The RAND Corporation
The Optimality of a Competitive Stock Market
Author(s): Robert C. Merton and Marti G. Subrahmanyam
Source: The Bell Journal of Economics and Management Science, Vol. 5, No. 1 (Spring, 1974),
pp. 145-170
Published by: The RAND Corporation
Stable URL: http://www.jstor.org/stable/3003096
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http://www.jstor.org
of a competitive
The optimality
stockmarket
Robert C. Merton
Professor of Finance
Sloan School of Management
Massachusetts Instituteof Technology
and
Marti G. Subrahmanyam
Instructorof Finance
Sloan School of Management
Massachusetts Instituteof Technology
The findingsof Jensen-Longand Stiglitzon the optimalityof the
stockmarketallocationhave led to a controversy
over whetherthe
sourcesof thenonoptimality
of value maximization
are noncompetitiveassumptionsabout thecapitalmarketor are inherentexternalities associatedwithuncertainty
whichdo not disappear even in a
competitivemarket.At least, for the mean-variancemodel with
constantreturnsto scale technologies,
we claim thatthe answeris
theformer,
and thatthesourcesof thenonoptimality
are nonpricetakingbehaviorbyfirms,
restrictions
on theavailabilityof technoloand restrictions
gies to firms,
on thenumberof firmsthatcan enter.
and nontatonnement
Usingbothtatonnement
processes,it is shown
that if firmsvalue maximize,then the Jensen-Longand Stiglitz
equilibriaare unstablewithrespectto the numberof firms.If the
on theavailabilityof technologies
and on thenumberof
restrictions
firmsthat can enterare relaxed, then the equilibriumwill be a
Pareto optimumin mostcases, and in no case will the aggregate
be less than the Pareto optimalamount.If
amountof investment
firmsact as price takers(with or withoutthe otherrestrictions),
thentheequilibriumis a Paretooptimum.
of the stockmarket
* In theirimportant
paperson theoptimality
allocation, Jensenand Long and Stiglitz1use a mean-variance
model of the capital marketto demonstratethat the investment
allocationby value-maximizing
firmswill not in generalbe Pareto
optimal.While both analysesare technicallycorrect,the interprefrom
mathematics
RobertC. Mertonreceivedthe B.S. in engineering
and AppliedScience (1966),
ColumbiaUniversity's
School of Engineering
of Technology
fromtheCaliforniaInstitute
theM.S. in appliedmathematics
(1967), and the Ph.D. fromthe MassachusettsInstituteof Technology
(1970). Currently,
he is conductingresearchin capital theoryunder uncertainty.
receivedthe B.A. fromthe Indian Institute
Marti G. Subrahmanyam
of Technology,Madras, in 1967, the Post GraduateDiploma in Business
Administration
from the Indian Instituteof Management,Ahmedabad,
1. Introduction
OPTIMALITY OF A
COMPETITIVE
STOCK MARKET / 145
over whetherthe
tationof theirfindingshas led to a controversy
sources of the nonoptimalityof value maximizationare noncompetitiveassumptionsabout the capital marketor are inherent
externalitiesassociated with uncertaintywhich do not disappear
even in a competitivemarket.If the answer is the former,then
theirfindingsare consistentwithcertaintytheoryanalysis,where
value or profitmaximizationby firmswith some positivedegree
of monopolypower need not lead to Pareto optimal allocations.
If the answer is the latter,then, indeed, the findingsare more
fundamental.In his commenton the Jensen-Longand Stiglitz
papers, Fama2 argues that because both analyses use noncompetitiveassumptions,theydo not answerthe questionwhetherin
the mean-variancemodel, a competitiveequilibriumis a Pareto
optimum.Fama does performthe analysisbased on whathe calls
the appropriatecompetitiveassumptionsand concludes that,due
to inherentexternalities
a competitiveequicaused by uncertainty,
libriumwill not, in general,be a Pareto optimum.
Our paper addressesthe issue of whethera competitiveequilibriumin the mean-variancemodel is a Pareto optimumor not,
and its principalconclusionis thatit is. The reason thatour conclusion differsfrom the previous analyses is not technical,but
in interpretation
of whatthe competitive
rathera difference
market
assumptionsare. Because of the apparent disagreementin the
literatureas to what the "correct"competitiveassumptionsare in
with incompletemarkets,we considera
a model of uncertainty
numberof alternativeassumptionsabout firms'beliefs and the
market structureand examine the optimalityof the resulting
equilibria.We believe that these assumptionsspan the range of
what most people would be willingto accept as descriptiveof a
"competitive"market.There are two distinctissuesto be resolved:
(1) if firmsact as "price takers"and value-maximize(the usual
competitiveassumptions),is the resultingequilibriuma Pareto
optimum?(2) if firmsdo not act as "price takers"and do valuemaximizebut the marketstructure
is such thatentryis free,is the
resultingequilibriuma Pareto optimum?
We definea firmas a "price taker"if it perceivesa horizontal
supplycurveforcapital: i.e., it believesthatthe requiredexpected
returnon a project(the "cost of capital") is unaffected
by actions
taken by the firm.As will be shownfor perfectlycorrelatedoutputs,the"pricetaker"assumptionis equivalentto the firm'staking
the aggregateamountof investment
in a projectas "fixed" since
thecost of capitaldependsonlyon the aggregateamountof investment taken by all firmsin a given project. In all three of the
aforementioned
papers, firmsdo not act as "price takers"in this
sense. Whileour definition
of price-taking
may be open to debate,
it is difficultto imagine a definitionfor a competitivemarket
R. C. MERTON AND
146 / M. G. SUBRAHMANYAM
and the Ph.D. in financefromthe Sloan School of Management,M.I.T.
His currentresearchis in capital markettheory.
Aid fromthe NationalScienceFoundationis gratefully
acknowledged.
We thankS. C. Myers and M. B. Goldman for helpfulcomments.We
especiallyappreciatethe many suggestionsby P. A. Diamond, E. Fama,
M. Jensen,and J. Long to improvean earlierdraft.
1 In [4] and [13], respectively.
2In [3].
whereentryinto that marketis restricted.Since all threepapers
restrict
entry,eitherexplicitlyin the case of Jensenand Long, and
in the case of Fama, theirfindingsdo not
Stiglitz,or implicitly,
demonstratethat a competitiveequilibrium is not a Pareto
optimum.
If new firmsare restrictedfromraisingfundsin the capital
marketon thesame termsas alreadyexistingfirms,
thenthecapital
marketdoes not satisfythe pure competitionassumptions.This
restriction
is implicitly
builtintoa model whichholds constantthe
numberof (noncolluding)firms.3
To restrict
entryintoan industry
by assumingthat all technologiesare not freelyavailable to all
firms(both existentand potential) is also a violationof the pure
can be builtintoa model
competition
assumptions.This restriction
by holding constantthe number of firms,or more subtly,by
assumingthat the correlationbetween returnson the "same"
projecttakenby different
firmsis not perfect.Since in a portfolio
context,projectsand technologies,or for that matter,assets and
firms,
are definedby theirprobability
distributions,
it is misleading
and inaccurateto referto a projectbeing consideredby two (or
more) firmsas the "same" projectif forthe same inputs,the cash
flowsfromtheprojectin each stateof natureare not the same for
each firm(i.e., ifthecash flowsfromthesame projecttakenby differentfirmsare notperfectly
correlated).For example,twoprojects
or technologieswhose cash flowsor outputsare identicallyand
independently
distributed
shouldnotbe consideredthesame project
for the same reason that an investordoes not treat two assets
whose returnshappen to be identicallyand independently
distributed as the same asset or even as membersof the same "risk
class."4
Jensenand Long are carefulto acknowledgethat eitherthe
assumptionof holdingthenumberof firmsfixedor the assumption
thatthe correlationbetweencash flowson the same projecttaken
by different
firmsis not perfectmay not be consistentwith free
entryand purecompetition.
Stiglitzrecognizes5thatby holdingthe
numberof firmsfixed,he is restricting
entry,but because he often
assumes in his analysisthat the distributions
of returnsfor the
"sameC'projectacross firmsare independent,
he is led to conclude
that even if the numberof firmsallowed to enteris expanded,
the Pareto optimalallocationis not achieved. Fama6 shows that
when the returnsfromprojectsare less than perfectlycorrelated
across firms,there are natural externalitiesin their production
3 In the usual analysisof a competitive
equilibriumunder certainty
(See, for example,[1] or [2]), it is assumedthatthereis a fixednumber
of firms,and, therefore,
entryis restrictedin the sense that we use it.
