N9 214
ON THE DIFFERENTlABILITY OF THE CONSUMER DEMAND FUNCTION Prof: Paulo Klinger Monteiro
Mário Rui Páscoa
Sergio Ribeiro da Costa Werlang Maio de 1993
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Paulo Klinger Monteiro
!MP A, Rio de Janeiro
Mário Rui Páscoa
Faculdade de Economia, Universidade Nova de Lisboa
Sergio Ribeiro da Costa Werlang
Fundação Getúlio Vargas, Rio de Janeiro
ABSTRACI'
For strictly quasi concave differentiable utility functions, demand is shown to be differentiable almost everywhere if marginal utilities are pointwise Lipschitzian. For concave utility functions, demand is differentiable almost everywhere in the case of differentiable additively separable utility or in the case of quasi-linear utility.
ADDRESS FOR CORRESPONDENCE: Mário R. Páscoa
Faculdade de Economia Universidade Nova de Lisboa
Travessa Estevão Pinto (CampolideJ 1000 Lisboa
L INTRODUCTION
Katzner (1968) showed that demand functions derived from quite nice utility functions, even of class C2, need not be differentiable everywhere. For strict1y quasi-concave C2 utility functions, such that the marginal utilities are always strictly positive and the indifference hypersurfaces never intersect the boundary of the positive orthant, the demand function is of class C 1 at a bundle x if and only if the bordered Hessian determinant does not vanish at x. This condition holds on an open dense set of bundles and a corresponding open dense set of fuII measure of prices and income. Differentiability of demand almost everywhere is therefore the most that can in general be expected, even for C2 utility functions.
A more general result by Debreu (1972) established that for C1 preferences, demand is smooth at the inverse image of a bundle x if and only if the indifference hypersurface has nonzero Gaussian curvature at x. Moreover, Debreu (1976) showed that continuous differentiability of demand does not require the existence of a C2 utility function without criticaI point; the necessary condition is that utility is of class C1 with nonvanishing derivative.
Rader (1973, 1979) established sufficient conditions on utility functions for demand to be difIerentiable on almost every price vector and income. Using Sard's (1958) lemma on criticaI values of differentiable mappings, not necessarily of class C1, Rader (1979) was able to replace the C2 condition on utility by the assumption of twice - differentiability.
Our first result in this paper goes a step further by assuming that utility functions are differentiable, with strictly positive pointwise Lipschitz marginal utilities. Our other results address the case of concave utility functions, showing that the desired almost everywhere differentiability of demand holds for quasi-linear concave functions and for additively separable strictly concave differentiable functions. Rader (1973) had addressed also the concave case and showed that here it
lI. DEFINITIONS AND AUXILIARY RESULTS
The consumer utility function u has IR~ as its domain and income is normalized to be one. The demand correspondence is given by p ~ d(p) = {x ~ O: x maximizes u(y) subject to p.y = I}. In this section we enunciate some basic results to be used later on.
We start with a pointwise version ofRader's (1973) theorem 1:
Lemma 1: Suppose u: IR: ~ IR is strictly quasi concave and continuously differentiable. Let f: IR: ~ IR ~ be given by f(x)=Vu(x)/(x. Vu(x». If p E 1R:+ is such that u is twice differentiable at d(p) » O and
f' (d(p» is nonsingular, then demand is differentiable at p.
Proof: This lemma is a direct consequence of the facts that d = fI, fis an homeomorfism and f ' (d(p») is nonsingular. In fact, an homeomorfism with nonsingular derivative at a point is such that the inverse is differentiable at the inverse image of this point (see Dieudonné (1960), for example).
Now we recall the strongest result by Sard on the measure of the set of criticaI points of a function.
Lemma 2: (Sard (1958»: Suppose n c IR n is open and f:
n
~ IR n.Let B={x E Q: f '(x) exists and is singular}. Then f(B) is a null set.
(This is a corollary to theorem 1 in page 254 of Sard (1958»; notice that f does not even need to be continuous outside ofB).
Next, we enunciate a recent result that will be used in our discussion of Rader's (1973) condition that demand maps null sets into null sets.
