Subadditive probabilities and portfolio inertia

\

t!~

162

SUBADDITIVE PROBABILITIES-AND PORTFOLIL 1:NERTIA

I

Ma~~0 Henrique S1"nsen Sirgio Ribeiro da Costa ~er1ang

-SUBADDITIVE PROBABILITIES AND PORTFOLIO INERTIA

Mario Henrique Sirronsen *

Sérgio Ribeiro da Costa iverlang*

1) Introduction

Why small changes in relative asset prices do not necessarily lead to an immediate change in individual portfolios is an intriguing question in the theory of financiaI decisions. Portfolio rigidities may obviously be caused by transaction costs. Corner equilibria, which may emerge when short sales are ruled out can ~e another reason for inertia. Yet, in many empirical cases, a more poweY ···...l1 source ~:::- friction appear.·" to be responsible for t , ~' ."

lack of respon~s of ~~~ntities to ~rices.

Subaddit-:"Je' probabilitü':> provide the proper anrl'~" ~"é!.l

a subadditive probability space ~~G; ,~; p) is a 'non-void set

(universe), ~n algebra ~ úf subsets of n(measurable event~·. and 2

rE:'01 valued fUL·~tion p defined for every element of::l, such that:

i) p (0) = O

ii) P (n) = 1

k k

P (n~l An) ~ _n=l L p(A ) _n for every finite collection of

iii)

measurable and disjoint events.

.2.

Subadditive Probabilities have been studied from the probabilistic viewpoint by Dempster(1967) and Shafer(1976) •

A random variable X in a subadditive probability space is a real valued measurable function defined for every point w € ~.

From the point of view of decision twory, Schmeidler (1982,1984), Gilboa(1987) and Gilboa and Schmeidler(1986) analyse subadditive probabilities. They describe a mix of pure risk and pure uncertainty according to Frank Knight's classical distinction

(Knight(1921)). It means, in fact, incomplete information about the true probabilities. AlI that is known about the true probability p' (B) of event B is that i t is greater or equal than p(B). Since

BUBc

=

Q, ',~"'~i:!re BC str.ads for the complt<"üent of B, p(B)+p(Bc) ~l,

be~ause of Siyi:6. '3uL~,r'1di t i vi ty. As

information on the ~':Y''..lp' :):.,'!)ar:!.~_::'t:y :" B ~_;:- that r\r:)~p' (B)~l-p(Bc).

'rho ., ::s::-.:..:. 0f. l.n~ert" . .Lr~_.i of eveü~ :' can be mCdsured by l-p (B) -~ ~:ci

.

measure of uncertainty aversion (,..~-. ~t:!gree of uncertainty) •

An t.-:-sy way to create a subaddi t i ve probabili ty space is to take a family (J,d,Pj)

universe and the sane

of probability spaces, for the same algebra of measurable

different probability functions; then take:

p(B)

=

inf

j

{p. (B) }

J

.3 .

Conversely, one may easily prove that every subadditive probabitity space can be generated by a family of probability

functions as above, as shown by Gilboa and Schmeidler(1986) •

The attractiveness of the concept of subadditive probabilities is that i t possibily provides the best possible

description for what is behind the widespread notion of subjective probabilities in the theory of financiaI decisions. Probability theory gained its scientific prestige because phycisists, insurance companies and card players realized that the law of large numbers works in the real world. Now, the whole power of the theory was based on the existence of objective probabilities. Subjective

probabilities where invented to translate uncertainty into a language

·':,'~.:;:2 the theot"y of choi.ce of involving risk could be easily used, an"l. +".hiR '·as the bas',: for the developrr.·~lt of the theorv _{0~ fi~a- ..:::'al}

translation of un('s":"i:"')~"_<1 ':'i1tn ri.~\ 1-:)' t.i1e USE" \''':: additive subject.ive

under very stringent cCDQicions. Sn!::'j;:;~tive probabilities suggest a lack of empjrical information that, at best, allows the individual t0 ~stimate the probability of an event within a ~Artain

interval, and not as a precise real nurnber. Now, this means describing subjective probabilities not by a conventional, but by a subadditive probability space, where the true probability of an event B lies in the interval [O<B), l-p (BcQ.

Incidentally, the extreme cases of pure risk and pure uncertainty can also be describErl by particular subaddi tive probability spaces. The case of pure risk, of course, corresponds to a

except for the universe and the void set, the probability of any event B" may lie in any point of the closed

fO;

lJ intervalo

The crucial point is that optimization under pure risk and under pure uncertainty follows different cri teria both incidentally explored in the von Neumann-Morgenstern seminal work on game theory (Von Neumann and Morgenstern(1947)). Under pure

.4 .

risk, the individual should maximize his expected utility given his budget constraint. Under pure uncertainty, there is no other

possible advice but prudence, that leads to a maximin choice: the portfolio should be selected so as to maximize the individual's wealth in the worst possible state of nature.