However,because it is also assumedthatfirmsare price takers,the entry
restriction
is neverbinding.Hence, the equilibriumis the same as if there
wereno such restriction.
However,whenfirmsdo not act as price takers,
restricting
the numberof firmsto any finitenumbercan affectthe equilibriumallocationand allow for non-Paretooptimalresults.
4 Risk class is definedas in Miller and Modigliani[8]: namely,two
assetsare in the same riskclass if theyare consideredperfectsubstitutes
for each otherby investors.
5 In [13].
6 In [3].
OPTIMALITY OF A
COMPETITIVE
STOCK MARKET / 147
decisions that violate the conditionsof a perfectlycompetitive
market.He thenuses the linear marketmodel whichsatisfiesthe
conditionsof perfectcompetition,to show that if the numberof
firmsis finiteand returnsacross firmsare less than perfectly
correlated,the resultingequilibriumis non-Paretooptimal.As shown
later,for the technologieswithimperfectcorrelationexaminedin
our paper, the equilibriumsolutionto his equations (12)-(14)
requiresan infinite
numberof firms.For othertechnologies,there
is some questionas to the existenceof a solutionto his equations.
In Section2, we reexaminethe Jensen-Longmodel7and show
that if firmsare price takers,then the equilibriumallocation of
investment
is Pareto optimal.It is also shownthatif firmsare not
price takersbut entryis free,thenthe equilibriumlevel of investmentwillbe no smallerthantheParetooptimalamount.In Section
3, we modifythe Jensen-Long
model to allow forreversibleinvestmentand show thatthe competitiveequilibriumis a Pareto optimum.In Section4, we discusstheJensen-Long
alternative
criterion
of social wealth maximization.Because the strictassumptionof
technologies'being freelyavailable to all firms(i.e., perfectcorrelation for the same project across firms) is not empirically
descriptive,in Section 5, we examinethe nonperfectcompetition
case of imperfect(but positive) correlationand conclude that
providednew firmscan enterand raise capital on the same terms
as preexistingfirms,the resultingequilibriumwill be a Pareto
optimum.
2. Jensen-Long
model
* In this section,we use the same model as Jensenand Long:
namely,all investorsare risk-averseand characterizetheirdecisions based on the mean and variance of end-of-periodwealth;
the marketis perfectwithall assetsinfinitely
divisible;all investors
have homogeneousexpectations;transactionscosts and taxes are
zero. Undertheseconditions,theyshow8thattheequilibriumvalue
of the firmis
Vj
-
1r
[Dj
-
XoAjmIfor all j
(1)
where
cash flow
Dj _ the expectedvalue of the total end-of-period
to the shareholdersof firmj,
r
1 + i, wherei is the one-periodrisklessrate of interest
at whicheveryinvestorcan borrowor lend,
- covariance
of thetotalcash flowsof thefirm,
'TjM
jk
D,
withDM thetotalcash flowsfromall firms,
cojk - Cov (D , Vk) forall j and k,
q-2 -Var(Dm),
VM -l
- -rVm]/-2- marketpriceper unitof risk,and
[Dm
[
V - totalmarketvalue of all firms.
k
R. C. MERTON AND
148 / M. G. SUBRAHMANYAM
In [4].
8See [4], p. 153, (1).
7
We also assumethatX is a fixednumber.This assumptionis consistentwithconstantabsoluteriskaversionon thepartof investors.
or projectwithcash flow
opportunity,
Considera new investment
returnper unitinput,p where
E(pv)-
Var ()
Cov(p,
Dj) =
0p2
p.,
of ,'is independentof whichfirm
It is assumedthatthedistribution
takesit and thatthereare stochasticconstantreturnsto scale. I.e.,
in the projectby the jth
if Ij denotesthe amountof investment
firm,thenI,p is the randomvariablecash flowto the jth firmas a
resultof takingthe projectand the randomvariablep' is independentof the choice of Ij. As Jensenand Long do, we assume that
the cash flowsfromall otherassets remainfixed,independentof
I.e., it is
made in the new opportunity.
the amountof investment
in other
assumedthatfirmscannotchangethe level of investment
projectsin responseto investmentin the new technology.Thus,
but,in addition,
assumedto be irreversible,
not onlyis investment
While
cannotbe made in the "old" technologies.9
new investment
such an assumptionis probablynot veryrealistic,the qualitative
resultsof theJensen-Longmodel are preservedwhenthisassumption is weakened to allow for reversibleinvestmentas is shown
in Section 3.
As Jensenand Long have shown,'0the Pareto optimalamount
in the new projectis"
of investment
I*w
max
Xo-2 [P
r- Xo-p]}.
(2)
Equation (2) is derivedfromthe conditionthat,at the Pareto optimallevel of investment,
p
r+Xo-M,p,
(3)
9 Thus, it is assumedthat thereis "putty"capital for the new technologyonlyand "clay" forall othertechnologies,
i.e., a typeof "short-run"
model.We believethisdescription
to be consistent
withthatin Jensenand
Long. The analysisused throughout
the paper is that of comparingequilibria (comparativestatics;see [1], Chapter 10). I.e., we compare the
alternative
equilibriaresultingfromthe introduction
of a new technology
and changesin assumptions
about firms'perceptions.
Under the alternative
interpretation
of lookingat a cross-section
of firmsfora givenequilibrium,
the distinctionbetween"putty"and-"clay" (long-run/short-run)
is not
necessary.Similarly,our notionof "new" versus"old" firmsand our definitionof "entry"suggestsa kindof dynamicswhichis not necessaryfor
examinationof the conditionssatisfiedby firmsinvesting
in the (N + 1)st
technology
fora givenequilibrium.
Nor is it necessaryto assumethatX is
a constantfor this type analysis.However,since much of our discussion
centerson the approachesto (stabilityof) the equilibria,thislatterinterpretationis of limitedusefulness.
10In [4], p. 156, (5).
11As discussedin Section 1, a projectis definedby its probability
distribution,
and hence,thecorrelation
of returns
on thesame projectacross
firmsis perfect.Hence, the aggregatelevel of investment
for a Pareto
optimumis independentof the distribution
of investment
in the project
across firms.
OPTIMALITY OF A
COMPETITIVE
STOCK MARKET / 149
ev
eV
where DMi'= Dm + I5 is the total cash flowfor all firmswhen
in the new projectis I and
the amountof (aggregate) investment
where 0(M'p -= MP + JO-p2 iS the covariancebetween'; and the
returnon thenew marketportfolio.
level
Jensenand Long'2 also show that the value-maximizing
in the projectfor the jth firmis
of investment
I
max
{,
2XO-2 [
r-
X(OMp
+ Orjp+ 1'
2)
(4)
in the projectby
where1' is the aggregateamountof investment
all firmsotherthan the jth firm.'3
Because the value-maximizinglevel of investmentdepends
upon the otherprojectsalreadytakenby the firm,expression(4)
seemsto conflictwiththenotionof riskindependenceof projects.'4
Suppose thata projectis available whichwill not affectthe distributionof cash flowsof alreadyexistingprojectsand the distribution of the project'scash flowis the same, no matterwhichfirm
takes the project,then the doctrineof risk independencestates
that (1) the marketvalues of the projectand all otherassetswill
be the same, independentof which firmtakes it; (2) the firm
can determinewhetherto take a project or not, independentof
otherassets held by the firm;i.e., it can act as if it were a new
assets.
firmwithno otherpreexisting
Actually,part (1) of the statementdoes hold forthe JensenLong model in the sense thatfora givenlevel of aggregateinvestin the project
of investment
mentin the project,the distribution
across firmsdoes not affectthe marketvalue of the projector of
other assets. The reason that part (2) does not obtain can be
tracedto theassumptionby Jensenand Long thatthefirmbelieves
that its investmentdecision will affectthe aggregateamount of
in the project. Further,throughthis belief,the firm
investment
that
decisionwill not only affectthe cost
its investment
perceives
but in addition,the cost of capital
for
the
new
project,
of capital
taken
by the firm.This is in contrast
on otherprojectspreviously
to Fama,'5 whose "reactionprinciple"requiresthata firmact as
a price-takerin the sense of takingthe aggregatelevel of investmentin the projectas fixed.
To see this,we use equation (1) to rewritethe formulafor
the value of the jth firmexplicitlyas a functionof the amountof
in the new technology,Ij, and the aggregate
its own investment
1, as
by all firmsin the new technology,
amountof investment
hVe(O;
Ir)
Vj(wj;
l)
+ Ijg(I),
(S)
where
12In [4], p. 161, (18).
