❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s
❊s❝♦❧❛ ❞❡
Pós✲●r❛❞✉❛çã♦
❡♠ ❊❝♦♥♦♠✐❛
❞❛ ❋✉♥❞❛çã♦
●❡t✉❧✐♦ ❱❛r❣❛s
◆◦ ✸✾✽ ■❙❙◆ ✵✶✵✹✲✽✾✶✵
■♥t❡❣r❛❜✐❧✐t② ❛♥❞ t❤❡ ❞❡♠❛♥❞ ❢♦r ♠♦♥❡t❛r②
❛ss❡ts✿ ❛♥ ❛❧t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤ t♦ ❛♥ ♦❧❞
♣r♦❜❧❡♠
❘✉❜❡♥s P❡♥❤❛ ❈②s♥❡
❖s ❛rt✐❣♦s ♣✉❜❧✐❝❛❞♦s sã♦ ❞❡ ✐♥t❡✐r❛ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡ ❞❡ s❡✉s ❛✉t♦r❡s✳ ❆s
♦♣✐♥✐õ❡s ♥❡❧❡s ❡♠✐t✐❞❛s ♥ã♦ ❡①♣r✐♠❡♠✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛
❋✉♥❞❛çã♦ ●❡t✉❧✐♦ ❱❛r❣❛s✳
❊❙❈❖▲❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊❈❖◆❖▼■❆ ❉✐r❡t♦r ●❡r❛❧✿ ❘❡♥❛t♦ ❋r❛❣❡❧❧✐ ❈❛r❞♦s♦
❉✐r❡t♦r ❞❡ ❊♥s✐♥♦✿ ▲✉✐s ❍❡♥r✐q✉❡ ❇❡rt♦❧✐♥♦ ❇r❛✐❞♦ ❉✐r❡t♦r ❞❡ P❡sq✉✐s❛✿ ❏♦ã♦ ❱✐❝t♦r ■ss❧❡r
❉✐r❡t♦r ❞❡ P✉❜❧✐❝❛çõ❡s ❈✐❡♥tí✜❝❛s✿ ❘✐❝❛r❞♦ ❞❡ ❖❧✐✈❡✐r❛ ❈❛✈❛❧❝❛♥t✐
P❡♥❤❛ ❈②s♥❡✱ ❘✉❜❡♥s
■♥t❡❣r❛❜✐❧✐t② ❛♥❞ t❤❡ ❞❡♠❛♥❞ ❢♦r ♠♦♥❡t❛r② ❛ss❡ts✿ ❛♥ ❛❧t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤ t♦ ❛♥ ♦❧❞ ♣r♦❜❧❡♠✴ ❘✉❜❡♥s P❡♥❤❛ ❈②s♥❡ ✕ ❘✐♦ ❞❡ ❏❛♥❡✐r♦ ✿ ❋●❱✱❊P●❊✱ ✷✵✶✵
✭❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s❀ ✸✾✽✮
■♥❝❧✉✐ ❜✐❜❧✐♦❣r❛❢✐❛✳
Integrability and The Demand for Monetary
Assets: An Alternative Approach to an Old
Problem
1
.
Rubens Penha Cysne
2July 28, 2000
1T his work bene…ted from conversat ions wit h Robert Lucas Jr, t o whom I am
indebt ed. Humberto Moreira, Andrei Draganescu, Paulo Klinger Mont eiro and Samuel Pessoa made valuable comment s and suggest ions. Remaining errors are my responsabilit y. Financial support from Capes is acknowledged.
2Professor at t he Getulio Vargas Foundat ion. Present ly, visit ing scholar at t he
A bst r act
1
I nt r od uct ion
The answer t o t he quest ion of under what condit ions t he M arshallian con-sumer’s surplus int egral furnishes accept able measures of welfar e change has been given, among ot hers, by Chipman and M oore (1976). The two well known cases ar e t hose of homot het ic preferences and of parallel preferences wit h respect t o t he numeraire.