Lemm
a 3: (Villani (1984»: Let U ç: ]R n be open and q>: U -+ ]R nbe continuous, one-to-one and differentiable almost everywhere. Then q>-l takes null sets into null sets if and on1y if q>' is nonsingular almost everywhere.
We wiIl appeal in our proofs to the following consequence of Fubini theorem:
Lemma 4: Consider the measure spaces (X, A ,A) and (Y, B, Il) such that X and Y are Polish spaces and Il is a nonatomic measure. If f:X-+ Y is measurable, then graph (f) has nuIl AXIl measure.
Proof: Let G: X -+ XxY, given by x -+ (x, ftx». Now G is one to one and by Kuratovsky theorem Im(G)
=
graph (f) is a mesurable set. Now O .. xll) (graph (f) =J
1l({Y: (x, y) E graph(D}) d A(X) =J
Il({f\x)}) d A(X) =o.
Finally, let us recall that a vector-valued function f defined or an open set of ]R n is said to be pointwise Lipschitzian at Xo if there exits a
constant K and a neighborhood N of x such that lIf(x) - f(xo)1I ~ K IIx - xolI, for every x E N.
m.RESULTS
Our first result extends Rader's (1973) theorem 1, by replacing the assumption of twice - differentiable of utility with the weaker condition of pointwise Lipschitzian marginal utilities.
Theorem 1: If the utility function is differentiable on R:+ with pointwise Lipschitzian deriva tive and such that \l u(x) » O for every x E
R:+, then the demand function is differentiable in almost every price p in (p» O: d(p) » O}.
Proof: Demand is given by a function whose inverse is the mapping f: x -+ \l u(x)/(x. \l u(x», which is weIl defined since the denominator is never
zero. Our first claim is that f is pointwise Lipschitzian. In fact, lIf(x) -f(xo)1I
=
(Vu(x) - Vu(xo»/
(xo .Vu(xo
» + Vu(x) [(x.vu(x)r1 - (xo.vu(xo)r1] =:;;(1)
Kxo IIx - xo" + V u(x) [(x. V u(x))(xo. V u(x o
))r
1 [x. V u(xo) - Xo . V u(x)] =:;;(1) (2) (1) (3)
Kxo IIx - xolI + Kxo [lIx ., xo" Vu(x) + xo(Vu(x) - Vu(xo))] =:;; (Kxo + Kxo) IIx -xo"·
N ow, a consequence of the claim is that f is differentiable almost everywhere and that the set of nondifferentiability of f is mapped into a null set N. By lemma 2, f has a null set C of criticaI values and therefore the set of regular values of f is
1R:+ \
(NU
C), a set of fuH measure. Then, by lemma 1, d=
fI diflerentiable for every p E1R:+
\(N U C) such thatd(p»>O.
D
As a coroHary we obtain Rader's (1973) theorem 1: when the utility function twice differentiable, demand is differentiable in almost every price in (p » O: d(p) » O}. In fact, a differentiable function is pointwise
Lipschitzian.
If we assume u differentiable in
1R:
then a stronger result is true (which extends Rader's (1979) theorem 2 and its first corollary onn
2 utility to the more general case of pointwise Lipschitian marginal utilities).Theorem 1': Add to the assumptions of therem, the hypothesis that u is continously differentiable on
1R:.
Then the demand function is differentiable for almost every price.Proof: from theorem 1 it suffices to prove that demand is a.e. differentiable in (p » O: d(p) » O}'. Using Kuhn-Tucker's theorem we have
O. Therefore, for any set of indices I, the set AI
=
{p » O: fi (d(p» < Pi for i e I and d1(p) > O for 1 $ I} is open. In this set, d(p) is the solution of max {u(x): xi=
O for i e I, Ll1xl=
I}. Hence theorem 1 applied to uI IR:'
ensures thatdemand is
diff~tIentiable
almost everywhere in AI'n
It remains to consider the set E
= .