A convenient definition of expected value EX of a random variable X ~,.. a subadài t i ve probabi li ty :-"pace unifies the two

The trick is 1:0 defi:-,p' F.': <"3 r : k .lOW-:"~ ~ T"~'"""'J.Lble r ..<y ... ected value of

X ID 3;"' ~,:J"'.~:".J..ve pror:...:r'-:'.Lity Spá~_ :-vlapatible vlith the

availah1r-informati0 .... 0 •• r--_übabilities:

BA -" ~ nl

P.

J

E.X J

where EjX is the expected value of X for an additive probability function consistent with p. Equation (1) means solving the

uncertainty issue by the most prudent rule: since the true expected value of X is uncertain, let us take its infimum.

Pl = 0,2 P 2 = 0,2 P 3 = 0,2 P12= 0,5 P

13= 0,6 P 2 3= 0,6 P123= 1

and that the values of the random variable in the three states are

Indicating by pi, P2' P3 the true probabilities of the state of nature (pi + P2 + P

3

=

1), the true expected value of X is .5.

pi Xl+

P2

X2+ P3 X3

=

pl + Sp: . ~

l.J).

SinCl'~ _{pi, P 2 , p) are uncertain,}

the J~;=~t pos~101e value f~r EX J.~ 0~t~in~d ~y taking P

2 as ]0~ as pC'->plble, namely, P

2 =0,2; then, p) as lúw as poss':'~Jp. oi.'?2n T"'~< which yields p'

=

P __

n'

= (\

li. It rema:i.ns ::-... l-y ;l:<.e j?ossibi l~ t j -':Jr

3

::J

~ L .

EX =

°

I 4

x

1 +

°

,:2 x!)

+

O : 4 x 3 = ~ ,6

The example above indicates ho~ the expected value of a random variable in a subadditive probability space should be computcd by the safest leveI criterium: given the available information,

states of nature with high realizations of the random variable should be wighted as less as possible. A general formula can be provided in terms of distribution functions. Given a random variable X, for every real number

w

let us define:

• G •

F(U) is the maximum possible probality of the event (X >U). Then, compute EX according to the usual formula:

EX

=

I O F(u)du + lO co (l-F(u))dU (2 )

-co

This the formula of Choquet's(1955) integral. The general relation between the two ways of defining the expected value of a random variable is decribed in Gilboa and Schmeidler(1986) .

Some important propositions about expected values in subadditive probability spaces are listed below:

Proposition 1: If X and Y are random variables in a subadditive probability space, then:

:;..: :x

+ Y) ~ EX -:

.si

(3 )

P~~Jf: According to (1),

F ('\~.L '.' ~ - ~

-

,- .:;. r .. I , , . \ ..:. _iil.i: .,

.

y -! ., ....,)

~inf E v

₊

.; .d _E .Y =EX +EY

_,-"'4..1- - I \ >J r. ,

1-'- j p. J J T" - ,i J p.

J

.J

..

J;

QED.

Propositiof'! _2_: It X ;,s a random var':'~ .. .'Jle in a subadditive probability space, c a positive constant; then:

E(cX) = cEX

Proof: E(cX) =inf{E.cX} =inf{cE.X}=c inf{E.X}=cEX

p. J p. J p. J

• 7 •

Proposition 3: rf X and Y are random variables in a subadditive probability space and if X dominates stochastically Y, i.e,

x

(w) '?Y (w) for every w E r2, then EX

?

EY.

Proof: We first note that E.X ?E.Y for any additive probability J J

function p .• Then use (1).

J QED.

Proposition 4: rf X is a random variable is a subadditive probability space, then:

E(X) + E(-X) ~ O

Proof: Since EO =0, the result follows immediately from proposition

1. QED •

.I':le imp ::!-: , r oi' t!1is proposl tiün is that i t sets D0t.>:

~'r-:ç ',~P?çr l,·'.cr ·.J.'· ... nd anr ·_ne lower upper bound of the true expected

value ot Á: E X, tbat IIlUSt be such Pj

"::.: .... ú_ .t,;

... -;

(-'-'.~ _{_, - v . ~'.}

,

definitlo:l, is the ~pn~~ ~v~~~ Lüuna of the tr~2 eXp2ct~~ v~l~~

-E(-X) =inf E (-X) =-inf -(E X) =+ sup E X

Pj Pj Pj Pj Pj Pj

That is to say that the true expected value of X lies in the uncertainty internaI EX::; E X ~ -E (-X) •

Pj

.8.

Proposition 5: Let X be a random variable in a subadditive

probabili ty space i R (x) a real vaI ued increasing and differentiable function of the real variable Xi then, if:

~ indicating a real variable, F(~) is differentiable for every ~

f

O. For ~

=

O the right side derivative is F~(O)

=

RI (O)EX, the left side derivative F~(O)

=

-H I (O)E(-X).