The at t empt t o int roduce money int o t he analysis of int egrability, wit h money in t he ut ility funct ion, raised several di¢ cult ies. In t he words of Samuelson and Sat o (1984, pp. 591): “ once money ent ers int o t he model in an essent ial way, demand t heory not only loses it s crown jewels (of t est able well behaved curvat ure and perhaps reciprocity relat ions), but worse t han t hat , in a sense it loses it s raison d’et re as a t heory.” These conclusions are a consequence of cont rast ing homogeneit ies of t he demands for goods and for money. I n Samuelson and Sat os’s analysis, money is deduced t o be homogeneous of degree one - not zero - in prices and income. Samuelson and Sat o showed t hat t he problem can be …xed by considering ut ilit y funct ions t hat are weakly separable in goods and money. By assuming t his hypot hesis, int egrabilit y wit h money in t he ut ilit y funct ion can be t reat ed in t he same way as when only goods and services are considered, and we are back t o t he classical goods-and-services approach.
In t his not e, we consider speci…cally t he int egrability of t he demand for monet ar y asset s, and approach t he problem wit h a di¤erent framework. We derive t he demand for monet ary asset s under t he shopping-t ime perspect ive popularized by t he work of M cCallum-Goodfriend (1983). M onet ary asset s are demanded because t hey save agent s t ime which can be allocat ed t o t he product ion of t he consumpt ion good. Money does not ent er int o t he ut ilit y funct ion direct ly.
We do not rely on compensat ed demands and we are not direct ly con-cerned wit h t he t ype of problem raised by Samuelson and Sat o. I ndeed, we do not consider monet ary asset s in t he ut ilit y funct ion.
We mot ivat e our st udy of int egrability by considering part ial-equilibrium measures of t he welfar e cost s of in‡at ion in economies where more t han one asset performs monet ary funct ions. Some examples of works in t he lit era-t ure era-t haera-t use mulera-t idimensional consumer’s surplus measures wiera-t h era-t his inera-t enera-t are M arty and Chaloupka (1988), Marty (1994, 1999) and Balt ensperger and Jordan (1997). In such works, which in a cert ain sense ext end Bailey’s (1956) original cont ribut ion, t he measurement of welfare variat ions is made by cal-culat ing t he areas under t he inverse demand of each asset and t hen adding t he result s. These aut hors, however, concent rat ed on ot her issues, not on in-t egrabiliin-ty. T he resulin-t s here derived supporin-t in-t he pioneering conin-t ribuin-t ions of Mart y and Chaloupka (1988) and Mart y (1984, 1999), when t he t ransact ing t echnology is separ able.
Besides t he applicat ion in t he st udy of t he welfare cost s of in‡at ion, our result s can also be useful when checking for pat h independence of alt ernat ive de…nit ions of Divisia indexes. Cysne ( 2000) invest igat es t his issue.
The cont roversy about t he exact ness of consumer’s surpluses as welfare measures or, relat edly1, of analyzing t he problem of pat h dependence when performing an int egrat ion in t he n-dimensional space, is by no means a new issue in economics. An example t hat dat es from t he mid-ninet eent h cent ury is given by Walras’ (1874 [1954] ; p:443) crit ique of Dupuit ’s (1844) ingenious observat ion t hat demand schedules can be used t o infer welfare cost of price changes. As Hines Jr. (1999) reminds us, Walras, comment ing on Dupuit ’s work, felt compelled t o “ call at t ent ion t o an egregious error which Dupuit commit t ed in a mat t er of capit al import ance.” Walras was referring t o t he quest ionability of assuming t he marginal ut ilit y of income t o be const ant as prices changed. Hot elling (1938) provided a pract ical defense of Dupuit ’s met hod, by analyzing it s implicat ions in a sit uat ion in which more t han one price changes simult aneously (which is going t o be t he same type of quest ion t hat we invest igat e in t his work, regarding t he opport unit y cost of holding di¤erent monet ary asset s).
Hot elling showed t hat , provided t hat t he vect or …eld generat ed by t he simult aneous consumer’s surplus calculat ions is conservat ive, Dupuit ’s anal-ysis could be applied t o each commodity separat ely, for a given pat h in t he n-dimensional Euclidean space, and t he result s could t hen be added. Empir-ically, Hot t elling did not consider income e¤ect s t o be large enough t o make ordinary demand curves inappropriat e for Dupuit ’s pur poses. Hicks(1939, 1942, 1946) t ook anot her approach, showing t hat t he condit ions associat ed wit h t he conservat iveness of t he respect ive vect or …eld, also called int egrabil-ity condit ions, are t rivially sat is…ed by using compensat ed demand curves.