U 1 {p » O: f(d(p»=
p. and d.(p)=
I
=
I 1 1O}. However, if d.(p)
=
O it is true that d is a function dI ofp . alone, where1 ~
P . -1
=
(P1' ... , P1 " p. -1 1+ l' ... , Pn)' But in this case d is mapping a null set because each set {p » O: p.=
f (dI (p .))} is the graph of a measurable1 1 -1
function and is therefore a null set, by lemma 4.
D
In the proof of theorem 1 it was fundamental that f = d- 1 takes null sets into null sets (this is usually called Lusin's condition). The following example shows that in general f may not have this property.
Example: Take a function g: [O, +00] ~ [O, +00] continuous, strictly decreasing such that g'
=
O a.e. (for example, let g be the inverse of the00
increasing function H in Sacks(1937), p. 101, given by H(x) = LF(nx)/2n ,
n=1
where F is Cantor's function on (O, 1»).
Now, let G be a primitive of g and define the utility function by u(x, y)
=
G(x) + G(y). Clearly, u is concave and continously differentiable. The function f is given by f(x, y)=
(g(x), g(y))/ (x g(x) + y g(y)). It is easy to check that f ' is singular a.e .. Since f is an homeomorphism, f is open and therefore its range has positive measure. But lemma 2 implies that the image of the set of criticaI points of f is a null set. This proves that f takes a null set (i.e., the complement of the criticaI points ofD
onto a non-null set.As discussed in Rader (1973) it is desirable that demand takes null sets into null sets. The above example gives us also a demand function that does not satisfy Lusin's property and is an alternative to Rader's (1973) example (pp. 924-925). In fact, the above example faiIs the following necessary and sufficient condition for Lusin's property on demando
Theorem2: Suppose f given by f(x)
=
"Vu(x)/(x."Vu(x)) is differentiable almost everywhere. The demand function d takes null sets into null sets if and only if f ' is non singular almost everywhere.(Proof: this is an immediate consequence of lemma 3)
Our next results deal with the case of concave utility functions. We wiIl show that in this context it is possible to drop, in some special cases,
(i) the assumption made an marginal utilities or even (ii) the assumption that utility it self is always differentiable. In case (i) we will look at additively separable utility and in case (ii) we will look at quasi-linear utility.
Theorem 3: Suppose u: IR: --t IR, given by u(x, y)
=
G(x) + H(y), is strictly concave, differentiab1e and that g == G' > O, h == H' > O. Then the demand function is differentiable almost everywhere.Before proving this theorem we have to introduce some notation. Let g: [O, 00] --t IR and x ~ O, we define Dg(x)
=
lim sup (g(x') - g(x»/(x' - x) andx' ~ x
Dg(x) = lim inf (g(x') - g(x»1 (Xl - x). We have that g is differentiable at x if
x' ~ x
and on1y ifDg(x)
=
Dg(x).Proof of theorem 3: (i) Let (x, y)
=
d(p, q) and (x', y')=
d(p', q'). We wiIl start by estab1ishing the following identity:(g(xl) • g(x) +
~
• h(y') . h(y» (Xl. x)=
Xl. x h2(y) y.y
h(y') . h(y) g(x) [ I ( I ) I( I )] h(y') ( I g(x) ( I
»
In fact, g(x)fh(y)
=
p/q, gCx')fh(y')=
p'/q' and let dg=
g(x') - g(x), dh=
h(y') - h(y); then dg=
(p'/q') h(y') - (p/q) h(y)=
(p'/q' - p/q) h(y') + (p/q)dh, that is , (x' - x)dg/(x' - x)=
(p'q - pq') h(y')/q'q + (p/q) (y' - y) dh/(y' - y). Now 1=
p'x' + q'y'=
(p' - p)x' + (q' - q)y' + p(x' - x) + q(y' - y) +1 and substituting above we get: (x' - x)[(g(x') - g(x»/(x' - x) + (p/q)2 (h(y') - h(y»/(y' - y)]=
