Bailey's rule for the welfare of inflation: a theoretical foundation

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■♥❝❧✉✐ ❜✐❜❧✐♦❣r❛❢✐❛✳

Bailey’s Rule For The Welfare Of In‡at ion: A

Theoret ical Foundation

¤

Samuel de Abreu Pessoa

E-mail: pessoa@fgv.br, pessoa@ssc.upenn.edu

Working Paper

A b st r act

T his paper demonst rat es t hat for a very general class of monet ary

models (t he Sidrauski type models and t he cash-in-advance models),

Bai-ley’s rule t o evaluat e t he welfare e¤ect of in‡at ion is indeed accurate. T he

result applies for any t echnology or preference, if t he long-run capit al st ock

does not depend on t he in‡at ion rat e. In general, a dynamic version of

Bailey’s rule is established. In part icular, t he result ext ends t o models in

which t here is a banking sect or t hat supplies money subst it ut es services.

A ddit ionally, it is argued t hat t he relevant money demand concept for

t his issue - t he impact of in‡at ion under welfare - is t he monet ary base.

¤_{T his paper bene…ts from conversat ions wit h M arcos de Barros L isboa. Evident ly}

remain-ing errors are t he responsibilit y of t he aut hor. I acknowledge t he superb guidance of M s. Elizabet h D arby t o t he writ ing and her t olerance and underst anding of my ‘weird L at in-language-speaker’ st ile of writ ing in English. T he aut hor t hanks t he Brazilian Government ’s research assist ance agencies - CA PES and CN Pq - for …nancial support .

1

I nt r oduct ion

Since Bailey’s (1956) classic paper, we have been accust omed t o measuring t he

welfare cost of t he perfect ly foreseen in‡at ion by t he area under t he inverse

money demand. Notwit hst anding, t here has not been much e¤ort in t rying t o

gat her a more solid t heoret ical foundat ion for t his approach. T he aim of t his

paper is t o show t hat it is quit e simple t o …nd t hat t heoret ical foundat ion which

is lacking. It is demonst rat ed t hat for any model a¢ liat ed t o Sidrauski or t o

t he cash-and-advance families of monet ary models, which present a st at

ionary-st at e capit al ionary-st ock t hat is not sensit ive t o in‡at ion, “ Bailey’s rule1_{” provides t he}

accurat e measurement of t he impact of in‡at ion under welfare. When in‡at ion

a¤ect s t he st at ionary-st at e capit al st ock, it is possible t o derive a dynamic

version of Bailey’s rule. In part icular, t his result applies t o models in which

it is t aken int o considerat ion t hat t here is a second sect or, called t he banking

sect or, which provides services t hat are subst it ut es for money services. T his last

class of models present s t he observable phenomenon of t he increase of t he share

in t he product of t he banking sect or, along wit h t he in‡at ion rat e.

T he second cont ribut ion of t he paper is t o est ablish t hat t he relevant concept

of money, as far as t he impact of in‡at ion upon welfare is concerned, is t he

narrow monet ary aggregat e, t he monet ary base. To t he best of my knowledge,

it seems t hat t his point has not been at t ract ing t he deserved at t ent ion by t he

scholars. Bailey’s discussion is not very clear in t his respect . He begins his paper

1_{D ue t o t he generalit y of t he result and of it being a consequence of a very general propert y}

supposing t hat banks are not present . Aft erwards, he int roduces t he banks2. According t o him, if t he bank works rat ionally, t hen t he correct concept is t he

monet ary base; ot herwise, t he M1 demand should be considered, alt hough it is

not very clear what he means by a bank not “ behaving absolut ely rat ionally.”

Lucas (1981), Cooley and Hansen (1989) and (1990) and Lucas (1997) employ

M1; Barro (1972), Fischer (1981) and Aiyagari, Braun and Eckst ein (1998) use

M0.

T he import ance of …nding a t heoret ical foundat ion for Bailey’s rule is t hat

t he alt ernat ive approach, t o calibrat e a dynamic general equilibrium monet ary

model t o evaluat e it3_{, is not robust t o paramet ers calibrat ion}4_{. T he area under}

t he inverse money demand funct ion, a direct ly observable funct ion, does not

present a lack of robust ness. Alt hough, for very low in‡at ion rat es t his measure

could be inexpressive, it can assume quit e high values for high in‡at ions5_{, being}

a reliable lower-bound est imat ion of t he impact of in‡at ion under welfare. A

d-dit ionally, abst ract ing from capit al accumulat ion e¤ect s, t his measure is a t rue

general equilibrium one, and t he speci…c role played by money or t he alt

erna-t ives which are open erna-t o erna-t he economy in order erna-t o adjuserna-t erna-t o a higher in‡aerna-t ion raerna-t e

are not a very import ant issue.