Proof: Let us first assume ~ > O. In this case H(~X) is a increasing function of X, except for ~

=

O, where i t becomes a constant. Hence, if for two additive probability functions E X < E X, then

Pi Pj

E R (~X) < E R(~IX)

.

In short, alI calculations _-~-~ be made Pi p. _J

according tr \~"-'+;nn

-.

, "" _{"the scüne aQ('i tivp p'!:'0hability ~-; }iac..:::: •}

Ir--~-"'-- --- ... ~

h~..:. ::'::tf',' ~ C'"""t De -:"r"..;ated dt', a linear operator and, as a consequence:

For ~ < O i t suffices to make ~X ,= ÀY,where À and

y

=

-X.

Proposition 6: Let p I

=

(Pi, P

_{2, ... ,}

P~) be an n-dimensional random vector in a subadditive probability space; y

=

(YI' Y2' ... , Yn) a

. t ' Rn

pOln ln ;

concave function of the real variabIe W. Then:

G (y)

=

EU (P I • y)

.9 .

Proof: Lef y I and y" i!!llicate two points in Rn and a real

number such that O ~ ~~ 1, since U(W) is concave and PI.y linear:

U(P'.((l-~)y' +~y"» ~ (l-~)U(P'.y') +~U(P~y")

Hence, by proposition 3 (stochastic dominance) :

G((l-~)y' +~y")=EU(PI.((l-~)yl +~y")~ E((l-l1)U(P~y') +~U(PI.y")

Now, as a result os propositions 2 and 3:

E ( (1-_~ ) U ( P.' Y I ) + _~ U (P .1 Y 11 ) ) > ( 1-~ ) EU ( P .1 Y I ) + ~ E U ( P .1 Y " )

=

(1-~ ) G (y I ) + ~ G (y 11 )

.10.

2) How subadditivity may lead to inertia

Before we proceed with the general discussion of portfolio choice, let us clarify why subadditive probabilities may lead to inertia. The key point is that the expected value operator E is no longer Gateaux differentiable. As a result, eveh if the von

Neumann-Morgenstern utility function is differentiable, the portfolio indifference curves may be kinked.

As an example let us assume that a risk neutra I individual is to distribute his wealth between two assets, whose unit present prices are P

1 and P2. His initial endowment are quantities xl' x2 of these assets, which is to say that his initial wealth is measured 11" }': ~- P I xl +P 2x2' These assets can be purchased or sold without

transacti.:!l costs, ir-.'~l.ying that the i~J~ividual can choose anv

p 1 Y I + P 2Y 2 ~ w :: ' ] X I + P 2 x 2 l 4 ) To guarantee the exiFtel"r"'''' ~;. an equilibrium, we shall r :,,-,

prolLiin t S'.'J.L C bctl.eS, ad/!i.!l':] to tne buâget constrain"t (4) i...:'lt:!

inequalities:

Given these constraints the individual, who is non satiable and risk neutral, will select his portfolio so as to maximize the expected value E(YIPi+Y2P2) of his fature wealth, Pi'P

₂

indicating the randorn future asset prices.

.11.

We now assume that the individual is uncertain about the probability function in the event space. It may be either PI' P2 or a convex combination of both. If i t is P

I , the expected future asset prices are EIPi, EIPi. If i t is P2' these expected values change to E2Pi, E2Pi. To make the discussion interesting we shall assume that:

>

>

What is the expecte:1 'lélue of the individual 's future (6 )

wealth with a portfolio (Yl'Y2)? All available information is that i t may be either ylEIPi +y:E₁P

2'

ylE2Pi + Y2E2Pi or some convex combination of t:-':.::se ·",;::i..ues. The ".'-::Aimin rulp ~s to choose (Yl_'~-2) 1n ~~d budget constraint area so as to m~~1m1~~·

The portfolio indifference curves are. very ea&y.tO draw since the.l' are hc,7'0(-iJ.etical, as shm:l • in figure 1. In region I,

> E 2P

i -

EIPi

EIPi - E2 P i

In region 11, _{y 1E2P i} +Y2 E 2 P

:i

> y1E1Pi +Y2E1P:i, which is to saythat:

.12.

with E(y P' +y P') =y E p' +y E p' and with indifference curve slopes such that:

dy 2

=

Inequa1ities (6) imp1y, since expected asset prices shou1d be non negative:

E :::-'

1. i.

E p'

1 2

>

.·'\ich is to say that indifference curVl':.<O: 2rf' E- :-.·:::2r:-;r L~ )'00'_".1 1":'

Now 1et us discuss the prob1em of portfo1io irerti :,\

conc1usion is that as lOi1S as the presé ·~t asset prices P 1 and P 2 are such that:

> >

the optimurn portfolio wil1 be the point of the budget frointcr AG. P₁ (Yl-x₁) +P2 (Y2- x 2) = O in the borderline of regions I and 11, namely, where:

---~

i I

i

-.13.