(p'q -pq') h(y')/q'q - (p/q2) (dh/(y' - y)) [Cp' - p)x' + (q' - q)y'].(ii) Now, to prove the differentiability of demand it suffices to prove the differentiability ofthe mapping (p, q) ~ x(p, q)
=
d1(p, q). By Stepanov's theorem (see Saks (1937), IX, p. 310), it is enough to show, for almost every (p, q), that lim sup (x' - x)/( I p' - p I + I q' -q I < 00. From the above(p', q') ~ (p, q)
identity and the fact that g and h are decreasing we have the following implication: lim sup I x' - x I I( I p' - p I + I q' - q I)
=
00 ~ Dg(x)=
O=
(p', q') ~ (p, q)
Dh(y). Therefore it suffices to prove that the set P
=
{Cp, q): Dg(x)=
O=
Dh(y), where (x, y)=
d(p, q)} is nul!.If g'(x) and h'(y) exist and d-1(x, y) E P, then g'(x)
=
O=
h'(y). Thisimplies that the Jacobian of f at (x, y) is zero (recall the definition of f, lemma 1) and, Sard's (1958) theorem, the set {Cp, q): (x, y)
=
d(p, q), g'(x)=
O=
h'(y) } is nul!. Therefore, it suffices to prove that P g=
{(p, q): (x, y)=
d(p, q), Dg(x) > :Qg(x)} and Ph=
(Cp, q): (x, y)=
d(p, q), Dh(y) > :Qh(y)} are null.(iii) We wiU prove that P g is a null set. The proof for Ph is similar.
g(x)
-Notice that P g = {(h(y) q, q): q(xg(x) + yh(y» = h(y), Dg(x) > :Qg(x)}. Define ",(y)
=
h(y) (1 - qy). The function '" is strictly decreasing and continuous;1 ~
-call e
= ",- .
Then P g= {(
x ' q): e(qx g(x»=
y, Dg(x) > :Qg(x)}. By Fubini1\
1- qy
theorem it suffices to prove for almost every q > O that P g
= {
x . e(q x g(x»=
y, Dg(x) > l.?g(x)} is null.Since g is differentiable almost everywhere, the set E = (x: Dg(x) >
~g(x)} is null. Now, by item 406, po 271 of Saks (1937), if a function has at each point of a set a finite Dini deriv ativ e , than the function satisfies Lusin's property on the seto So, if the mapping x ~ R(x) = [1 - q8(q x
1\
g(x»]lx has for each point ofE a finite Dini derivative, then P g is null.
Let us calculate one such derivativeo Call z
=
xg(x), and z'=
x'g(x') Then it is true thatR(x') - R(x) , - 2 8(qz') - 8(qz) z' - z
x , x-x
= -
R(x ) -q qz-z ( ' ) . x-x , (B)In fact, x'R(x') - xR(x) = -q8(qz') + q8(qz), then (x' - x) Rex') + x(Rex') -R(x»
=
q(8(qz) - 8 (qz'» and the above identity followsoNotice that z' - z
=
(x' - x)g(x') + x(g(x') - g(x», so (z' - z)/(x' - x)=
g(x') + x(g(x') - g(x»/(x' - x)o Then,R(x') - R(x) _ R( ') -2 8(qz') - 8(qz) [ (') g(x') - g(x)]
x , x-x _- x -q qz-z ( ' ) . gx +x x-x , (C)
Now, let u
=
8(qz) and u'=
8(qz'), then (u' - u)/q(z' - z)=
«<p(u') -<p(u»/(u' - u»-l=
[«h(u') - h(u»/(u' - u» (1 - qu') - qh(u)rl < 00 80 I (u' -u)/q(z' - z) I ~ (qh(u)rlo Therefore for each x E E, identity (C) implies thatxDR(x) ~ - R(x) + qh(ur l (g(x) + xDg(x» < 00
D
Our last result addresses the case of a quasi-linear utility function and does not require differentiability of this function on its entire domain:
Theorem 4: If the utility function u: IR: ~ IR is given by u(x)
=
xl + v(x2, 000' xn), where v is a strictly concave function, then, for almost every price in {p » O: d(p) » O}, the demand function is differentiableo
Proof: This result can easily be established appealing to results on concave conjungacy and to Alexandroff 's (1939) theorem. Let v* be the concave conjugate function of v, that is, v *(z)
=
sup {v(x. l ) - < x. l ' z >}; thex -I
function V * is concave. Let us denote by Gg(x) the set of supergradients of a
concave function g at x, that is Gg(x)
=
{z e IR n: gCy) ~ g(x) + <z, y - x>;1fy}. Now, x.l e Gv *(z) if and only if z e Gv(x. l ), (as can be easily seen usingknown results as convex conjugacy, see Rockafellar (1970) 23.5.1).