1
Bot h cases demand t hat t he int egrand represent s an exact di¤er ent ial of ut ilit y.
Beyond t hese achievement s, many works have emphasized, aft er t he 1970s, t he import ance of using line int egrals in t he discussion of consumer’s sur-plus. This happens when one want s t o t ake int o considerat ion t he in‡uences of t he market for one good in t he mar ket for anot her good, or service. T wo examples are Silberberg (1972, 1990), and Chipman and Moore ( 1980).
The int egrability problem can be described as follows. Suppose t hat one makes t wo comput at ions, c1 and c2; of t he areas under t he demand curves
of t he asset s. I n comput at ion c1, one …rst calculat es t he area under t he
inver se demand curve of t he …rst asset , holding t he price of t he second asset const ant , and allowing t he demand of t he second asset t o shift t o a new posit ion. T hen one calculat es t he area under t he demand curve of t he second asset and adds it t o t he previous result . Comput at ion c2is made by means of
a symmet ric procedure. One st art s by changing t he price of t he second asset and calculat ing t he area under it s demand, while allowing t he demand of t he …rst asset t o shift . Then, one calculat es t he area under t he demand of t he …rst asset , and adds it t o t he previous ar ea. I f t he int egrabilit y condit ions are not sat is…ed, we can have c16= c2:
We use Simonsen and Cysne’s (1994, 1999) model, which, in t urn, draws on Lucas’ (1993, 2000) previous analysis of welfare cost s of in‡at ion in shopping-t ime economies. To simplify, we present a model using currency and one kind of alt ernat ive monet ary asset , which we call deposit s. Cysne (2000) uses a version of t his model wit h several di¤erent types of deposit s.
The work is st r uct ured as follows: I n sect ion 2, we present t he basic model. Households are assumed t o maximize discount ed ut ility subject t o t heir budget const raint and t o t he const raint t hat t he t ot al t ime spent pro-ducing t he consumpt ion good and shopping must sum t o one. A t ransact ing t echnology speci…es how cur rency and deposit s permit agent s t o economize on t he amount of t ime spent on t ransact ions in t he goods market . Ut ilit y-maximizing behavior generat es t he shopping-t ime curve as well as st andar d asset s’ demand curves. Bot h ar e funct ions of t he nominal int erest rat e on bonds, which is t he opport unit y cost of holding cur rency, and of t he oppor-t unioppor-t y cosoppor-t of holding deposioppor-t s.
2
T he M odel
Household prefer ences are det ermined by: Z 1
0
e¡ gt U(c)dt (1)
where U : X0! R; X0½ R+; is a st rict ly concave funct ion of t he
consump-t ion aconsump-t insconsump-t anconsump-t consump-t and g &gconsump-t; 0. The household is endowed wiconsump-t h one uniconsump-t of t ime t hat can be used t o t r ansact or t o produce t he consumpt ion good wit h const ant ret urns t o scale:
y + s = 1 (2) Here, y st ands for t he product ion of t he consumpt ion good and s for t he fract ion of t he init ial endowment spent as t ransact ing t ime. Households can accumulat e t hree asset s: currency (M ); bonds (B) and deposit s ( X ). To simplify, we assume t hat all asset s are issued by t he government .
In t heir maximizat ion, households t ake as given t he nominal int erest rat e on bonds, i; and t he opport unit y cost of holding deposit s, j = i ¡ ix , where
ix is t he int erest rat e paid by X and 0 j i. Let t ing P = P(t ) be t he
price of t he consumpt ion good, t he household faces t he budget const raint :
_
M + _B + _X = iB + ixX + P (y ¡ c) + H
H indicat es t he (exogenous) ‡ow of currency t ransferred t o t he household by t he government and t he dot over t he variable it s t ime derivat ive. Making ¼= _P=P (in‡at ion rat e), m = M =P; b = B=P; x = X =P and h = H=P; and t aking (2) int o account , t he budget const raint reads:
_
m + _b+ _x = 1 ¡ (c + s) + h + (i ¡ ¼) b + (ix¡ ¼)x ¡ ¼m (3)
Compared t o currency, or t o deposit s, bonds are obviously preferable as a reserve of value. However, currency, as well as deposit s, are useful because t hey save t ransact ion t ime, as t he t ransact ing funct ion describes:
c = F (m; x; s) (4) Here, F is supposed t o be di¤erent iable and st rict ly increasing in each of it s variables, wit h decreasing marginal ret urns.