If the initial endowment of the individual is already on this ON straight line, a small change in present asset prices, which means a rotation of his budget constraint with the fixed point N, will not change the individual optimum portfolio, as long as inequalities (7) are fulfilled. Out of these inequalities, the individual will concentrate his portfolio in a single asset, moving to a corner equilibrium, since short sales are prohibited.

Y2 Region 11

A

\

,

\/

; /

,

'"

\

,\

\

I

--

"- .c\.er;ion 1

I

/ _.--...:...,-.

/

~

V

u

,..., ,J y

1 Figure 1

Why a portfolio along the straight line ON is inertial, as opposed to chose in region I or region 11, deserves both

proportions, ylElPi +Y2 E l P 2 =ylE2Pi +Y2E2P2' namely, its future expected value is no longer uncertain. The general case is much

.14.

more complicated. Section 4 explores the case of pure uncertainty.

How subadditive probabilities can lead to portfolio inertia was shown for a particular case by Dow and Werlang(1989) . An individual must distribute his wealth W between two assets: money, which is certain, riskless but that has no yieldi and a risky and uncertain stock the present price of which is indicated by P. The individual is non satiable, and risk averse. Short sales are unrestricted, so that his budgetconstraint is given by:

y, afi~ ~_ in~i~?~~~~ che quantities 0~ mnnpv ~n:1 nf thc Li3ky ~~set .

.L Lo

Since utility is ê.'1 increa'::'~""''::' L.r...:tir~-, v I ".e:ath h.1Ç" individual vlill

\..0 maxi~ize:

\ - . . . . - ... , . ~I­

'. -'"~ J _. . =

w-p,·

choosing. Y2 so as

~I ~L.'

~ .. r.~re p I is th,"' random future price of the stock. NCNv', U(\oJ+Y2(P'-P» is an increasing iunction of pI-P. Hence, according to proposition 5, F(Y2) is differentiable for every Y2

t

O, but may be non

differentiable for y 2

=

O. For y 2 = O, the right sid and Ieft sido derivatives are:

F ~ ( O )

=

U I (W) (EP' -P)

F~(Ol

=

-UI (W) E(P'-P)=-U' (W) (P + E(-P ' »

---~---~

.15.

namely, if and only if:

EP' ~ P ~ -E (-P' ) ( 8)

Inertia occurs if the present stock price P lies in this intervalo The individual will neither buy nor sell short the risky asset, concentrating alI his wealth on rnoney. The reason is easy to understand: the individual is uncertain whether the true expected price of the stock in the future will be higher or lower than its present price.

Outside the interval (8) the individual will either buy

,-,1.: sell sho ... ; 30me quaT)-::i ty of the stock> Now inertia is no longe)'""

pc~sible, ~i~~~ ~ iy.~ is differentia~'e

.!.

figure ~

~. __ __ ' r \

.16.

3) Portfolio optimization

We can now present the general portfolio selection problem with subadditive probabilities. A non satiable and risk averse

individual has a von Neumann-Morgenstern utility function U(W),

namely U is an increasing and strictly concave function of wealth W. There are n assets, the initial prices of which are described by the n-dimensional vector P. The individual is initially endowed with quantities x E R n of these assets, so that his initial wealth is given by W

=

P-x. Assets can be purchased and sold without transaction costs. Moreover, short sales are unrestricted. This is to say that any y E Rn represents a feasible portfolio as long as:

P.y ~ P.x (Q)

Fut-o.lre ?8set -o~ I.'-C': ,".X.::; i

r.>

~atr

,í

'.Jy th',: .l-dimensional

feas~b~c p0~:~0j;~ y so as to maximtze oi:" his

future wealth:

F(y)

=

EU(P'.y)

Formally, the problem appears the same of portfolio choice with additive probabilities. This is due to the definition of expected value of a random variable with subadditive probabilities. The difference now is that even if the utility function is

differentiable, the expected utility F(y) may not be. The

hypothesis os risk aversion, however, avoids most of the CanpliCéltiOns

of non differentiability. In fact, according to proposition 6, F(y) is a concave function of y, defined in alI Rn, since shorts sales

.17.

a derivative in every possible direction.

