it t ≤ t t
(cid:20)Z0 (cid:21) (cid:20)Z0 (cid:21)
Here, EPi[] stands for the expectation with respect to the subjective probability
·
measure P of agent i. The exact form of P is specified in (2.15). We assume that
i i
all agents are represented by ’catching up with the Joneses7’ preferences:
1 cit 1−γi
U (c )= .
i it
1 γ H
− i (cid:18) it(cid:19)
The subjective ’standard of living’ index (H ) is defined through a certain
it t [0, )
∈ ∞
geometric average of the aggregate endowment process. We consider here a more
general specification for H than the one in Chan and Kogan [9]. Namely, we set
it
Hit =eβixt, for some βi 0, where
≥
t
(2.5) x =e λt x +λ eλslog(D )ds ,
t − 0 s
(cid:18) Z0 (cid:19)
or equivalently, (x ) solves the SDE
t t [0, )
∈ ∞
dx =λ(log(D ) x )dt.
t t t
−
For eachagent i, the number β measures the impact of the index x on the agent;
i t
inparticular,whenβ =0,theagentisnotinfluencedbytheindexatall. Forlarge
i
β ,theinfluenceissomewhatheavy. Incompletemarkets,theoptimalconsumption
i
stream can be easily derived as in the following statement.
Proposition 2.1. The optimal consumption stream of agent i, in a complete mar-
ket represented by a state price density (M ) , is given by
t t [0, )
∈ ∞
cit =eγρiitMt−γ1iZiγt1iHiγtiγ−i1ci0,
and
E ∞c ξ =E ∞ǫiM ,
it t t t
(cid:20)Z0 (cid:21) (cid:20)Z0 (cid:21)
7ThisparadigmofautilityfunctionwasfirstintroducedinAbel[1],andiscommonlyreferred
to in the literature as a utility with exogenous habits. This specification describes a decision
makerwhoexperiences animpactofthe’standardofliving’index.
MarketselectionwithlearningandcatchingupwiththeJoneses 7
where the density process (Z ) is given in (2.15).
it t [0, )
∈ ∞
Proof. The assertionfollows by standardduality arguments involving the first-
order conditions. (cid:3)
Finally, we introduce the notion of Arrow-Debreuequilibrium.
Definition 2.1. An equilibrium is a pair (c ) ,(M ) such that:
it t [0, ) t t [0, )
∈ ∞ ∈ ∞
a. Each process (c ) is the optimal consumption stream of agent i and
it t [0, ) (cid:0) (cid:1)
∈ ∞
(M ) is the state price density that represents the market.
t t [0, )
∈ ∞
b. The market clearing condition is satisfied:
N
(2.6) c =D ,
it t
i=1
X
for all t>0.
2.3. Diverse Beliefs and Learning. The are two processes in the economy that
are observable by all agents. The first one is the aggregate endowment process
(D ) , and the second one is a certain public signal
t t [0, )
∈ ∞
s =φW(2)+ 1 φ2W(3),
t t − t
p
forsomeφ [0,1).Thatis,thepublicsignalexhibitsanon-negativecorrelationφ
∈ ∈
[0,1) with the shock governing the mean growth-rate process. The corresponding
filtration is denoted by
(2.7) :=σ s ;u t D ;u t .
t u u
G { ≤ } { ≤ }
(cid:16) [ (cid:17)
In contrast to this, the mean growth-rate process is unobservable. That is, neither
of the agents possesses access to the data revealing the dynamics of the process
(µD) . Furthermore, agents may have diverse beliefs concerning the average
t t [0, )
∈ ∞
and initial mean growth-rate. More precisely, each agent i believes that the initial
mean growth-rate is some µ R and that the average mean growth-rate is some
0i
∈
µ R. That is to say, before filtering, agent i assigns in his mind the following
i ∈
model for µD:
t
t
(2.8) µi+(µi0−µi)e−ξt+σµe−ξt eξsdWs(2).
Z0
Furthermore, individuals may have an irrational perception of the signal. Con-
cretely, each agent i believes that the public signal (s ) has a correlation
t t [0, )
∈ ∞
φ [ 1,1) with (W(2)) , when if fact, the correlation is φ [0,1). There-
i ∈ − t t∈[0,∞) ∈
fore, under the belief of agent i, the following model is attributed to the signal
s :
t
(2.9) φ W(2)+ 1 φ2W(3).
i t − i t
q
8 R.Muraviev
We denote by Qi the measure corresponding to agent’s i th beliefs regarding the
−
models in (2.8) and (2.9), where W(1),W(2) and W(3) are independent Wiener
t t t
processesunderQi. Consequently,agentsareintheprocessoflearningandfiltering
out the dynamics of the mean growth-rate, which is deduced by using the theory
of optimal filtering.
Definition 2.2. The process
t
µD :=EQi µ +(µ µ )e ξt+σµe ξt eξsdW(2)
it i i0− i − − s Gt
(cid:20) Z0 (cid:21)
(cid:12)
is called the subjective mean growth-rate of agent i. (cid:12)
Proposition 2.2. We have
µ ξµ t 1 t ν y σµφ t
(2.10) µD = i0 + i y du+ iu iudD + i y ds ,
it yit yit Z0 iu (σD)2yit Z0 Du u yit Z0 iu u
where
1 t
(2.11) y =exp ξt+ ν ds ,
it (σD)2 is
(cid:18) Z0 (cid:19)
and the variance process
2
ν :=EQi µD EQi µD
it t − t Gt Gt
(cid:20)(cid:16) (cid:2) (cid:12) (cid:3)(cid:17) (cid:12) (cid:21)
is deterministic and given by (cid:12) (cid:12)
(2.12) ν =α (σD)2 e(αi2−αi1)t−1 ,
it i2 e(αi2−αi1)t−αi2/αi1
where
α = ξ2+(σµ/σD)2(1 φ2) ξ,
i2 − i −
q
and
α = ξ2+(σµ/σD)2(1 φ2) ξ.