Lucas’ (1997) and A iyagari, Braun and Eckst ein’s (1998) are t he most relat ed

work t o t his one. T he main di¤erence between t he formulat ion accomplished

in t his paper wit h Lucas’ paper is t he speci…c way t he impact of t he in‡at ion

on welfare is calculat ed. Lucas evaluat ed it by t he proport ional increase in

consumpt ion, which makes t he household indi¤erent between t he two sit uat ions

- in t he presence of or wit hout in‡at ion. In t his paper two concept s are adopt ed.

First ly, t he marginal impact of in‡at ion under welfare, measured in t erms of

goods, is evaluat ed. T he t ot al impact of t he in‡at ion under welfare proceeds

from t he int egrat ion of t his marginal e¤ect . Secondly, t he compensat e income

t hat should be given t o t he household in order t o keep it in t he same ut ility level

as under Friedman rule it is considered. A ddit ionally, Lucas does not consider

t he exist ence of a banking sect or which supplies money subst it ut es services.

Aiyagari, Braun, and Eckst ein (1998) examine a cash-in-advance economy

in presence of credit goods. T here is a cont inuum of goods, which could be

acquired in t he market in exchange for money or a credit service. Under t his

second possibility, t he price of a good is t he money price plus a cost which varies,

depending on t he good. T he higher t he in‡at ion rat e, t he larger t he range of

goods acquired by credit and, consequent ly, t he higher t he money velocity is6_.

Similar t o t he present work, t heir model cont emplat es t hat t he provision of t his

money subst it ut es services by t he banking sect or requires t he employment of

product ion fact ors, which have been divert ed from t he real sect or. T he

cash-in-advance model which is invest igat ed in t his paper is a generalizat ion of t heirs. It

is argued, in disagreement wit h t hem, t hat generally t he share in t he product of

t he banking sect or is not t he precise measure of t he allocat ion impact of in‡at ion

under welfare, alt hough t he area under t he inverse money demand funct ion is.

6_{T his manner of producing a variable money velocit y in cash-in-advance models was int}

On t he ot her hand, t his paper generalizes t heir …ndings in many dimensions. It is

shown t hat t he result s depend neit her on t he speci…c monet ary model t aken int o

considerat ion, nor on t he int rat emporal elast icity of subst it ut ion if t he model

considers a cont inuum of goods. Furt hermore, if capit al accumulat ion t akes

place in succession of an increase of in‡at ion, t here is a simple close expression

between t he marginal impact of in‡at ion under welfare and t he marginal impact

of in‡at ion under t he money demand pat h.

In t his paper it is supposed, as it is st andard in t his lit erat ure, t hat t he

economy works under t he monet ary regime: t he unique role of t he government

is t o print money, and, consequent ly, t he seigniorage is rebat ed t o t he household

in a lump-sum fashion. For t his kind of economy t he Friedman rule is sat is…ed.

Alt hough it is an open quest ion7 _{whet her, in presence of ot her imperfect ions,}

t o in‡at e t he price index is a second-best policy or not , t he monet ary regime

provides a benchmark and, as it will be seen, an analyt ical workable solut ion.

T he …rst st ep in …nding a t heoret ical foundat ion for Bailey’s rule is t o work

wit h models t hat present a well-behaved, long-run-money demand. T he idea

behind Bailey’s rule, which is a st andard preference revelat ion argument , is

t hat t he reduct ion in t he consumpt ion surplus caused by t he in‡at ion is t he

correct measure of it s impact under welfare. Consequent ly, t he main ingredient

for Bailey’s rule is t he idea of a st able money demand funct ion. T he di¢ culty

is t hat usually t he monet ary models are dynamic in nat ure, and normally t he

such a way t hat t here is not a st able money demand but a st able money pat h.

To be fair t o Bailey’s rule, it is necessary t o work wit h models which do not

exhibit a t ransit ory dynamic, subsequent ly an alt erat ion of t he in‡at ion rat e

from a long-run equilibrium. Fort unat ely, t here is quit e a large set of models

which possess t his property. Speci…c t o t his class of monet ary models, aft er

a change in t he increase rat e of t he nominal quant ity of money, t he capit al

st ock does not change, t he real quant ity of money and t he consumpt ion ‡ow

jump, and, a new long-run equilibrium is inst ant aneously at t ained. Under t his

condit ion, t he pat h int egral of t he welfare funct ion t rivially becomes a st andard

one, and it is possible t o calculat e it wit hout any considerat ion wit h respect t o

t he speci…c pat h t aken by t he increase-rat e of t he nominal quant ity of money.