The household maximizes (1) subject t o t he budget const raint (3) and subject t o t he t ime-t ransact ing t echnology (4). We are int erest ed in st eady-st at e solut ions where m; x and b converge t o coneady-st ant …gures. In t his case, Euler’s equat ions lead t o t he equilibrium relat ions:
i = ¼+ g
Fm = i Fs (5)
Fx = j Fs (6)
In equilibrium, since t he consumpt ion good is non-st orable and since all households are equal, y = c: Using (2) and (4), we get t he fourt h equat ion t hat complet es t he descript ion of household behavior:
1 ¡ s = F (m; x; s) (7) We use equat ions (5), (6) and (7) t o det ermine, locally, i ; j and s; as funct ions of t he asset holdings of m and x: Lat er we will be int erest ed in assuring t hat t hese funct ions are globally int egrable, and t hat t he necessary and su¢ cient condit ions of t he Pot ent ial Funct ion Theorem (present ed in t he next Sect ion) are sat is…ed. Hence, we shall assume t hat i(m; x) and j (m; x) are de…ned over an open and connect ed set , U; which is also simply connect ed.
3
T he M ult id imen sional C onsu mer ’ s Su r p lus
M easur e
I n what follows we will be int erest ed in t he evaluat ion of i(m; x) and j (m; x) along pat hs C( t) = (m(t) ; x(t )); C(t ) ½ U; wit h a t b2: Wit h currency and int erest -bearing deposit s, t he part ial-equilibrium measur e of t he welfare cost s of in‡at ion is given by t he line int egral:
PE = ¡ Z
C
idm + j dx (8) PE can be t hought a generalizat ion of t he area under a demand curve, alt hough it is a di¤erent object from t he mat hemat ical point of view3.
2T he same necessar y and su¢ cient condit ions for int egrabilit y are achieved if one works,
when it is possible, wit h m; x and s det ermined as funct ions of i and j ; and considers P E along pat hs t ! (i , j )(t ):
3
4
Pat h I nd ep en dence
Our object ive here - par alleling t hederivat ion of special condit ions of t he ut il-ity funct ion t hat make t he marginal ut ilit y of income const ant in consumer t heory will be invest igat ing if t he same t ype of procedure can be fruit -ful in regards t o making part icular assumpt ions concerning t he t ransact ing t echnology.
It is nat ural t o hope t hat t he measure PE t akes a unique value for di¤er-ent pat hs of m and x; when t he init ial and …nal point s are t he same. I n or der t o assure necessary and su¢ cient condit ions for t his result t o hold, we need a well-known result from calculus generally called t he “ Pot ent ial Funct ion Theorem” .
T heor em 1 (Potential Function Theorem [PF T]): Let F = (A; D ) be a C1vector …eld in an open connected set L; which is also simply connected.
Then F is conser vative if and only if @A( i; j )
@j =
@D(i; j ) @i Pr oof. See any good t ext book on Calculus.