Equation (4) is valid onlyforperfectcorrelationof returnson the
projectacrossfirmswhichhas been assumed.
14 See S. C. Myers [9] and L. D. Schall [12] for a discussionand
of projects.
proofsof riskindependence
15 In [3], pp. 513-514, assumption(C6).
13
R. C. MERTON AND
150 / M. G. SUBRAHMANYAM
Vj(O;I)
-
1 [Dj
-
X(-MJj+ l-jp)]
(6a)
and
r [P - X(O- + Io2)]
g(l)
(6b)
Vj(0; I) is the marketvalue of all otherassets owned by the jth
firm,giventhatthe aggregateinvestment
in the projectis 1, and
g(l) is the value per unit input of investmentin the new technology (the same for all firms).Ijg(l) is the value of firmj's
ownershipin thenew technologyand it is the amountthatit would
receiveif it sold its part of the projectto anotherfirmor "spins
it off"as a separatefirm.The total marketvalue of the project,
of the distribution
of investment
E Ijg(l) = Ig(l), is independent
i
in
across firms.The requiredexpectedreturnper unit investment
theprojectequals p/g(l), the same forall firms.Hence the "cost
of capital" for the new project depends only on the aggregate
amountof investment.
Thus, fora givencost of capital,the value
of the project (as a specifiedlevel of investment)is the same,
independentof whichfirmtakes it.
It will throwlighton the analysisto computethe value-maximizingsolutionusing the representation
for firmvalue in (5).
Namely,to determinethe optimallevel of investmentto take in
a new project,the firmpicks Ij l=j so as to maximize[Vj(Ij;l)
- IJ],whichis the marketvalue of the firmnet of the cost of
inputs.For Ij
conl-j to be an interiorsolution,the first-order
dition is
d[Vj(lj: 1) -Ij,] =&aV, _ 01"+,I
dl =
forIj = I*j,
(7)
where
b1i
1
-r
-[p-
X(op + I-J2)]] independentof Ij,
(8)
and
aVj
01
A [C + Ij,o2]
r
(9)
If we assume,as Jensenand Long do, thatthe firmtakes investment by other firmsin the project as fixed (i.e., I'
I - Ij
fixed) then,dIl/dIj 1, and the value-maximizing
solutionl*j is
as definedin (4). If, instead,Fama's "reactionprinciple"is used,
thendl/dlj 0 and theoptimalsolutionforeach firmis thesame.
r - X(o-MP + lO-p2)] < 0, the firm(s)will invest
Namely,if [nothing;if [p- r - X(o-MP+ lO-p2)] > 0, each firmwill be willing to investan indefiniteamount; if [p - r - X(oMP + lO-p2)]
0, each firmis indifferent
to the amountit invests.Obviously,
the only equilibriumsolutionin thiscase is
OPTIMALITY
OF A
COMPETITIVE
STOCK MARKET / 151
I =Max {
, Xo2 [P- r-Xa-MP]}
(10)
as definedin
whichis the Pareto optimumamountof investment
(2). As is usual forstandardcompetitivemodels,whilethe aggregate amountof investment
in the projectis determinate,
the scale
and hence,thenumberof firmsinvestingin theprojectis not. Note
that from (5), each firmbelieves that its marketvalue expands
in proportionto the amountof investment
it takes. Therefore,if
firmsbelieve theycannot affectthe aggregateamount of investmentin the project (or equivalently,its cost of capital), thenthe
numberof firmsnecessaryto achieve a Pareto optimalallocation
is finiteand can be as few as one.'6
We now turnto the Jensen-Longcase wherefirmsact to maximize marketvalue under the assumptionthat investmentby all
otherfirms,1', is fixed.Unlike the price-takercase, the resulting
equilibriumallocation will depend on the numberof firmsthat
can investin the new technology.In particular,Jensenand Long
have shownthatif thereis a finitenumberof such firms,thenthe
equilibriumallocationwill not be an optimum,althoughit will be
an optimumwithan infinitenumberof firms.
However,as we shall argue,the assumptionof a finitenumber
of firmsseemssomewhatinconsistent
withthe notionof freeentry
and infinitely-divisible
technologieswithconstantreturnsto scale.
To allow foreasier comparisonwiththe Jensen-Longanalysisand
in keepingwiththe spiritof our analysiswhichis one of comparing equilibria,we will call thosefirmswhichhave previousinvestmentsin the othertechnologies,"old" (or preexisting)firmsand
those firmswhichhave no previousinvestments
but can investin
the new technology,"new" firms.If, in equilibrium,a new firm
takesa positiveamountof investment,
it is said to have "entered,"
and if it has not,thenit is called a "potential"new firm.
For the moment,let us make the Jensen-Longassumptionthat
thereare a finitenumber(N + K) of firmswherethe firstN(j =
1, 2, . . . , N) firmsare old firmsand the last K(j = N + 1,
N + K) are new firms.From the value-maximization
assumption,it must be that in equilibrium,there can be no firm
whichbelieves that it can raise its marketvalue by a (feasible)
change in its investment.'7
Assume an equilibriumwithrespectto
these assumptionswith aggregateinvestment,1. Now, consider
whatwouldhappenifwe allowedone more (potential) new firmthe (K + 1)st. From equation (4), its value-maximizing
investmentdemand would be
I*N+K+l
Max{O,
2Ao 2
[
r
-rX
(-Mp+ lO-p2)]}
(11)
R. C. MERTON AND
152 / M. G. SUBRAHMANYAM
16 Hence, as is the case for the usual competitive
analysis (see note
3), if firmsare price takers,thenthe restriction
of entryby postulatinga
fixednumberof firmshas no effecton the investment
allocationnor on
the Pareto optimality
of the allocation.
17 This equilibrium
conditionholds for all the models in the paper.
However,whatconstitutes
a "feasible"changein investment
variesaccording to the particularmodel beingdiscussed.
Now, if I*N+K+1 > 0, thenthe firmwould enterif it could. Why
does it not?Formally,because it is assumednot to exist.However,
if it is also assumed that the constant-returns-to-scale
technology
is freelyavailable and thatboth old and new firmscan raise funds
in the capital marketon the same terms,we see no apparentreason (other than implicitrestrictions
to entryhidden from the
model's explicitassumptions)why such a firmwould not be created by a profit-seeking
entrepreneur.
If, on the otherhand, I*N+?K1
0 (as would be the case,
forexample,if otherfirmsacted as price takers),thenthe (potential) new firmwouldhave no incentiveto enterand theequilibrium
would be unaffected
by expandingthe numberof firmswhichcan
enter.Thus, at least with respectto stability,the assumptionof
a fixednumberof firmsdoes not matterif the restriction
is not
bindingin equilibriumas is the case forthe standardcompetitive
model.
Of course, our oppositionto restricting
the numberof firms
in a competitivemodel is an argumentover an assumption,and
hence, we cannot "prove" that it is inconsistent
withfree entry.
The methodof argumentis stabilityof the equilibriumwith respect to the numberof firms,and we have shown that if in an
equilibrium,profitopportunitiesexist for additionalfirmsif the
numberof firmsallowed were expanded,thensuch an equilibrium
is not stable.
Since the assumptionof a fixednumberof firmsseems to conflictwith the notion of free entry,we replace it with the more
compatibleassumptionthatthereis potentially
an indefinitely
large
numberof firms.From the value-maximization
assumption,a necessaryconditionforequilibriumis thatno potentialnew firmwill
have incentiveto enter.From (11), thisconditionimpliesthatin
the aggregateamountof investment
mustsatisfy[ptequilibrium,
r -X(-cIP + lorP2)] < 0, or in otherwords,the conditionfor no
entryby an additionalnew firmis that1 ) l,. Thus, the market
cannotbe in equilibriumunless aggregateinvestment
is at least as
large as the Pareto optimalamount.This resultis in sharp contrastto the previousanalyseswhereit was deduced thatthe equilibriumamount of investmentwas less than the Pareto optimal
amount.
Using two different
dynamicprocesses of adjustmenttoward
equilibrium,we find the distributionof investmentacross firms
which makes the equilibriumstable. Having done so, we then
answerthe questionunderwhat conditionswill these stable equilibriabe Pareto optimal?