Let us first describe the geometrical solution to the .problem. Since the individual is non satiable, the portfolio

equilibrium yO' assuming i t exists, will lie in the budget frontier P-y

=

P-x. Let us indicate by:

B(yO) is the set of portfolio combinations with expected utility higher than yO' Obviously yO is an equilibrium portfolio if and

only if any point of B(yO) violates the individual's budget constraint. Moreover, since F(y) is concave, B(yO) is a convex subset of Rn•

These observations are summarized in:

Proposi tior_ -;: i\ ~.:::;- ·1;-:~ _.Jo-:tEol.i..o Y

O lsoptimal if and only

.Lr

';:'f'c r,;.!c::.:jet ~(,mt""~~·_ P. (v- .... I =0. is a supporting hyperplane of

Now, rhr ~~oposition above admits two b~~~~des:

a) The supporting hype~91ane O! B(yO) at yO is uLi'ue (figure 3). In this case, any normal of the suppc,:::: ting hyperplêi.r~,=

must be proportional to the present asset price vector P. This is the case where F(y) is differentiable at yO. This is the non

inertial case: any change in relative present asset prices must displace the portfolio equilibrium;

·18.

as a normal to one of these multiple supporting hyperplanes. This is the inertial case: within certain limits, a change the portfolio composition.

y 2

\

\

\

\

\

... _...

--I _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ~---~

I---~----~

Yl Yl

Figure 3 Figure 4

To provide an analytical solutior. to t~1E' l-iL"-}h~,,.~ill j-,:

~Vilcave fU!lcti0n :::-:"1 t--=, use the theorem of K ... i~n and Tucker.

l,et G

ex)

~e a real concav:> function defined in an open . hb h d f Rn t f R~"u~O) .

ne1g or 00 o y € ; U a vec or o T The directional

derivative G' (y,u), namely, the derivative of G at point y in the direction u is defined as:

G' (y ,u)

=

lim

À~O+

benig a positive real variable.

G(y+ u)-G(y)

.19.

Cne easily proves that:

i) G I (y, u) exists for every u =/ O and is a superlinear

function of u, i.e., GI (y,~IU) _{=llG ' (y,u) for every positive real}

ii) G(y) has its absolute maximum atyo if and only if

G I (yo' u) ~ O for every u=/ O in Rn;

iii) there exists at least one vectbr v=v(y)€Rn such that

. n

GI (y,u) ~(v;u) for every O=/uER , (ViU) indicating the euclidean

inner product of V and u. Any vector V fulfilling this condition

is called a supergradient of G at point y:

iv) G(y) is differentiable at yo if and only if GI (y,u)+

,.G I (y ;-u) =() !:e>r every 1:.=/0 in Rn• In th.t;..: case, the supergradient

.... -1= G at -. .. ~

v _ ' ~O.J..';"

supergradient of G ot: y.. _{, I} t~ w,; ~l1...i.e, S ~s "":.J ~ i:ereni-.L -4Dle at yO.

proh' U". Accvrding to Kuhn anu Tucker I s tnt:>o.,.. S!:"" d feasibIe portfolio

Yo will be optimal if and only if, for some non negative Lagrange

muI tipIier I.:

F(YO)+À P. (x-y)_--: F(y ) ~F(y)+ÀP. (x-y), for every y €Rn

The maximum of the Langrangean, in terms of directional

derivatives, can be expressed by the necessary and sufficient

condition:

FI (yo'u)- P.u. ~ O for every u=/O

Summing up, a feasible portfolio YO' located at the individual's budget frontier P y

=

P x is optimal if and only if

.20.

the present asset vector price P is proP9rtional to a supergradient of F(y) at yO. Inertia will occur if and only if this supergradient is not unique, namely, if F(y) is not differentiable at yO.

FUNDAÇAO GETULIO VARGAS

.21.

4) Pure uncertainty

A particular case to be discussed is that portfolio choice under pure uncertainty in the sense of Frank Knight. Let (nidiP) be the subadditive probability space of future events, P'=P(w) the

n-dimensional random vector descibing future asset prices. For any event B

~ ~

such that BC

~ ~,

p(B)=O, which is to say that its true probabili ty may lie at any point of the closed

@,

lJ intervalo

Accordingly, the expected value·and the expected utility of a portfolio y E Rn are given by:

H(y)=E(P!y) = inf {(P'~)}

WEn W

EU(P!y) = inf U(P'~)

WEn vi

or, since U(W) is an increasing continuous function of W:

EU (P ! y ) = U (H (y) )

Summing up, an optimal portfolio is one that max-:'j,'::"zes

R(y) ;::"7~~ thc individual's budget constraint. (The exi~'~~=~ ~f

both, H(y) and of an equilibrium portfolio can be assured if 3hort sales are restricted, namely, y+yo ~O for some non negative vector yO). The von Neumann-Morgenstern utility function is irrelevant, as

long as i t is increasing. This should be no surprise, since the role of the utility function is to arder preferences involving risk. Is the case under discussion there is no risk but only pure uncertainty.

.22.

possibilities are shown below.

In both cases

n

is the open interval

(0,1), and the individual must distribute his wealth between to assets,

namely, p

l

(w)=(Pi{w) iPi{w»).

Short sales are ruled out, meaning that

y= (y I' Y 2)

~

O.