Given (8), t he condit ions of t he P F T for t he pat h independence of t he welfare measure PE in our problem are met i¤:
@i
@x(m; x) = @j
@m(i ; j ) (9) is veri…ed for all i; j considered in C(m( t); x(t )):
The formal de…nit ion and an encompassing analysis of separability can be found in Leont ief (1947). For our purposes, t he funct ion F (m; x; s) is said t o be blockwise weakly separable when t here are funct ions G and H such t hat F (m; x; s) = H (G(m; x); s): It also follows from t he analysis made by Leont ief t hat t his condit ion is equivalent t o having t he marginal rat e of subst it ut ion bet ween m and x independent of s :
@
@s(Fm=Fx) = 0
Proposit ion 3 uses t his de…nit ion t o est ablish our main result :
Pr oposit ion 2 Suppose that both m(i; j ) and x(i; j ) satisfy the conditions required for t he applicat ion of the P F T: Then, it is a necessar y and su¢ cient condition, for the welfare measure PE to be well de…ned (pat h independent), that t he transacting technology F (m; x; s) is blockwise weakly separable with respect to the monetar y aggregator and the shopping time var iable, s:
Pr oof. Considering (4), t aking t he part ial derivat ives of t he …r st or der and equilibrium condit ions (5), 6), and (7) and making:
¢ = det 2 4
Fs 0 i Fss¡ Fms
0 Fs j Fss¡ F xs
0 0 1 + Fs
3 5 = F2
s(1 + s) > 0
We have ¢ @i
@x = ¡ Fs[(iFsx ¡ Fmx)(1 + Fs) ¡ (iFss¡ Fms) Fx] ¢ @j
@m = ¡ Fs[(j Fsm ¡ Fxm)(1 + Fs) ¡ (j Fss¡ Fx s)Fm] Ther efore,
Fs(1 + Fs)(
@i @x ¡
@j
@m) = (¡ iFsx + j Fsm)(1 + Fs) + Fss(i Fx ¡ j Fm) + (FmFxs¡ FxFms)
Using t he …rst order condit ions t he above equat ion reduces t o:
Fs( 1 + Fs)(
@i @x ¡
@j
@m) = (¡ iFsx + j Fsm) + Fs(i Fxs¡ j Fms) I t follows t hat @i
@x = @j
@m for any values of m and x if and only if FmsFx =
FmFxs: , @@s(Fm=Fx) = 0:4 Proposit ion 3 t hen follows from Lemma 2.
When m and x can be det ermined as funct ions of i and j ; t his result has a dual economic int erpret at ion. Under t he blockwise weak separability of t he t ransact ing t echnology, t he represent at ive consumer’s int ert emporal problem is formally equivalent t o a t wo-st age problem, in which t he shopping t ime and t ot al amount of monet ary services are decided in a …rst st age, and t he relat ive quant it ies of m and x are decided in a second st age. I n t he second st age t he consumer t akes t he opport unit y cost vect or (i; j ) as given and performs a minimizat ion of t he cost of holding monet ary asset s, which is given by R(m; x) = im + j x; subject t o t he const raint t hat t he t ot al quant it y of monet ary services equals t he one decided in st age one.
The set ©(m; x) j G(m; x) ¸ ¹Gª is closed and non-empt y, and t he expen-dit ure funct ion R is precisely it s support funct ion. Since (m; x) = gr ad
4
Once one knows t hat t he funct ions i (m; x ) and j (m; x) can be de…ned, t he condit ion Fm=Fx = Fm s=Fx s can be direct ly achieved by deriving i = Fs=Fm wit h respect t o x ,
R(i ; j ), t he reciprocit y condit ion @m=@j(i; j ) = @j=@i(i; j ) is a direct conse-quence of t he di¤er ent iability of R:
In t he following t wo examples, we consider t he funct ions i (m; x) and j (m; x) de…ned in R2
+ +:
Exam ple 3 Here we consider an economy with a non-weakly separable trans-acting technology, and show t hat the multidimensional consumer s‘ sur plus measure is dependent on the path followed by the monetar y assets. The trans-acting technology is given by:
F (m; x; s) = A(m1=2+ s)x1=2; A > 0:
Fir st notice that this technology is not weakly separable, since Fm=Fx 6=
Fms=Fxs. Given the …rst order conditions, (5) , ( 6) and (7), we have the
functions:
i(m; x) = 1 2m1=2
j (x; m) = 1 + m
1=2
2x(1 + Ax1=2)
and t he par tial-equilibr ium welfare measure:
PE = ¡ Z
C
1
2m1=2dm +
1 + m1=2
2x(1 + Ax1=2)dx ( 10)
Since in this example we are interested only in compar ing t he values of PE along di¤erent paths, we make A = 1 to simplify the calculations. Let us then suppose that this economy presents an initial value of (m; x) given by (0:04; 0:04), and a …nal value of (0:01; 0:01). We consider two di¤erent paths. I n the …rst, C1; the economy moves from (0:04; 0:04) to ( 0:01; 0:01) along the
straight line x = m = t : The second path is a sum of two inter mediate paths. I n the …r st , C21; in which m is always kept constant, the economy moves
from (0:04; 0:04) to (0:04; 0:01): I n the second, C22; in which x is always
kept constant in its new level, 0:01; t he economy moves from (0:04; 0:01) to (0:01; 0:01): By …r st calculat ing t he value of PE for C1; making x = m = t
and dx = dm = dt in (10) , we get PE1 = 0:1 + log 2: On the ot her hand,
along C22; PE22= 0:1; whereas along C21 we have:
PE21= 0:6
Z 0:04 0:01
1
x(1 + x1=2)dx
Since PE2 = PE21+ PE22; PE1= PE2 , P E21= log 2: However, this does
not occur, as the inequality below shows: PE21= 0:6
Z 0:04 0:01
1
x(1 + x1=2)dx > 0:6
Z 0:04 0:01
1
1:2xdx = 0:5 Z 0:04
0:01
1
xdx = log 2
Any evaluation of PE for this economy would therefore have to consider the path followed by the var iables.