To determinestability,
we firstconsidera sequential-type
analysis wherefirmsarrivein the marketplace in a randomfashion
and make an investment
decisionaccordingto (4). However,once
theycommitthemselvestheycannot reversethe decision (disinvest) althoughtheycan "come back" to the marketplace to make
more investment.On completionof this analysis,we then considerthemoreconventionaltatonnementprocessforreachingequilibrium.For simplicity,
we assume thatold firmsdo not investin
-
OPTIMALITY OF A
COMPETITIVE
STOCK MARKET / 153
the project (i.e., Ij
0, j
1, . . . , N) althoughthe analysis
would be virtuallyidenticalprovidedthat I1Nl*j <
The analysisforthe case whereboth old and new firmsinvest
in the new project (again, providedX1N1* ? I*") is performed
in the Appendix.
The processforgettingto an equilibriumis as follows:a new
firm(chosen at random) arrivesat the marketplace and is given
the following"macro" information(in addition to the "micro"
information
whichit alreadyhas: namely,the distribution
forp):
r, X, and the probabilitydistribution
for the marketportfolioat
the "time" of the firm'sarrival."Time" is countedhere as incrementingby one with each arrivalof a new firmin the market
place. Thus, if thecurrentnew firmmakingan investment
decision
is the kthsuch firmto do so, thenthe distribution
forthe market
cash flowat timek is denotedby the randomvariable Dm(k)
in the
Dm + Il -IN+tp whereIN+t is the amountof investment
projectchosen by the tth (new) firmto arrive (1'-IIN+t
is the
aggregateamount of investmentin the project by all previously
arrived firms) and DM(1)
Dm. By choosing the information
providedin this way, the new firmneed not know what investmentsotherfirmshave taken whichis more in the spiritof decentralizeddecisionsby firmsand an impersonalmarketplace for
raisingor issuingcapital. Since it is assumed that each firmacts
so as to maximizemarketvalue, the optimal investmentby the
kth (new) firmwill be
=
I*N+k
Max{,
r-Xo-mp(k)]},
2AO2 [P-
(12)
where o-m(p
Cov[DI (k) , opmp
(k)
+ (Ylk llN? + t)Op2. Formally,(12) and (4) are thesame equations.However,(12) shows
that directknowledgeof otherfirms'investmentdecisionsis not
necessary.The processof arrivalsby firmscontinuesuntilno new
firmwould want to enterin which case the marketwill then be
in equilibrium.
To determinethe amount of investmenttaken by the kth
new firm,we have that AkoMIp(k)
orMp(k + 1)
-mp (k)
I*
l*N+kOp2
2and therefore,Akl*N+k
-
-
I *N+k+1
~~~~~~~~~~~~~1
I*N+k, or
2
2IN+k
/
k
I
-
whereI*N+1 = Max {
r[oAmp]
p
(13)
N+1,
Further,the
takenby thefirstk firms,
aggregateamountofinvestment
I(k), is
I(k)
R. C. MERTON AND
154 / M. G. SUBRAHMANYAM
18 See
Max {O,
the Appendix.
2
p
[p
r-Acrp}
=L1 (
2
)
(14)
Il*W.
By thisprocess,it actuallytakes an infinitenumberof new firms
(k - oc) to reach equilibriumwhich also correspondsto the
Pareto optimal amount of investment.We did not assume that
manynew firmsenter,but deduced it as the resultof free entry
exist,firms
and the assumptionthatas long as profitopportunities
will enter.Note, however,that the convergenceto equilibriumis
rapid: forexample,afterentryby ten firms,the aggregateamount
of investment
takenis 99.9 percentof the Pareto optimalamount.
Because of the nontatonnement
natureof the approachto equilibeffectsif tradesactuallytake
rium,thereare wealthdistributional
place at the "false" interimprices.However,giventhe assumption
of a constantprice of risk,we have that the equilibriumwill be
unaffectedby these transfersof initial wealth, and it will be a
Pareto optimum.
of the process could be a seAn alternativeinterpretation
equilibriawhere in each
quence of (unstable) Jensen-Long-type
subsequentequilibrium,investmenttaken in each previousequiin
libriumis treatedas "clay" in the same manneras investments
othertechnologiesare in theirmodel. It should also be noted that
the dynamicprocess describedcorrespondsto the staticbehavior
of a perfectlydiscriminating
monopolistwhose behavior,in the
standardtheoryunder certainty,will produce a Pareto optimal
allocation.
We now look at the equilibriumsolutionachieved when the
process,but where
approach is by the conventionaltatonnement
we still allow entryby new firmsas long as profitopportunities
exist.We can writethe equilibriumvalue-maximizing
level of investmentfor the jth firmfrom (4)
I*j
Max{O,
A[P-
Max {O,(I*w. I*)
-.
r
_
(rMp+ ijp+I*.p2)]}
0jp/0p2),
(15)
where I* is the equilibriumaggregateamount of investmentin
theprojectand I*w is the Pareto optimalamountof investment
as
definedin (2).
To determinetheequilibrium,
thefirms
we startby partitioning
into threegroups: group 1 containsall firmssuch that o-jp> 0;
group2 containsall firmssuch thato-jp< 0; group 3 containsall
firmssuch that o-jp= 0. Groups 1 and 2 contain only old firms
whilegroup3 containsall new firmsand in addition,any old firms
whose o-jp 0. For notationalsimplicity
and withoutloss of generality,we assume that there are no old firmsin group 3. By
renumbering
firmsif necessary,let the firmsin group 1 be numbered 1, 2, . . . , N1; the firmsin group 2 be numberedN1 + 1,
N; the firmsin group 3 to be numberedN + 1, . .
N + K (where K is the numberof new firmswhichenter).
By inspectionof (15), all new firmswill choose the same
I N+K.
= I*N+2
.
amount of investment,Inew = I'N+1
*. .
Further,for l* to be an equilibriumamountof aggregateinvest-
OPTIMALITY
OF A
COMPETITIVE
STOCK MARKET / 155
mentwhen free entryis allowed, theremustbe no incentivefor
any firmto change its chosen level of investment.In particular,
considerthe (K + 1)st (potential) new firm:from (15), it will
not enteronly if 1P is such that
p-~rAX(.prMP +
I*P2)
<O.
(16)
Hence, from(15) and (16), we have thattheequilibriumamount
of investment
by all firmsin groups 1 will be zero, i.e., 1*1
1*2
*. .
0N1 O. Further,strictinequalityin (16) can only
obtainif =new 0 and no new firmsenter.
Equation (16) is equivalentto the conditionthat 1* > I*,
withl* > I*, only if =*new 0. As in our analysisof the nontatonnementapproach to equilibrium,we have found that free
entryensuresthat the equilibriumamount of investmentin the
projectwill neverbe less than the Pareto optimalamount.A sufficientcondition
forI* = I*w is that- XN (o-P/CP2) ?I*W
N1+1
If the strictinequalityholds,thennew firmswill enterand KlInew
-I*w
_
[
EN,N+1
(jP/P2)
Inspectionof the Jensen-Long'9formulafor the equilibrium
amountof investment
whenprojectreturnsare perfectly
correlated
across firmsshowsthatif any new firmsenter,an infinitenumber
will (i.e., K= oo).
Hence, as in the previousnontatonnement
analysis,if at least
one new firmenters,thenthe numberof new firmsenteringwill
be infinite,
and the stable equilibriumwill be a Pareto optimum.
Again, we emphasize that it was not assumed a priori that the
number of firmsenteringwas infinite,but deduced from the
assumptionsof value-maximizing
by firms,free entry,and infinitelydivisibleassets. The convergenceto the Pareto optimumis
less rapid for the tatonnement
process than in the previouscase
because entryby a new firminduces previouslyentered firms
to contracttheir (contingent)investment,
i.e., 1 > I*/I*w > K/
(K+ 1).
Actually,theconvergenceto theParetooptimumdoes noteven
requirethatall firmsare value-maximizers.
Essentially,theanalysis
has shownthatas long as the aggregateamountof investment
is
less than the Pareto optimalamountand as long as entryis not
restricted,
profitopportunities
will exist and new firmswill enter.
In the precedinganalysis,it was shown that if, at least, one
new firmenters,thenthe equilibriumis a Pareto optimum.However, it is possible that under our definitionof free entry,the
can be largerthanthe Pareto
equilibriumallocationof investment
optimumamount.Further,it is difficult
to findan additionalmechanismwhichwillensurethatin all cases theequilibriumis a Pareto
optimum.One mechanismwhich would substantiallyreduce the
forsuch "overinvestment"
possibility
would be to assumethatnew
firms,in additionto lookingforprofitopportunities
for directinvestmentin the new technology,also considerthe possibilityof
takingover old firmsand rearranging
theirinvestment
in the new
technologyso as to make a profit.