1\
If at prices p, the optimal solution x is interior, then p == P./Pl e
1\
Gv(x_l) and, therefore, x_I E Gv
*
(p). Now, V is strictly concave on everyconvex subset of {x_I: Gv(x -1) :I:- 0} ç dom V and therefore v * is
differentiable throughout int (dom v *) (see Rockafellar (1970) 26.3). Then,
1\
Now, by Alexandroff 's (1939) theorem, the concave function v * is twice differentiable almost everywhere on int(dom v *) and therefore, the demand function will be differentiable in almost every price in {p » O: d(p)
1\
» Ol. In fact, outside of a null set, V x. l
=
V2v * (p) and xl=
1· I,xjP/Pl'D
References
Alexandroff, A. D. (1939): "Existence of the Second DifIerential of a Convex Function AImost Everywhere in its Domain and Some ReIated Properties of Convex Surfaces" (Russian), Leningrad State University, Annals 37, Mathematical Series, 6, pp. 3-35.
Debreu, G. (1970): "Economies with a Finitive Set of Equilibria" , Econometrica 38.
Debreu, G. (1972): "Smooth Preferences", Econometrica 40.
Debreu, G. (1976): "smooth Preferences: A Corrigendum", Econometrica 44.
Dieudonné (1960): "Foundations of Modern Analysis", Prentice - HalI. Katzner, D. (1968): "A Note on the DifIerentiability of Consumer Demand
Functions", Econometrica, voI. 36, No. 2.
Rockafellar, T. (1970): "Convex Analysis", Princeton University Press. Sard, A. (1958): "Images of CriticaI sets", Annals of Mathematics, vol.68,
No. 2.
Sard, A. (1942): "The Measure of the CriticaI Points of DifIerentiable Maps", Bulletin af the American mathematical Society, 48.
Saks, S. (1937), Theory of the integral, Hopner Publishing Company, 22 ed.
Villani, A. (1984), On Lusin's condition for the inverse function, Rendiconti deI Circolo Matematico di Palermo, serie II, tomo XXXIII, pp 331-335.
A
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147. FISCAL REFORM AND STABILIlATION: THE BRAZIIlAN EXPERlENCE - Femmdo de HolaodaBarbosa. Antonio Salazar PeBBoaBnmdão e CloviB de Faro - 1989 (esgotado)
148. RETORNOS EM EDUCAÇÃO NO BRASIL: 1976/1986 - C.-IoB Ivan SimoDBen Leal e Sérgio Ribeiro daColda WerllD&-1989 (ngotado)
149. PREFERENCES, COMMON KNOWLEDGE AND SPECULA11VE TRADE - James Dow, Viçente Madrigal e Sérgio Ribeiro da Costa Werlaog - 1990 (esgotado)
ISO. EDUCAÇÃO E DISTRlBUIÇÃO DE RENDA - C.los Ivan SimoDBen Leal e Sérgio Ribeiro da Costa Werbq - 1990 (ngotado)
1051. OBSERVAÇÕES A MARGEM DO TRABAUlO liA AMAZONIA BRASILEIRA" - Ney Coe de Oliveinl- 1990 (esgotado)
153. O EFEIl'l'O DA TAXA DE JUROS E DA lNCERTF2.A SOBRE A CURVA DE PHILLIPS DA ECONOMIA BRASILEIRA - Riv.do de Oliveira Cavalcaoti -1990 (eesotado)
154. PLANO COlLOR: CONTRA A FACTUALIDADE E SUGESTÕES SOBRE A CONDUçÃO DA POLÍTICA MONEI'ÁRIA-FISCAL - Rub8D8 Penha Cyme -1990 (esgotado)