Exam ple 4 Here we consider the t echnology F (m; x; s) = G(B ; m; x)s = B m®x1¡ ®s; B > 0: This technology is separable, since F
m=Fx = Fms=Fx s=
(®=(1 ¡ ®))(x=m): Along a path C; this t echnology leads to the following measure of PE:
PE = ¡ Z
C
idm + j dx = ¡ Z
C
® m
1
m®x1¡ ®+ 1dm +
1 ¡ ® x
1
m®x1¡ ®+ 1dx
( 11) We prove that this int egral is path independent in t wo di¤erent ways. I n the …rst, we show that for all closed path Cc its value is equal to zero. This
is equivalent to path independence ( Lang, 1997, p. 399) . We consider the gener ic closed path Cc . I f K is the region in R2 that is bounded by Cc; it
follows from Green’s theorem that : I
Cc
idm + j dx = Z Z K (@j @x ¡ @i @m)dmdx
Checking the partial cross der ivatives in (11), one easily concludes t hat @x@j ¡
@i
@m = 0: Hence, PE is always equal to zero for closed paths, and therefore
path independent.
As a second way to prove the path independence, we show that this problem admits a pot ential function. By making G(m; x) = B m®x1¡ ®and solving the par tial di¤erential equations:
@¡ (m; x) @m =
® m
1 1 + G @¡ (m; x)
@x =
1 ¡ ® x
1 1 + G
one easily …nds t he potent ial function to be ¡ (m; x) = log1+ GG : Hence, for ini-tial and …nal values of m and x given by, respectively, (m1; x1) and (m2; x2) :
PE = ¡ (¡ (m2; x2) ¡ ¡ (m1; x1))
5
C on clu din g R em ar ks
Tradit ional demand t heory approaches t he int egrabilit y problem by consider-ing compensat ed demand funct ions or special feat ures of t he ut ility funct ion. Here we follow t his second t ype of approach wit h respect t o a t ransact ing t echnology. The demands for asset s are derived from a shopping-t ime per-spect ive, in an int ert emporal ut ilit y maximizat ion. We show t hat t he neces-sary and su¢ cient condit ions for int egrability of non-compensat ed demands for monet ary asset s are given by t he blockwise weak separability of t he t rans-act ing t echnology. This out come, derived in t his not e, est ablishes under what condit ions P E is well de…ned. Elsewhere (Simonsen and Cysne (2000) and (2000) ) we also show (assuming weak separabilit y) t hat PE can be regarded as an approximat ion t o t he welfare measur e (variable s) which endogenously emerges from t he model t hat we present here. Added t oget her, t hese result s est ablish t he grounds for, like suggest ed by t he pioneering cont ribut ions of Mart y and Chaloupka (1988) and Mart y (1984, 1999), using PE in or der t o assess t he welfare cost s of in‡at ion.
R efer ences
R efer en ces
[1] Bailey, Mart in J., 1956, Welfare cost of in‡at ionary …nance, Journal of Polit ical Economy 64, 93-110.
[2] Balt ensperger, E. and T homas J. Jordan, 1997 Seigniorage, banking, and t he opt imal quant it y of money, Journal of Banking and Finance 21, 781-796.