Example A:

Pi(w)=Wi Pi{w)=l-w

In this case, H{y)=inf {wY I+{I-w)Y2} = min {Yl iY 2}

O<w<l

This is case ,of total inertia. The optimum portfolio

combines equal quantities of the two assets, independently of the

present asset prices.

Portfolio indifference curves are straight

angles.

Example B:

Pi (w)= -

log w

p' (w) = - log (l-w)

2

Now, H(y)= inf {-YI logw - y

2

Iog{l-w)}

O<w<l

a differeni: iable fUf':'. "'J.on of y.

The example shows that uncertainty

.23.

5)

Conc1usions

This paper inteded to address the following question: can

an investor stay put under small variations of the prices of assets?

The very general answer is yes: as long as there is

uncertàinty, as opposed to risk.

The theory of uncertainty, as viewed by Savage(1954), was

a simp1e reduction to subjective risk.

A1ternative1y, we used here

Schmeid1er-Gilboa's uncertainty theory: the consumer maximizes

expected utility with respect to subadditive probabi1ities.

This theory can be app1ied to several apparent1y paradoocica1

phenomena:

(1) Why do some 1arge creditor banks do not se11 or buy

debt in the secondary market?

(2) The theory of inf1ationary inertia assumes that peop1e

are not Nash-equilibrium p1ayers.

(See Simonsen(1987».

In a

certain

se~~~,

this view is that inertia is always an out of

equi1ic.:-.: ...

~h~nompno.... ,Ey

hê

TT

; n']

Nash equilibria with non additive

probabi1iti~;s,

we

fi~.J

out that inf1ationary inertia may arise in

equi1ibrium.

(3) Keyne's theory of investment, where "animal spirits"·

p1ay a rô1e, may be viewed as a ref1ex of uncertainty in the sense

described here.

(4) The

i~vestigation

of Bayesian 1earning of independent

tria1s of a random variab1es (as, for examp1e, tossing the saroe coin

over and over), in connection with a subadditive prior, roay 1ead to

.24.

the more the uncertain phenomenous is observed, the less uncertain

it becomes.

We expect to

pursue

some of these problems in the

.25.

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.26.

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,.,.-ENSAIOS ECOí;üMICOS DA EPGE

(a partir do nº 100)

100. JUROS, PREÇOS E DíVIDA PÚBLICA VOLUME I: ASPECTOS TEÓRICOS - Marco Antonio C. Martins e Clovis de Faro - 1987 (esgotatiu)

101. JUROS, PREÇOS E DíVIDA PÚBLICA \/oLUME 11: A ECONOMIA BRASILEIRA -l97]/es

- Anto n i o Sal a z a r P. Brandão, ClÓvis cE Faro e M:m:u A. C. M3rtins - l~ (est;:ptaiJ)

102. 103. 104. 105. 106. 107. 108. 109. 110.

, U ..

113.

114.

MACRoECoNoMIA KALECKIANA - Rubens Penha Cysne - 1987

O PRÊMIO DO DÓLAR NO MERCADO PARALELO, O SUB FATURAMENTO

DE

EXPORTAÇÕES E O SUPERFATURAMENTO DE IMPORTAÇÕES - Fernando de Holanda 8arbosa - Rubens Penha Cysne e Marcos Costa Holanda - 1987 (esgotado)

BRAZILIAN EXPERIENCE WITH EXTERNAl DEBT ANO PRoSPECTS FDR

GROWTH-Fernando de Holanda Barbosa and Manuel Sanchez de La Cal - 1967 (esgotado) :<EVNES NA SEDIÇÃO DA ESCOLHA PÚBLICA - Antonio M.da Silveira-1987(esgatado) O TEOREMA DE FRoBENIUS-PERRON - Carlos Ivan Simonsen Leal - 1987

POPULAÇÃO BRASILEIRA - Jessé Montel1o-1987 (esgotado)

MACRoECONoMIA - CAPíTULO VI: "DEMANDA POR MOEDA E A CURVA LM" - Mario Henrique Simonsen e Rubens Penha Cysne-1987 (esgotado) MACRoECoNoMIA - CAPíTULO VII: "DEMANDA AGREGADA E A CURVA lS" - Mario Henrique Simonsen e Rubens Penha Cysne - 1987 - (esgotado)

MACROECoNoMIA - MoDFI S~ S~ EQUI~1BRIO A~RfS~TIVo A CURTO PRAZO

- Mario Henr~.:;Je :":nonsen e R':,wp.p ,-; i-:r-: ..

;.a

::.,:!sne-1987 (esgotar..·;

-IHt. BAVE.:J!·AN F,':,iNOATlijN::l

LJr

::"JLi;7:C~: :C'-:.;EPTS DF GAMES - :·~r.f);n

Ribelro da Costa Werlang e Tommy Chlr~-ChiUlc..- - 1987 (esgot· ... J) ~

PREÇOS LíqUIDOS (PREÇOS DE VALOR ADIC!.ONAO[l! =- !Jr ·.!S ~t:TF":"":rJJlP:', ':..1;

DE PRODUTOS SELECIONADOS, NO PER1oDo:9P':,·.:o ~::'T'..,.,st:-l'!/.· ..