155. DEPÓsrrOS DO TESOURO: NO BANCO CENTRAL OU NOS BANCOS COMERCIAIS? -Rub9D8 Peoba Cyane - 1990 (esgotado)
156. SISTEMA FINANCEIRO DE HABITAÇÃO: A QUESTÃO DO DESEQun.tBRIO DO FCVS -Clovis de Faro - 1990 (esgotado)
157. COMPLEMENTO DO FAScíCULO ND 151 DOS "ENSAIOS ECONÔMICOS" (A AMAZôNIA BRASILEIRA) - Ney Coe de Oliveira - 1990 (esgotado)
158. POLÍTICA MONEI'ÁRIA ÓTIMA NO COMBATE A INFLAçÃO - Fernando de Holamla Barbosa - 1990 (esgotado)
159. TEORIA DOS JOGOS - CONCEITOS BÁSICOS - Mario Heurique SimODBell- 1990 (esgotado) 160. O MERCADO ABERTO BRASILEIRO: ANALIsE DOS PROCEDIMENTOS OPERACIONAIS
- Fmumdo de Holanda Barbosa - 1990 (esgotado)
161. A RELAçÃO ARBITRAGEM ENTRE A ORTN CAMBIAL E A ORTN MONETÁRIA - Luiz
Guilherme Schymura de Oliveira - 1990 (elgotado)
162. SUBADDlTIVE PROBABILlTIES AND PORTFOUO INERTIA - Mario Henrique SimODSn e Sérgio Ribeiro da Costa Werlq - 1990 (esgotado)
163. MACROECONOMIA COM M4 - Carlos Ivan Simoosen Leal e Sérgio Ribeiro da Costa Werlaog - 1990 (eagotado)
164. A RE-EXAMlNA'I10N OF SOLOW'S
GRowm
MODEL wrrH APPUCA'I10NS TO CAPlTALMOVEMENTS - Nnntro SuvedraRiV1Jl1o -1990 (esgotado)16S. 1lIE PUBUC CHOICE SEDmON: VARIA'I10NS ON
nm
THEME OF SCIENTlFIC W ARF ARE - Aotooio Maria da Silveira - 1990 (esgotado)166. 11IE PUBUC CHOPICE PERSPECnVE AND KNlGlIT'S INS'ITI'UTIONAIJST BENT -Aotonio Maria da Silveira - 1990 (esgotado)
167. nJEINDETERMINA'I10N OF SENIOR - ADtonio MariadaSilveira-I990 (esgotado)
168. JAPANESEDIR.EcrINVESTMENTlNBRAZIL- Neaotro SaavedraRivano -1990 (esgotado) 169. A CARTEIRA DE AÇÕES DA CORRETORA: UMA ANALIsE ECONÔMICA - Luiz
Ouilherme Schymura de Oliveira - 1991 (esgotado)
170. PLANO COlLOR: OS PRIMEIROS NOVE MESES - Clovis de Faro - 1991 (elgotado) 171. PERCALÇOS DA INDEXAÇÃO EX-ANTE - Clovis de Faro - 1991 (eBgOtado)
173. A DINAMICA DA HlPERINFLAçAO - Fenumdo de Holaoda Bno... Waldyr Muoiz Oliva e Elvia M .... b Salbm - 1991 (eagotado)
174. LOCAL CONCA VIFIABILlTY OF PREFERENCES AND DETERMINACY OF EQUlLlBRIUM - Mario Rui Péscoae Sérgio Ribeiro daCoBta Werlq- maio de 1991 (esgotado)
175. A CONI'ABTT,IDADE DOS AGREGADOS MONETÁRIOS NO BR.ASlL -Carlo.IV8Il Simcuen Leal e Sérgio Ribeiro daCoBta Werlq - maio de 1991 (esgotado)
176. HOMOTHE'IlC PREFERENCES - James Dow e Sft'gio Ribeiro da Costa Werlq - 1991 (esgotado)
177. BARREIRAS A ENTRADA NAS INDÚSTRIAS: O PAPEL DA FIRMA PIONEIRA - Luiz Guilherme Schymurade Oliveira-I991 (eagotado)
178. POUPANÇA E CRESCIMENTO ECONÔMICO - CASO BRASILEIRO - Mario Heorique SimoDSen - agosto 1991 (esgotado)
179. EXCESS VOLATIUrY OF STOCK PRICES AND KNIGHrIAN UNCERTAIN'IY - lamel Dow e Sérgio Ribeiro da Costa Werlq - 1991 (esgotado)