To see the effectof this assumption,we examine the case
R. C. MERTON AND
156 / M. G. SUBRAHMANYAM
l9In [4], p. 163, (27).
where the project's returnsare sufficiently
negativelycorrelated
withsome of the currently
existingassets so that the investment
taken by firmsin group 2 is greaterthan the Pareto optimum
considerthe case wheregroup 2 contains
amount.For simplicity,
onlyone firm,the Nth one, and the value maximizingP ( l,)
> I*,,.whichimpliesthat-CoNp > Iw*O-p2.
The change in marketvalue of the kthfirm,k 1, 2,
in
N -1, goingfroman "equilibrium"withaggregateinvestment
the projectof I*N to one withaggregateinvestment
equal to l%w
(with1; = 0 in eithercase) would be
Vk(O;I*W)
-
Vk(0;1
N)
I*w]Ckp > 0,
r[I*N
r
k
1, ... ,N
-1
(17)
and the change in value of the Nth firmif it divesteditselfof
in theprojectand (throughtheentryof new firms)
theinvestment
in the projectwere PI , would be
the aggregateinvestment
VN(O;I*W)
+ I*N -
VN(I*N;I*N)
-1[I*N
r
+ I*Nc12]
I*W][o-Np
-
[I*N
-
I*W][Np
(18)
+ I*WOP2] < 0.
Now, if the totalchange in the marketvalue of all existingfirms
were positiveby goingfromlIPNto I*w, thena profitopportunity
would exist for a "new" firmto buy out all existingfirmsat the
values associated with the I*N- "equilibrium"and then to divest
the Nth firmof its investment
in the project and sell everything
out at the new values associatedwiththe I
equilibrium.
By summing(17) fromk 1 to (N - 1) and adding it to
(18), we have that the conditionfor the sum of the change in
the marketvalues of all firmsto be positiveis
IV-1
p
r+ 2
k7p
> 0.
(19)
Hence, providedthat (19) is satisfiedand that tenderoffersor
mergersare allowed,the competitiveequilibrium(where firmsact
so as to maximizetheirmarketvalue) will be a Pareto optimum.
* In the previoussection,we assumed as did Jensenand Long
thatinvestment
could only take place in the new technologyand
that investmentin "old" technologieswas held fixed.Both the
investmentdemand by firmsand the Pareto optimal amount of
investment
were computedon the basis of thisassumption.While
it maybe realisticto assumethatinvestment
alreadytakenis nonreversible(except possibly throughsome limited depreciation),
thereis no apparentreason to assume thatinvestment
in previous
technologiesmightnot be expanded.
In this section,we go to the otherextremeand assume that
all investment
is reversible,i.e. thereare no fixedcosts,and cap-
3. A modified
Jensen-Long
model
OPTIMALITY OF A
COMPETITIVE
STOCK MARKET / 157
ital can be freelytransferred
fromone technologyto another.20
An
equivalentintertemporal
assumptionwould be that capital depreciates completelyin a single period. We choose the completely
reversibleassumptionover the more realistic"mixed" reversibleirreversible
assumptiondescribedabove, not only for the obvious
analyticalsimplifications,
but also to preservethe one-periodnature of the problem which would not be adequate if complete
depreciationwere not allowed.21
Because we are only interestedin the Pareto optimal allocation and because we maintainthe assumptionof a fixedprice of
risk X, we can assume withoutfurtherloss of generalitya single
representative
investorwithconstantabsoluterisk aversion.
To determinethe Pareto optimalallocation,we eliminatefirms
fromthe economyand derivethe optimalinvestment
allocationto
each technology
by the investor.One can view the problemas that
faced by an economywith a single input,wheat (seed), which
can be planted in various ways so as to produce a finaloutput,
wheat (grain) at the end of the period. The probabilitydistributionsforoutputfromeach technologyare given and the assumption of stochasticconstantreturnsto scale is maintained.The investorhas some giveninitialendowmentof wheatand mustchoose
an optimalallocationacross the varioustechnologies.
First, considerthe allocation when there are N technologies
withexpectedreturnper unitinput,ai, and covariancesof returns
per unit input vij, i, j = 1, 2, . . ., N. The optimal allocation
of investment
will be22
1
/j
A
\N
/jUk(ak-
r),j
1, . . ., N,
(20)
where,uj is the ijth entryof the inverseof the N X N variancecovariancematrixof returns.
Now considerhow the allocationwill change if a new (N +
1)st technologyis introducedwithexpectedreturnper unitinput,
a -N+l =, andwithcovariances
ofreturns
vjN+1
Vj, j 1, ....
N, (VN+1 N+1
The
new
optimal
allocation
of investment
o-p2).
will be
If *j
1
A2
V
N+1
jk(ak-
r), j
1,.. .,N + 1,
(21)
where,u'tj is the ijth entryof the inverseof the (new) (N + 1)
matrixof returns.
X (N + 1) variance-covariance
To investigatethe changesin the optimalallocationof investmentcaused by the introduction
of the new technology,
we derive
some relationships
betweenthe entriesin the inverseof the "old"
R. C. MERTON AND
158 / M. G. SUBRAHMANYAM
20 In contrastto the analysisin Section2 (see note 9), the assumptionsof thissectioncorrespondto a "long-run"model and the comparative
staticsis betweena N-technologies"putty"equilibriumallocation and a
(N + 1)-technologies"putty"one.
21 See Merton [7] for a discussionof an intertemporal
capital asset
pricingmodel wheresuppliesare takenas variable.
22 See Merton [6], p. 1863. To avoid bothersomeinequalities,it is
productiveso that Pi > O,
assumedthat all technologiesare sufficiently
i=1
. N..
.
- [,uj] and theinverseof
matrix,Q
N X N variance-covariance
the new (N + 1) X (N + 1) variance-covariance
matrix,F-I
[,u'ij].Define the N X N matrix,y; the N-vectorsB and E; and
the scalar D by
B'
[V1p . . ., VATP]
and
IN11~~B
rr['P
[E'
p2]
D]
where1K is the K X K identitymatrix.Then, y, E, and D must
satisfy
fly + BE'
'N
B'E + OP2D
'1
fQE+ BD
0
O.
B'y+O-2E'
(22)
By manipulatingthe conditionsin (22), we have that
D =
y
E =
(23a)
(23b)
B'fl-:1B]-1
-
[o2
=f1?
+ DfV-1BB'-:L
DQ1:-B.
(23C)
.,
[E', D], we have
/uN?1 N+1]
Noting that [,ut'1+L, .
in the
fromequation (21) thatthe optimalamountof investment
new technologyis
1X+1j}
[D(av+l
=A
Z
r
-r
[aN+?
7lpujN+l(ai
,=1 (
r-
ON+I-
[aN+iAN+
-A
r) +
-
(agj- r)]
=_viP/4j)
E
vip
r)]
(z;
from (23c)
1Ij(aj-r))]
=,TvpI*, from(20).
-
(24)
To compare (24) with the Pareto optimal level (2) in the
previous section,we note that, as definedin Section 2, o0Mp*. Hence, rewriting(24) in termsof the notationof the
,1
previoussection,
D
(25)
Xo-Mp]
Ap[P-r-
IN1
(Do- 2)I*W as definedin (2).
Hence, unlessDO-p2 - 1, the Pareto optimalamountof investment
in the new technologywhen investmentin the old technologies
in the
can be changedwill not be the same as when investment
old technologiesis held fixed.Noting that B'h-1B > 0 because
V-1is positivedefinite,we have from(23a) that
Do-p [12
-(B'Q-1B)
[1
-
/p2]
-1
N N
-
(I
1
1
VkpVjpFtjk)/p2]
?
1,
(26)
OPTIMALITY OF A
COMPETITIVE
STOCK MARKET / 159
with strictequalityholdingif and only if Vk - 0 for k - 1, 2,
N. Thus, the Pareto optimalamountof investmentin the
new technologywill be largerin the unconstrainedcase than in
theconstrainedone unlessthenew technologyis uncorrelatedwith
all other technologies.23
(Note that while 0MIp C0is necessary,
it is not sufficient
to ensurethatDo-P2- 1.)