Hr:

- Baul ~'.~illlan - 1 QH7

t ~ ~ ~\ ~.:J fIM O c; n,.,: :. .'.; , .L u ~ E S A L D O -M

t

~ =- :J = :J C A S O DE P R E S T A

ç

Õ f S

~~OV1S de Faro - 198Q (~_~u~auuJ

A DINÂMICA DA INFLAÇÃO - Mario Henrique Simonsen - J~~: ~~sgotado)

11 S. ~ ~~ ::

c:

R T .". ! ~! T V A" E R S T n N A ~!D T : ! [: 8 P T ! M

.o '-

C ~ !J J S E O F P O ~ T F Q L :': n

-Jame~ - Dow ~ ~;rgio Ribeiro d~ ~osta Werlang-1988 (esgotadG)

116. O CICLO ECONÔMICO - ~2rio Henriqu~ Simonsen - 1988 (esgotado)

117. FOREIGN CAPITAL AND ECoNCMIC GRoWTH- THE BRAZILIAN CASE STUDV-Mario Henrique Simonsen - 1988

118. CoMMoN KNoWLEDGE - Sérgio Ribeiro da Costa Werlang - 198B{~txb)

119. OS FUNDAMENTOS DA ANALISE MACROECoNÔMICA-Prof.Mario Henrique Simonsen e Prof. Rubens Penha Cysne - 19B8 (esgotado)

120. CAPíTULO XII - EXPECTATIVASS RACIONAIS - Mario Henrique

Simonsen - 1988 (esgotado)

121. A OFERTA AGREGADA E O MERCADO DE TRABALHD - Prof. Mario Henrique Simonsen e Prof. Rubens Penha Cysne - 19B8 (esgotado)

122. -INlRCIA INFLACIONARIA E INFLAÇÃO INERCIAL - Prof. Mario Henrit: Simonen - 1988 (esgotado)

123. MODELOS DO HOMEM: ECONOMIA E ADMINISTRAÇAo - Antonio Kari~ da

Silveira - 1988

124. UNDERINVoICING OF EXPoRTS, OVERINVoIC=\~ DF IMPORTS, A~B lHE

DJLU\R PRl=.:r'-:Iur·1 O"! THE BL t:'CK MARKET - ;::;:.f. Fernando de Hol..a(\d~· 5:,rr"E;J,

,.

17.~.

(J

R(]NO r'iAcJcO DO CHOQlI[ HE1[RODD)W - Fernando

dp

Holanda

Borbo~~

Antora) o SLllLlZiJI

PI:!~S(Jó

Drond~o

(' C1cwis

de

FarO - 1908

(~S~lDtudo)

12L.

PLAND CRUZADD:

tO'JCEPÇ~O

[ O [RRO D[ POL111CA fISCAL - Rubens

Penha Cy5np - 190B

127.

lAXA DE JUROS

rLUTUAI~l[

VERSUS

CORREÇ~O

MD~[lARIA

DAS

PRESTAÇ~E5:

UMA

COKPAR~Ç~D

I~O

CASO DO SAC [ JNFLAÇnO

CONST~NTE

- Clovis

de

12B.

129.

130.

131.

132.

133.

13l; •

135.

136.

~ 57

FoTO -

19E:lli

CAPiTULO 1J _

f',OI~ETAP.Y

CORRECl10f\ AIW REAL IN1[RESl

ACCOUlnlf~G

-Rubens PFnhe Cysne - 198B

CAPilULO l11 _ INCOME ANDDEMANO POLICIES

IN

BRAZIL -

Ruben~

Pen~G Cy~ne

- 19GB

CAPíTULO

IV _

BRAZ1LIAN ECONOMY IN lHE EIGHTl[S AND lHE

0[81

CR151S -

Ruber.~

Penha Cysne - 1908

lHE 8RAZIlIAN AGRICULTURAL POLICY EXPERIENCE: RATIONALE AND

fUTUR[ D!RECTIONS - Antonio 5elazar Pessoa

Brand~o

- 19B8

'-10 R f'. í Ú

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S~rgio

Ribeiro da Costa Werlang - 198B

CAPiTULO

IX _

TEORIA DO CRESCIMENTD ECON5KICO - Mario

Henriqu~

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R ANDO E

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CE 550 DE D [f'IAND A

-_ Joaquim vielr8 Ferreira Levy e SÉrgio Ribeiro

d-a

Costa Werlang -

lc;[;

AS ORIGENS E CONSEQUtNCIAS DA

INFLAÇ~O

NA AMERICA LATINA

-Fernando

~~

Holanda Barbosa - 1988

A CON1;,

S~PQrt>.)Tr

DO

GO'J[~~IS

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:';"L.MI~ÇO

DE PAGAI',ENTOS:

U!'~A

ABur.P~Gftvl

:'11"II-'L11-:'d ..