180. BRAZIL - CONDmONS FOR RECOVER,Y - Mario Heorique SimODSen - 1991 (esgotado) 181.
nm
BRAZILIAN EXPERIENCEwrm
ECONOMY POUCY REFORMS AND PROSPECTSFOR
nmFUI'URE -
Femaodo de HolaodaBarboa- Dezembro de 1991 (eBgotado)182. MACRODINÂMICA: OS SISTEMAS DINÂMICOS NA MACROECONOMIA - Fernando de HohmdaBarbon- Dezembao de 1991 (esgotado)
183. A EFIcrFNCIA DA INTERVENçAO DO ESTADO NA ECONOMIA - Fernando de Holaoda Barbosa - Dezembro de 1991 (eI80tad0)
184. ASPECTOS ECONÔMICOS DAS EMPRESAS ESTATAIS NO BRASIL:
TELECOMUNICAÇÕES, ELETRICIDADE - Femando de Holaoda Barbosa, Maouel Jeremiu Leite Caldas, Mario Jorge Pina e Hélio LechugaArteiro - Dezembro de 1991 (esgotado)
18S.
nm
EX-ANTE NON-OPTIMAllTY OFnm
DEMPSTER.-SCHAFER UPDATlNG RULEFOR AMBlGUOUS BEIlEFS - Sérgio Ribeiro da COBta Werlq e J .... Dow - Fever.iro de 1992 (esgotado)
186. NASH EQUlLlBRIUM UNDER KNIGHrIAN UNCERTAINTY: BREAKING DOWN BACKWARD INDUrnON - James Dow e Sérgio Ribeiro da Colta Werlq - Fevereiro de 1992 (eagotado)
187. REFORMA DO SISTEMA FINANCEIRO NO BRASIL E "CENTRAL BANKING" NA ALEMANHA E NA ÁUSTRIA - Rubens Peuha Cyaoe - Fevereiro de 1992 (esgotado)
-I j 190. ! I I 119l. ,192. I 1 '193. 194.
I
1195. 1196. !I
1197. ! I 1198.,
1199. I 1200. 1201. ! ! 202. 203. I I 204. 20S.dIPERlNFI..AçAO E O REGIME DAS POUnCAS MONETÁRIA-F1SCAL - Femanào dei
Holanda Barbosa e Elvia Mureb Salhm - Março de 1992 (esgotado)
I
t
A CONSTIrUlçAo, OS JUROS E A ECONOMIA - Clovis de Faro -A~ril de 1992 (esgotado)
I
APUCABILIDADE DE TEORIAS: MICROECOXOMIA E ESTRATEGlA EMPRESARIAL-I
.J\ntonio Maria da Silveira - Maio de 1992 (esgotado)~ I
INFLAÇAO E CIDADANIA - Fernando de Holanda Barbosa - Julho de 1992 I
A INDEXAÇAO DOS ATIVOS FINANCEIROS; A EXPERltNCIA BRASn..E1RA - Fernando de Holanda Barbosa - Agosto de 1992
A n-."FLAçÃO E CREDmILIDADE - Sérgio Ribeiro da Costa Werlang - Agosto de 1992
I
A RESPOSTA JAPONESA AOS CHOQUES DE OFERTA 1973/1981 - Fernando Antonio
I
I
Hadba - Agosto de 1992
!