Althoughnot directlyrelated to the question of whethera
competitiveequilibriumis a Pareto optimum,an importantdifference betweenthe Pareto optimumsolutionof thissectionand the
one foundby Jensenand Long is that to computeF*+ requires
knowledgenot onlyof the distribution
forthe new technologyand
itsrelationship
to thedistribution
of the (prenewtechnology)marstrucket portfolio,but also of the completevariance-covariance
ture for the new technologyand all the othertechnologies(i.e.,
unlikein equation (2), whereone could computethe Pareto optimumwithonly information
about the projectand the aggregate
marketportfolio,
to computetheParetooptimumin equation (25),
forall other
one would have to knowthecompletemicro-structure
assets). The only statementthatcan be made withoutsuch informationis thatif [p - r - XoA4P]> 0, thensome positiveamount
r - XMP]/XOP2should be
of investment
at least as large as [taken in the new technology.
In a similarfashion,we can deduce the changesin investment
in the other technologiesto be
I*
-I
(
/N
vkpkjI)*
j-
1,2,.. ., N. (27)
As in computinglF*1, a knowledgeof the completemicro-structureforall assets is requiredto computeF'.* Equation (27) also
demonstrates
thatthe assumptionmade by Jensenand Long that
the risklessasset is available in unlimitedamountsis not a sufficient conditionto ensure that investment
in the old technologies
will not change in responseto the introductionof a new technology.
The importantquestionsto be answeredare: (1) if we now
introducethe value-maximizing
firmsand allow for free entry,
can the resultingequilibriumbe a Pareto optimum?(2) How
much information
is requiredby the firmsso that the answerto
(1 ) is yes?
Note thatbecause capital is freelymobile among technologies,
thereis no need to distinguish
betweennew and old firms.Hence,
about
considerthe kth firmwhich only has "micro" information
the jth technology.Suppose it acts so as to maximizeits market
value using the same assumptionas in the previous section:
namely,it takes investment
by all otherfirmsas given. Then, it
will choose its level of investment
Ijk so as to
R. C. MERTON AND
160 / M. G. SUBRAHMANYAM
23 While,at firstglance,it may not be clear why the Pareto optimal
in the new technologywill always be largerin the
amountof investment
"long-run"than in the "short-run"(independentof the covariancestructure), this resultis just a special case of the Le ChatelierPrincipleas
system.See Samuelson[10] for a disapplied to a constrained-maximum
cussionof the Le ChatelierPrincipleas applied in economics.
Max
{Vkj-kMax
(cx.-r)
Max -(aej
(jk
Max j-[(aj
~x(ZhV~N+1
_ X
LlVtj
-r)
-
r)
-
X(crmj(k)
3
+ Ijk Vj2
+
Ijk vj2)]
(28)
by all otherfirmsin the ith
whereI, is the aggregateinvestment
technologyat the "time" of firmk's decision and crmj(k) is the
covarianceof thereturn(per dollar) investedin thejthtechnology
by otherfirmsat the "time"
withthe cash flowof all investments
of firmk's decision(o-Mj(k) is definedin an analogousfashionto
omp(k) in the previoussection). The solutionto (28) is
N? 1
[,
j9k= Max
2Xvr2
-Max [,
2XvA2
(29)
I
(
{Ijvrj
r-Xom(k)}].
{a-
available
Note thatthe firmuses preciselythe same information
to firmsin Section2. (I.e., it did notneed to knowthemicrocharacteristicsof otherfirmsor technologies.)Because we allow for
freeentryand exit,the marketcan only be in equilibriumif the
entryintoany technology
1hare suchthatanynew firmconsidering
no firmwhichhas
will have no incentiveto do so and similarly,
willhave any incenin anytechnology
takeninvestment
previously
tiveto exit.From (29), theseconditionsrequirethatthe equilibriuminvestments,
Pi, satisfy
N+1
a-
r AX(
I*.
A
I*ivtij)_ O,j
1, 2, ...,N+
1 (30)
or that
N+1
',j(aj -r),i =1, 2, . .
1
N ,+ 1,
(31)
fromequation
whichare theParetooptimalamountsof investment
(21).
To completethe analysis,we go to the other extremeand
assume thateach firmis aware of all the available technologies,
athoughit does not know what otherfirmsare doing. Then, the
levels, (1k,... , lN+), so as to
kthfirmwill choose its investment
I
f
Max {Vk
-
P~N+1
IZ1
{N+1
x(Z/%
-
k
Max
kIjlmj(k) +
1r
N+1
N+1
N+1
[
-
ZZJ
N+1
Ijk(aj - r)
1
Ijkj,kv.)
}
(32)
whichcan be solvedto yield
OPTIMALITY OF A
COMPETITIVE
STOCK MARKET / 161
Ik
Max
L{
Ikij(aj
-r -Xoj(k))}],
i-1i...,N+
1.
(33)
As before,equilibriumwould requirethata new firmwould have
no incentiveto enterwhichfrom(33) requiresthat
fN+1
O=
u-. a!,
/
/
N+1
j -X r At-
\
kVkj
,i=-
,...
N + 1,
(34)
N+1
wherewe have substituted
for o-Mj(k). Notingthat
-O of i# k, and - 1 if i = k, we can rewrite(34) as
?
N+1
1
U'ij(aj
-
r)
-
XI*j, i - 1, . . . ,N+
U'jjvk
1, (35)
whichis the Pareto optimumsolutionas writtenin (21) or (31).
with respect
Thus, even if each firmhas limitedinformation
to othertechnologiesand what otherfirmsare doing, as long as
entryis freeand firmsvalue-maximize,
the resulting(competitive)
equilibriumis a Pareto optimum.
4. On the social
wealth maximizing
criterion
* In an analysisparallel to the ones deducing the amount of
investment
associatedwithPareto optimalityand value maximization of firmvalue, Jensenand Long examine the social wealth
maximizingcriterionand derive the amount of investmentthat
maximizesthe marketvalue of all firms.The solution can be
writtenas
1*S
Max 0, 2XOj2 [(
-r)-22Xo-Mp.
(36)
Comparing(36) with (4), the social wealthmaximizingsolution
is the same as fora firmwhichis the only one withaccess to the
new projectwhen in additionthat firmowns all the otherassets
in theeconomy(i.e., when -jp cr=,,p). Thus, if (4) is thesolution
for a monopolistwho has sole rightto the new technology,but
does not controlall technologies,then one mightinterpret(36)
as the solutionfor a "super" monopolistwho not only has sole
but also owns all othertechnologies.
rightto the new technology,
then (4)
If one interprets
such an entityto be the government,
could be termedthe "private"monopolysolutionand (.36) termed
the "public" monopolysolution.
Viewed in thislight,it is not surprising
that Jensenand Long
diffoundthatthe social wealthmaximizinglevels of investment
feredfromthe Pareto optimalamount.24
A naturalquestionto raise is, how "nonoptimal"is the social
wealthmaximizingsolution?To answerthisquestion,we consider
a representative
investorwitha mean-varianceutilityfunctionof
R. C. MERTON AND
162 / M. G. SUBRAHMANYAM
24 See, forexample,Long [5], wherehe showsthatwealthmaximization
withmonopolydoes not lead to a Pareto optimum.
terminalwealth,H[E(W), Var (W)], withH1 = OH/IOE(W) > 0
and H2 = OH/OVar(W) < 0. For analyticalsimplicity,suppose
thatthereis a single (composite) riskyasset withexpectedendof-periodcash flow,la and varianceof the cash flow,12o2, where
in the riskyasset. Then, let
I is the amountof investment
ac eaI/V
op2 =_2j2/
2
WO + V -I
WO(i)
(37)
whereV is the marketvalue of the riskyasset; ap is the expected
returnper dollarinvestedin the riskyasset; op2 iS the varianceof
thatreturn;WO(I) is theinitialwealthof theinvestoras a function
takenin theriskyasset.For a given1,
of theamountof investment
the investor'sdemand forthe riskyasset in his optimalportfolio
can be writtenas
D = (a.
-
(38)
r)/crp2X
(2H2/H1) and is takento be constant.
whereX E(W) = Dp(a - r) + rWO(I) and var(W) By definition,
for Dp from(38), we have that for
Dp2o-2. Hence, substituting
theoptimalportfolio
+ rWO(I)
-(ap - r)2/XoAp2
E(W)
(39a)
and
(-ap r) 2/X2-p2.
Var(W)
(39b)
To determinethe effecton the investor'swelfareof a change
in 1, we firstnote that
dH
- dE(W)
X dVar(W) 1
dl -H1
L-dl
2
Second,the marketvalue of the riskyasset is Vand hence,
1
[a-
dV
-d
dI
r
(40)
dl
2XIo-2].
I[a-
Xlo-2]/r
(41)
Combining(37), (39), and (41), we have that,along optimal
portfolios,
dE( W)
(a
dI
r)
-
(42a)
and
dVar( W)
dl
2Io2.