ADA

-_

J~~D

Luiz Tenreiro Barroso - 1989

C O N TA 6 I L I DA D E C O M J U R O S R

~.

A i 5 - P.l'

~ n~

s

p

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P H A C Y S

~!:

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PERI-',At~li;;

I~~CO~·\E

i:~'r'OTHESIS"

- Vicen:< ..

~',a::lri9.::1.

Tommy Tan, Daniel Vicent, Sérgio Ribeiro da Costa we.-lang - 1989

liA At-'.AZONIA BRASILEIRA".- Ney Coe de Oliveira -

1989

1 lj 2 •

143.

DES~GIO

DAS LFTs E A PROBABILIDADE IMPLICITA DE

~ORA16~IA

Maria Silvia Bastos

~arques

e

S~rgio

Ribeiro da Costa

~crl~~r-144.

THE LDC DEBT PROBLEH: A GAME-THEORETICAL ANALYSIS

Mario Henrique Simonsen e Sérgio Ribeiro da Costa Werlang -

1989

.

145.

ANALISE CONVEXA NO Rn - Mario Henrique Simonsen -

1989

1~6. ~ CONTROVtRSIA MONETARISTA NO HEMISFtRIO NORTE

Fernando de Holanda Barbosa -

1989

147. F I SCAL REFORM ANO STAB I L I lAT I ON: THE BRf\.Z I L IAN EXPER I ENCE - Fern<lndo dI' f!o

I.l!l'~

)

148. RETORNOS EM EDUCAÇÃO ~O BRASIL: 1976-1986

Carlos Ivan Simonsen Leal e Sérgio Ribeiro da Costa Werlang - 1989 149. PREFERENCES, COMMON KNOWLEDGE, AND SPECULATIVE TRADE - James Dow,

Vicente Madriga1, Sérgio Ribeiro da Costa Werlang - 1990 150. EDUCAÇÃO E DISTRIBUIÇÃO DE RENDA

Carlos Ivan Simonsen Leal e Sérgio Ribeiro da Costa - 1990

151. OBSERVAÇÕES

A

MARGEN DO TRABALHO "A AMAZONIA BRASILEIRA" Ney Coe de Oliveira - 1990

152. PLANO COLLOR: UM GOLPE DE MESTRE CONTRA A INFLAÇÃO? - Fernando de Holanda Barbosa - 1990

153. O EFEITO DA TAXA DE JUROS E DA INCERTEZ~ SOBRE A CURVA DE PHILLIPS DA ECONOMIA BRASILEIRA - Ricardo de Oliveira Cavalcanti - 1990 154. PLANO COLLOR: CONTRA FACTUALIDADE E SUGESTÕES SOBRE

A

CONDUÇÃO

DA POLíTICA MONETÁRIA-FISCAL - Rubens Penha Cysne - 1990

155. DEPÓSITOS DO TESOURO: NO BANCO CENTRAL OU NOS BANCOS COMERCIAIS? Rubens Penha Cysne - 1)90

156. SISTEMA FINANCEIRO DE HABITAÇÃO: A QUESTÃO DO DESEQUILíBRIO DO FCVS - .Clovis de Faro - 1990

157. COMPLEMENrr,'~ DO FAScíCULO NQ 151 DOS ""'~.::;A:\':' ECONOr-nCO'",' (R f-U'JA-ZONIA pr.' s:rr.RT~A' - Npv ~oe rle Ol.L'.~eir.1 - 199v

-

.... _{PO~'~IC~ ~~~p~~r~A ÓT~~rl NO COMBATE}

_A

_INFLAÇÃO

.- eer, -dàr _{,~_} Ho1;:o1'"'.J.? Barbosa - 1990

]59. TEOR.i.". DOS JOGOS - CONCET,!,OS !:":.~:\''J.'' - :~;--:..:." t1efl:::~(1'.'''.? Sir.lOnsel"

- lOf'íJ

160. O "=-.~,,-.-1()O ABERTO BRASILEIRO: ANÁLISE"DOS PROCEDIMENTOS OPERACrO ... NAIS - Fernando de Holanda Barbosa - 1990

161. A RELAÇÃO ARBITRAGEM ENTRE A 01'~N C~MBi~~ ~ A ORTN MONET;~IA -- Luiz Guilherme Schymura de Oliveira - ~990

162. SUBADDITIVE PROBABILITIES AND PORTFOLIO INERTIA - Mario Henrique Simonsen e Sérgio Ribeiro da Costa Werlang - 1990

000075255