IlTM MODELO GERAL DE ~"EGOCIAÇAO EM U~Il\fERCADO DE CAPITAIS EM Qt:r: ~Aoj
EXIS'I'D-'Il?\"VESTIDORES
lRRACIO~AlS
- í..uiz GuiihenneSch~mura
de ()iiveira - Setemorolde 1992
I
SISTEMA FlNANCEIRO DE li4J3ITAÇAO; A NECESSIDADE DE REFO~\{A - Clovis d~ ! Faro - Setembro de 1992
BRASIT..: BASES PARA A RETOMADA DE DESffi'.i'VOLV1\!ENTO - Rubens Penha Cysne -j
Outubro de 1992 ,
I
AVISA0 TEORICA SOBRE MODELOS PREVID~CL.uuOS: <) CASO BRASILEIRO - Luiz
Guilherme Schymura de Oliveira - Outubro de 1992
HIPER.1NFLAÇAO: CÂL\4BIO, MOEDA E ÂNCORAS NOMINAIS - Fernando de Holanda
i
Barbosa - Novembro de 1992 -(es~ota.do)
!
PREVID~CIA
SOCIAL: CIDADA;.'lIA E PROVISÃO - Clovis de Faro - Novembro de 1992 IIOS BANCOS ESTADUAIS E <) DESCO~LROLE FlSCAL: ALGUNS ASPECTOS - Sérgio.
Ribeiro da Costa Weriang e Anmnio Fra,ga~eto -~ovembro de 1992 - (esgotado)
I
TEORIAS ECONÔMICAS: A MEIA-VERDADE TEMPORÁRIA - Antonio Maria da Silveira-IDezembro de 1992
I
nm
RICARDIAN VICE ANDnm
~1)ETER..\fiNATION OF S~10R - Antonio Maria da I Silveira -Dezembro de 1992206. HIPERlNFLAçÃO E A FORMA FUNCIONAL DA EQUAÇÃO DE DEMANDA DE MOEDA -Fernando de Holanda Barbosa -Jan~iro de 1993
207 REFORMA FIN&~CEIRA - ASPECTOS GERAIS E &~ÁUSE DO PROJETO DA LEI COMPLEMENTAR - Rubens PenhaCysne - fevereiro de 1993.
209. 210. 211. 212.
I
I 1213. I j II
1214. I IELEMDi'TOS DE lJMA ESTRATÉGIA PARA0 DESEN"VOLVIMENTO DA AGRIC'ULTh'"RA, BRASILEIRA - Antonio Salazar PessoaBrandlo e Eliseu Alves - Fevereiro de 1993
I
I
~~~C'IA.
SOCIAL PÜBUCA:A.E~EIill:NCIA B~~
-H~l.io
Portoçarrero dei castro. LuIZ GUIlherme Schymura de Ohverra. Renato Fragelh Cardoso ~ Unel de Magalhães -/Março de 1993. i
. I
OS SISTEMAS PREVIDENCIARIOS E UMA PROPOSTA PARA A REFORMU1..ACAO DO MODELO BRASILEIRO - Helio Portocarrero de Castro, Luiz Guilht"IlIle Schymura de Oliveira, Renato Fragelli Cardoso e Uriel de Magalhaes - Março de 1993.
TIIE INDETERJdlNATION OF SENIOR (OR TIIE ~TI~\flNATION OF WA<Th"ER) AND SCHMOLLER AS A SOCIAL EC'ONOMIST - Antonio Maria da Silveira - Março de 1993.
)1ASH EQUILIBRIUM
t~TIER M1GHTIAi~ t~CERTA1l\,,-y:
BREAKING DOWNIBACK\V ARD !NTI"GCTION (Extensively Revised Version) - lames Dow e Sérgio Ribeiro
dai
I
Costa Werlang - Abril de 1993. i
I ON THE DIFFERENTIABILlTY OF TIIE C'ONSUMER DEMAND FUNCTION - Paulo K1ingerl Monteiro. Mário Rui Páscoa e Sérgio Ribeiro da Costa Weriaog - Maio de 1993.