(42b)
from(42) into (40) we have that
Substituting
dH
dl
H [a-
which is positivefor I < (a
the Pareto optimallevel (a
-
r-
lo-2],
(43)
r) /XO-2 reachinga maximumat
r)/Xo-2. Therefore,investorwel-
OPTIMALITY OF A
COMPETITIVE
STOCK MARKET / 163
fare increases monotonicallyas investmentincreases until the
Paretooptimallevel of investment
is reached.
Under what condition will investorsbe worse off with a
"public" monopolythan witha "private"one? Inspectionof (4)
and (36) showsthatthe conditionis (Tjp< 0lp. How likelyis it
forthisconditionto obtain?If It is the aggregateamountof investmentby firmi, then OMlvp
thenO-Mp >
,N
lvp. Hence, if
,N
Z1Iivp > ?,
i#j
iftheviparegenerally
positiveandofapO-jp.Further,
proximatelythe same size, then 0-Mp> (Tjpsince Z1 I > Ij for
a large economy.Therefore,one mightexpect that the "public"
in the new technologysubstanmonopolistwill restrictinvestment
tiallymore than would the "private"monopolist.Thus, we conclude that the social wealthmaximizingrule has littlevalue as a
social welfarecriterion.
5. Imperfect
competition
R. C. MERTON AND
164 / M. G. SUBRAHMANYAM
* In the precedingsections,it was establishedthat given the
assumptionsrequiredfor perfectcompetition,the resultingequilibriumis a Pareto optimumwhenfirmsfollowthe value-maximization rule. One of the competitiveassumptionsis that the same
techniquesforproductionare available to all firmswhichimplies
that the returnsacross firmson the same project are perfectly
correlated.Because of thisstrongassumption,one mightreasonably
questionwhetherthe derivedoptimalityimplicationsof the competitivemodel are descriptiveof actual capital markets.To at least
partiallyanswerthisquestion,we now examinethe case of a noncompetitivestockmarketwhereit is assumedthatthe same technologyis not availableto all firms.However,we retainthe assumption of freeentryto the capital market(i.e., that new firmscan
raise capital on the same termsas old firms).Further,we assume
that all firmsfollow the value-maximization
rule, and therefore,
new firmswill enteras long as profitopportunities
exist.
The model we use is the same one used by Jensenand Long
in theirsectionthreealthoughit is also descriptiveof the models
used by Stiglitzand Fama. It is assumedthatthereexistsa set of
new projectswhose joint distribution
for per unit cash flowsis
and that each firmhas access to only one project.
symmetric,25
I.e., the expected cash flow per unit input for each firm'snew
project is p, the same for all firms,and similarly,the per unit
variance o-p2 is the same for all firms.The correlationcoefficient
for the cash flowsbetweenany two new projectsis /3,the same
forall firms.(Clearly,the /3- 1 case correspondsto all the new
projectsbeingthe same projectin whichcase the analysisreduces
to thatof the previoussections.)
Whiletheseassumptionsabout thejointprobability
distribution
may seem somewhatspecialized,theyare descriptiveof a (new)
industryor productwherethe basic technologyis freelyavailable
25 The randomvariablesX1, ....
joint probaX. have a symmetric
functionin its
bilitydistribution
P(x1, . . .x,X) if P ( ) is a symmetric
arguments.
See Samuelson[11] for furtherdiscussion.
firmsis
but wherethe correlationbetweenthe outputsof different
in each firm's
not perfectdue to (somewhatrandom) differences
applicationof the technologyor managerialtalents,etc.26
investment
Jensenand Long27deducethatthevalue-maximizing
forthe jth firmcan be writtenas
I*
1
Max {O, 2Xo-2
[
r -(crMp + crjp+ 160(p2I)]
1
- r -X(3rmp+
13)Xo 2 [P5
Max {O, (2
0jp
+ PO;,21)
(44)
takenby all firmsin the set of
whereI is the aggregateinvestment
Since we allow free access to the
new projectsand F'- I -P.
capitalmarketsby new firmsto raise capital to investin theirown
versionsof the new technology,a necessaryconditionfor equilibriumis thatI be such thatno additionalnew, value-maximizing
firmsshouldhave incentiveto enter.Hence, from(44) fora new
mustbe28
firm,the equilibriumaggregateinvestment
I*=
Max {O, fl-2
[P
r-
XO-MP]
(45)
If O-jp> 0 forj
1,... ,N, thenfrom(44) and (45), the equiby all old firmswill be zero, and by
libriumamountof investment
takenby each new
the equilibriumlevel of investment
symmetry,
firmwillbe thesame and equal to 1*/K whereK is theequilibrium
numberof new firms.29
By substituting
=I/K, k
1,.
I*N+k
K, into (44) and solving,we have thatforK new firms,the aggrewill be
gate amountof investment
I= Max{,
(
[2+(K
~KA
1)]X2
P-
r-Xo-MP]}
(46)
By comparing(46) withequilibriumcondition(45), we have that
thenumberof new firmsthatenteris infinite(K -oo ) as was the
case in the ,8 1 analysisof previoussections.
When the numberof new firmsis restrictedto be no larger
thanK, Jensenand Long30derivedthe (constrained)Pareto optito be
mal solutionfor aggregateinvestment
26 Althoughtheydo not explicitly
stateit, we believethatJensenand
Long had this basic descriptiveidea-in mind when theyformulatedthe
model,particularly,
since theyreferto the set of projectsas the "same"
projectalthoughstrictly
for /3< 1, it is not.
27In [4], p. 161, (18).
28 For [p - r - Xap] > 0, (45) is only valid for 0 </3 < 1. Inspection of (44) shows that for /3= 0, the only "equilibrium"is l* = oo,
althoughP* is finite.While for /3< 0, l* and J* are infinite.
However,
the /3< 0 cases are empirically
not relevant,particularly
whenthe amount
of investment
becomesunbounded.
29 The case where
ajp < 0 for some old firmscan be handledin a
similarfashionto the analysisin Section2.
30In [4], p. 164, (31).
OPTIMALITY OF A
COMPETITIVE
STOCK MARKET / 165
Iw()
Mal0
[1 + 8(K + N - 1)]Acp2
r
oJup
(47)
and the optimalamountof investment
for each firmto be
I,kj
(K +K)
(K +N)
1...
N, . . .,
(48)
N + K.
Therefore,the unconstrainedPareto optimumwill be (for 0 <
A3?11)
limr (K) = Max {O,
=l*
8
from(45).
2 (p -
r - XO-p)
}
(49)
Hence, for0 < /3< 1, theequilibriumamountof aggregateinvestmentwill equal the Pareto optimalamountas long as entryto the
capital marketis unrestricted.
Further,from (48) and the symof investment
theequilibriumdistribution
metryof theequilibrium,
across firmswill be Pareto optimal.3'
Since the model of this sectionis essentiallythe same as the
ones used by Fama and Stiglitz,why did they deduce different
results?Fama32 deduces fromhis model that unless returnsare
correlatedacrossfirms,
the marginalratesof substitutions
perfectly
for investorswill not equal the marginalrate of transformation
for the economy as a whole, and therefore,the "competitive"
equilibriumis not a Pareto optimum.In comparinghis analysis
withJensenand Long's, he goes on to state,33"They [Jensenand
Long] interpret
theresultas due in partto theexistenceof monopolisticcompetitionin the riskyindustry.
But thatexplanationis not
relevanthere since the analysisin thissectionhas assumedperfect
competition.The resultis thatthe competitiveequilibriumis not
Pareto optimal." Fama is mistakenin both his criticismof the
Jensen-Longinterpretationand his interpretationof his own
model's results.Throughoutour paper, it has been stressedthat
of the competitiveassumptionsimplythat (1)
strictinterpretation
returnson the same projectacross firmsmustbe perfectlycorrelated and (2) thenumberof enteringfirmscannotbe a priorifixed
withoutimplyingrestrictedentryto the capital market. While
Fama's modeldoes notviolatethesecondcondition,it does require
thatthe numberof firmsbe finiteif the non-Paretooptimalresult
is to obtain. However, withoutrestrictionson entry,we have
shownthatthe equilibriumnumberof firmsis infinite.If he had
applied this additionalconditionto his equation (21), he would
R. C. MERTON AND
166 / M. G. SUBRAHMANYAM
31 Jensenand Long also derivedthisresultas theylet the numberof
firmstendto infinity.