[3] Chipman, J. S. and J. C. Moore, 1976, The Scopeof Consumer ’s Surplus Argument s, in Ant hony M. Tang et al., eds, Evolution, Welfare and Time in Economics: Essays in Honor of Nicholas Georgescu-Roegen Lexingt on 1976, 69-123.
[4] Chipman, J. S. and J. C. Moore, 1980, Compensat ing variat ion, con-sumer’s surplus and welfare, American Economic Review (December ), 933-49.
[5] Cysne, R. April 2000, Divisia I ndexes of M onet ary Services as Welfare M easures of I nt erest Rat e Wedges and I n‡at ion, FGV and Universit y of Chicago Working Paper.
[6] Dupuit , Arsène Jules, Ét ienne Juvenal, 1844, De la mesure de l’ut ilit é des t ravaux publics, Annales des Pont s et chaussées. Second series, 8, t rans. R. H. Bar back. I n Readings in Welfare Economics, ed. K. J. Ar row and T Sat ovsky, 255:83, Homewood, I ll.: Richard D Irwin.
[7] Hicks, J. R, 1939, Value and Capit al, 2nd Edit ion, Oxford Universit y Press.
[8] _ _ _ _ _ _ _ _ _ , 1942, Consumer’s surplus and index numbers, Rev. Econ. St udies, 126-37.
[9] _ _ _ _ _ _ _ _ _ , 1946, A Revision of demand t heory, Oxfor d, 1956 [10] Hines Jr. J., 1999, T hree sides of Harberger t riangles, Journal of
Eco-nomic Perspect ives 13,167-188.
[12] Lang, S., Undergraduat e Analysis, 2nd ed. , Springer Verlag.
[13] Leont ief, W. , 1947, I nt roduct ion t o a Theory of t he I nt ernal St ruct ure of Funct ional Relat ionships, Vol. 15, Issue 4, 361-373.
[14] Lucas, R. E. Jr., July, 1993, The welfare cost s of in‡at ion, Univer sit y of Chicago working paper.
[15] _ _ _ _ _ _ _ _ _ , 2000, I n‡at ion and welfare. Economet rica 68, 247-274. [16] M art y, A.L., and F. Chaloupka, 1988, Opt imal in‡at ion rat es: a
gen-eralizat ion, Journal of M oney, Credit and Banking 20, 141-144.
[17] M art y, A.L., 1994, T he in‡at ion t ax and t he marginal welfare cost in a world of currency and deposit s, The Federal Reserve Bank of St . Louis Review 76, 67-71.
[18] _ _ _ _ _ _ _ _ _ , 1999, The welfare cost of in‡at ion: a crit ique of Bailey and Lucas, Federal Bank of Saint Louis Review, N. 1.
[19] M cCallum, Bennet t T . and Marvin S. Goodfriend. “ Demand for Money: T heoret ical St udies” I n T he New Palgrave: A Dict ionary of Economics, edit ed by John Eatwell M ilgrat e and Pet er Newman, p. 775-781, London, M cM illan, 1987.
[20] Samuelson, P. and R. Sat o, 1984, Unnat ainabilit y of int egrabilit y and de…nit eness condit ions in t he general case of demand for money and goods, The American Economic Review 74, 4, 588-604.
[21] Silberberg, E., Duality and t he many consumer’s surpluses, American Economic Review 62 (December): 942-52.
[22] _ _ _ _ _ _ _ _ _ _ , 1990, T he st ruct ure of economics, 2nd ed. New York: M cGraw-Hill 1st ed. 1978.
[23] Simonsen, Mario H. and R. P. Cysne, 1994 and 1999 (revised version), Welfare Cost of In‡at ion and Int erest Bearing Deposit s” . Ensaio Eco-nomico da EPGE/ FGV (1994), Fort hcoming in t he Journal of M oney Credit and Banking, February, 2001.
[24] Walras, Marie Éspirit Léon, 1926, Élement s d’economie polit ique pure, ou t héorie de la richesse sociale, 4t h ed. Lausanne: Corbaz; t rans. William Ja¤é, as Element s of pure economics or t he t heor y of social wealt h, I rwin, 1954.