i1 − − i −
q
Proof. Observe that Theorem 12.7 in Liptser and Shiryaev [24] implies that
µD satisfies the following SDE
it t [0, )
∈ ∞
(cid:0) (cid:1) ν dD
(2.13) dµD = ξ µD µ dt+ it t µDdt +σµφ ds ,
it − it − i (σD)2 (cid:18) Dt − it (cid:19) i t
(cid:0) (cid:1)
where the variance process ν is detected through the following Riccati ODE
it
1
ν = 2ξν +(σ )2 1 φ2 ν2,
i′t − it µ − i − (σD)2 it
(cid:0) (cid:1)
withν =0.Onecansolvetheaboveequationandverifythatν isgivenby(2.12).
i0 it
Now, we shall solve the SDE (2.13). By definition, we have yi′t = (ξ + (σνDit)2)yit,
MarketselectionwithlearningandcatchingupwiththeJoneses 9
and y = 1. Notice that the preceding observation combined with Ito’s formula
i0
implies that
ν dD
d y µD =ξµ y dt+ it y t +σµφ y ds ,
it it i it (σD)2 it Dt i it t
(cid:0) (cid:1)
completing the proof. (cid:3)
Remark2.1. Dumas,KurshevandUppal[14]consider thestaticversionof(2.10).
That is, the functions ν and y are substituted by the corresponding asymptotic
it it
limits. This can be justified by Lemma 5.3 of the current paper.
We denote by i=0 a fictional agent who is rational in the sense that he knows
the correct average,initial mean growth-rateand the correlationparameter φ. Let
us denote by µD :=EP µD the estimated mean growth-rate of this agent. As
0t t Gt
in Proposition 2.2, we have
(cid:2) (cid:12) (cid:3)
(cid:12)
µ ξµ t 1 t ν y σµφ t
µD = 0 + y du+ 0u 0udD + y ds ,
0t y0t y0t Z0 0u (σD)2y0t Z0 Du u y0t Z0 0u u
where y and ν are defined similarly to (2.11) and (2.12). It can be shown, as
0t 0t
in Theorem 8.1 in Liptser and Shiryaev [23], that Wt(0) =Wt(1)− 0t µD0sσ−DµDs ds is a
P Brownianmotion with respect to the filtration .
− G R
Remark2.2. Thefiltration isgeneratedbythepublicsignals andtheBrownian
t
G
motion W(0). To see this, note that
t
dD
t =µDdt+σDdW(0),
D 0t t
t
and
ν
dµD = ξ µD µ dt+ 0tdW(0)+σµφds .
0t − 0t− σD t t
(cid:0) (cid:1)
We set
µD µD
(2.14) δ := it − 0t
it σD
to be the i th agent’s error in the meangrowth-rateestimation. The dynamics of
−
(D ) from the i th agent’s perspective admit the form
t t [0, )
∈ ∞ −
dD
t =µDdt+σDdW(0),
D it it
t
where
(0) (0)
dW =dW δ dt
it t − it
10 R.Muraviev
is a Brownian motion (by Girsanov’s theorem) under the equivalent probability
measure8 P and the filtration , where
i
G
dPi t 1 t
(2.15) Z :=E =exp δ dW(0) δ2ds .
it dP Gt is s − 2 is
(cid:20) (cid:21) (cid:18)Z0 Z0 (cid:19)
(cid:12)
LetusstressthatW(0) isals(cid:12)oaQi Brownianmotionwithrespecttothefiltration
it −
.Inparticular,thisimpliesthatbyrestrictingthemeasureQitothesigma-algebra
G
generated by W(0), we get the measure Pi. Nevertheless, the measures Qi and P
it
(the physicalprobabilitymeasure)are singularonthe sigma-algebra(see (2.8) and
(2.9)) generated by the Brownian motions W(1),W(2) and W(3).
t t t
3. The Equilibrium State Price Density
In the current section we depict the structure of the equilibrium state price
density in both settings of homogeneous and heterogeneous economies.
3.1. Homogeneous Economy. Consider an economy where all agents are of the
same type i, and denote by (M ) the corresponding equilibrium state price
it t [0, )
∈ ∞
density. The homogeneity of the economy combined with the completeness of the
market allows to derive the corresponding state price density in a closed form.
Lemma 3.1. The equilibrium state price density in a market populated by one
agent of type i is given by
(3.1) Mit =e−ρitDt−γiZitHiγti−1 =
t 1 1
exp ρ +γ µD (σD)2 + δ2 ds
− i i 0s− 2 2 is ×
(cid:18) Z0 (cid:18) (cid:18) (cid:19) (cid:19) (cid:19)
t
exp (γ 1)β x + δ γ σD dW(0) .
i− i t is− i s
(cid:18) Z0 (cid:19)
(cid:0) (cid:1)
Proof. TheassertionfollowsbyusingthemarketclearingconditionandLemma
2.1. (cid:3)
We derive next the risk free-rate and the market price of risk in a homogeneous
economy.
Lemma 3.2. The risk free rate and the market price of risk in an economy popu-
lated by one agent of type i, are given respectively by
r :=ρ +γ µD 1 σD 2γ (γ +1) β (γ 1)(x log(D ))
it i i it − 2 i i − i i− t− t
and (cid:0) (cid:1)
θ :=γ σD δ .
it i it
−
8Onecancheckthattheprocess(Zit)t∈[0,∞) isatruemartingalebyverifyingNovikov’scon-
dition on a small interval and then applying a similar argument to the one used in Example 3,
page233,inLiptserandShiryaev[23].