Further,theydo state that as the numberof firms
becomelarge,the resulting
solutionis a competitive
equilibrium.
However,
theydid not emphasizethatvalue-maximization
by firmsplus unrestricted
entryto the capital marketmake thisthe only stable equilibriumsolution.
32In [3], p. 524, (21).
33 Fama [3], p. 525.
have foundthat the resultingequilibriumwas a Pareto optimum
providedthatthe correlationof returnsbetweenfirmsis positive.
Stiglitzrecognizesthat he is implicitlyrestricting
entryby
holdingthe numberof firmsfixed,but he concludes,34"Extensions
of the model,allowing,for instance,for freeentryof firms,reinforce the resultsreportedhere. In that case, not only will the
in the riskyindustriesbe too small,but
totallevel of investments
also the numberof firmswill not be optimal.The economywill
be operatingbelow its mean variance frontier(allowing for a
variablenumberof firms)even when the firmsare independent."
The reasons that Stiglitz'conclusionsdifferfrom ours even
whenhe allowsforfreeentryis thathe devotesmostof his analysis
to thecase wherereturnsacrossfirmsare independent(i.e. /3 0).
0 case. For
Therefore,we reexamineour analysis for the e
simplicity,considerthe case where there are no old firms(i.e.
N
0 and o-,p = 0) and > r. From (44), the equilibrium
by the jth new firmiS35
amountof investment
I
2Xo
(
r),
(50)
whichis half the Pareto optimalamount.Note, however,that as
the
long as p > r, new firmswill continueto enter,and therefore,
Further,unnumberof new firmsthatenterwill become infinite.
like in the /3> 0 case, each firmwill investa finiteamount,and
demandedwill become
hence,the aggregateamountof investment
infinite
whichimpliesthatno equilibriumexistsunderthe hypothesized conditions.
assumption
One source of the nonexistenceis the simplifying
that there exist unlimitedamounts of risklessborrowingat the
fixedrate r. While this assumptionmay be reasonablewhen the
aggregateamount of investmentis finite,it is absurd when the
investment
demand at that rate is infinite.Therefore,one would
expectthatr would tend towardp in the limit.To see why,note
that if Vj Ij (which is requiredin equilibriumfor no further
entry),thenthe returnper dollar investedin the resultingmarket
portfoliowill have expectedvalue and varianceop2/K whereK
is the numberof new firms.Clearly,as K
cc, the Law of Large
Numberstakes over and the marketportfoliodominatesthe riskless asset.On theotherhand,ifwe requirethat lVi/Vj r to avoid
thisdominance,thenVj > Ij, and as K -> cc, the marketvalue of
assets becomes infinite.Thus, for Pf- 0, and fixed r, no equilibriumexistsunderour definition
of entry,and thereis no stable
withrespectto the numberof firms.
Stiglitz-equilibrium
Alternatively,
we could rule out the ,B< 0 cases as simply
unrealisticwhen combinedwithentry.While thereis no natural
restriction
on the signof 0-l,p and /3
could reasonablybe less than
one due to slightlydifferent
applicationsof the same technology,
to imaginethatthe returnson firmsenteringthe
etc.,it is difficult
same industrycould be distributedsuch that /8< 0, particularly
underthe assumptionthatentranceby one firmdoes not affectthe
distribution
of cash flowfor anotherfirm.
34 Stiglitz[13], p.
55.
35 See Jensenand Long [4], p. 168.
OPTIMALITY OF A
COMPETITIVE
STOCK MARKET / 167
While it could also be argued thathavingan infinitenumber
of firmsenteris equally unrealistic,this resultfolloweddirectly
fromthe assumptionof infinitely-divisible
assets.Clearly,if investmentwere "lumpy,"the numberof firmsenteringwould be finite.
Further,if (as we believe) our nontatonnement
approach to stabilityused in Section2 is moredescriptiveof realitythanthe classical tatonnementprocess, then, even if a finite(and not very
large) numberof firmsenter,the discrepancybetweenthe aggretakenand thetheoreticalPareto optimal
gate amountof investment
amountwill be quite small.
6. Conclusion
* The cornerstonesof our analysisare the assumptionsthat as
long as firms(entrepreneurs)are value-maximizersand profit
opportunities
exist,thenif theycan enter,theywill enter,and in
a competitivemarket,entryis not restricted.
We have shownthat
if firmsact as price takers,the social welfareand value-maximizationcriteriaare consistent.If firmsdo not act as price takers
is suchthatentryis free,thenthe resulting
butthemarketstructure
allocationwill be no less than the Pareto
equilibriuminvestment
optimumamount. Further,this resultstill obtains for imperfect
competitionprovided that access to the capital marketsis free
and thatsimilartechnologiesare available to all firmsenteringan
industry.
Our paper shouldnotbe interpreted
as a criticismof theJensen
and Long, Stiglitz,and Fama articles,but ratheras a clarification
of whethertheirresultsapply to a competitiveor "near" competitivemarket.By using stabilityanalysis,we have shown that
theirequilibriaare not stablewithrespectto the numberof firms,
and we have questionedthe Stiglitzand Fama definitions
of "price
taker."Hence, withoutexplicitadditionalassumptionsof barriers
to entry,we also question the assumptionof a fixednumberof
firmsby Jensenand Long and Stiglitz,and of a finitenumberof
firmsby Fama. However, theirworksdo throwlighton the importantproblems associated with noncompetitivemarketswith
restrictedentry.
Appendix
R. C. MERTON AND
168 / M. G. SUBRAHMANYAM
* Considerthe case of sequentialinvestment
decisionmakingby
firms(both old and new) thatarrivein the marketplace in a random fashion.In general,the equilibriumdistribution
of investment
acrossfirmsand theequilibriumtotalinvestment
in theprojectwill
depend on the order in which firmsarrivein the marketplace.
However,independentof the ordering,the equilibriumtotalinvestment in the project will never be less than the Pareto optimal
amount.Moreover,the equilibriumtotal investment
in the project
will equal the Pareto optimalamountforany sequence of arrivals
witha partialsequencewhichincludesall old firmswitha negative
covariancewiththe project (i.e., o-jp> 0) such that that at that
in theprojectis less than
pointin thesequence,thetotalinvestment
or equal to the Pareto optimum.
From (4), theoptimalinvestment
forthejthfirmto arriveis
Max {O, 2XA-2
I*
-
Max
r-X(o-MP(J)?o-JP)]}
[P5
f0,1/2
I*tZi)
Cov(DM(j), p')
-
-,
Al
p2 }
j-1
where OMp(i)
lp
?
(Z
I* )O-2.
t=l
Since an equilibriumobtainsonly when no firmupon arrival
wouldinvestin theproject,we have from(Al) thattheequilibrium
amountof totalinvestment
cannotbe less thanthe Pareto optimal
amount. Otherwise,a new firmwould have incentiveto enter.
Inspectionof (Al) showsthatthelatera particularfirmarrives
in a given sequence,the smallerthe amountof investment
taken
in the projectby thatfirm.It is straightforward
to show thatthe
particularsequence of firmarrivalswhich produces the largest
amountof equilibriumtotal investment
in the projectis the one
wherefirmsarrivein the orderof nondecreasing
covarianceof the
if
projectwithits own previousprojects.I.e., if (by renumbering
necessary)firmsare labeled by the orderin whichtheyarrivefor
the maximal-investment
sequence,then0rlp< CT2p< .** <? Op <
.... If thereare no old firmswithnegativeown covarianceswith
theproject,thenin themaximal-investment
sequence,thefirst(and
in theprojectare new firms.
only) firmsto take positiveinvestment
But, thisis the case examinedin the text.Therefore,if thereare
no old firmswithnegativeown covariances,thenthe equilibrium
quantityof investmentwill be the Pareto optimal amountindependentof the orderof arrivals.
If thereare m firmswithnegativeown covariancesand if 1(m)
forthe firstm arrivals
denotesthe aggregateamountof investment
in themaximalinvestment
sequence,then,from(Al), we have that
1(m)
[1 - (1/2)J]I*w-
2I
[
J
(l/2)J*rjj,
(A2)
whereJ is the index of the last firmto undertakepositiveinvestmentin theproject(J < m). If 1(m) : 1*w,thenI(m) is a stable
equilibriumamountof investment.
If I(m) < I*w, thennew firms
will enter,and the equilibriumamountof investment
will be I*W.
Thus,from(A2), a sufficient
conditionfortheequilibriumamount
of investment
to equal the Pareto optimal amountfor everysequence of arrivalsis that
1
J
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