T he main result is t hat for a very general class of monet ary models, t he

impact on welfare of an alt erat ion on t he rat e of t he increase of t he nominal

quant ity of money is expressed by

dW d¾ =

Z₁

0

e¡ ½t_¸ dm

d¾dt; (1)

in which

W -... Welfare funct ion;

¾-... increase rat e of t he nominal quant ity of money;

½-... int ert emporal discount rat e;

m-... real quant ity of money.

T his result , a dynamic version of Bailey’s rule, is essent ially a consequence

of Samuelson’s envelop t heorem. In t his model t he social bene…t is equal t o

t he social cost for every choice variable besides money. Consequent ly, as will

be clear lat er, t he impact upon welfare of a changing in ¾, st emmed from t he

alt erat ions of t he choices variables, cancels out . What remains is t he t erm t hat

depends on t he variat ion of t he money demand, which is t he variable t hat has

privat e cost but does not have a social one. As a result , t he amount expressed

by (1) is left .

If addit ionally, it is supposed t hat t he st at ionary-st at e capit al st ock does

not depends on ¾, even if t he ot her real variables (for example consumpt ion,

labor supply, banking service demand, et c.) do, t hen, following a changing in ¾,

t here is no dynamic, and it makes sense t o t alk about a st able money demand

funct ion. Under t his condit ions it follows from (1) t hat

½dW d¾ = ¸

¤dm¤

d¾;

which means t hat t he ‡ow measure of t he marginal impact on welfare of a

changing in ¾, in unit s of capit al is

½ ¹¤

dW d¾ =

¸¤ ¹¤

dm¤

d¾ (2)

¹ -... shadow price of capit al.

Remembering t hat t he relat ive price of t he real quant ity of money in unit s

of capit al is t he nominal int erest rat e, Bailey’s rule follows from t he int egrat ion

of (2)

½¢ WU ni t s of C api t al =

Z _¾

¡ ½

R¤dm

¤

d¾0d¾ 0_{= ¡}

Z _m¤_{( ¡ ½)}

m¤_{( ¾)}

R(m)dm (3)

in which

R-... nominal int erest rat e.

Furt hermore, for t he same class of monet ary models which (3) applies, t his

paper shows t hat , if t he impact of in‡at ion under welfare is measured by t he

compensat e income which should be given t o t he household t o keep his in t he

same ut ility level, t han t he int egral in (3) should be t aken along t he compensat e

money demand.

In t he …nal part of t he paper it is argued t hat t he relevant concept of money

for t his issue - t he impact of in‡at ion under welfare - is t he monet ary base and

not M1. T he reason is t hat t he demand deposit is a service which belongs t o

t he bundle of services t hat are o¤ered by t he banking sect or. T he result follows

because t he area under t he inverse money demand grasp all t he general

equi-librium e¤ect s of a increase of in‡at ion on t he economy, including t he increase

of t he share of t he banking sect or. To put in anot her way, as far as t he e¤ect s

of in‡at ion under welfare are concerned, money is t he good which has privat e

should be excluded from t he concept of money. It is o¤ered by t he banking

inst it ut ions, and, consequent ly, has a posit ive social cost . To make t his point

clear, a model in which inside money t akes place can be found in t he sevent h

Sect ion of t he paper.

T he paper has t he following organizat ion. In t he subsequent Sect ion t o

t his int roduct ion, t he set up of t he model is exposed, and in t he t hird Sect ion

t he generality of Bailey’s rule is demonst rat ed. T he fourt h Sect ion deals wit h

t he sit uat ion in which unbound growt h is present , and t he validity of Bailey’s

rule for t he cash-in-advance class of monet ary models is discussed in t he …ft h

Sect ion. T he ensuing Sect ion present s t he validity of Bailey’s rule when t he

compensat e demand concept t o measure welfare variat ions is applied and t he

sevent h Sect ion discusses t he correct concept of money for t his subject - welfare

e¤ect s of in‡at ion. T he conclusion follows.

2

T he G ener al M odel

Usually money can be incorporat ed in an ot herwise st andard macroeconomic

dynamic real model in two ways: direct ly int o t he preference, t he st andard

Sidrauski (1967) model or as an argument of a t ransact ion cost funct ion int o

t he budget const raint , t he way popularized by McCallum and Goodfriend (1987)

in t heir ent ry in Palgrave’s dict ionary8_{. In order t o keep t he model exposed here}

8_{I t seems t o me t hat D razen (1979) was t he …rst t o suggest t his manner of building a}

as general as possible, it will be supposed t hat bot h possibilit ies are present . In

addit ion, it is considered t hat t here is anot her good, along wit h t he t radit ional

good which could be consumed and st ocked as capit al, called banking service

which helps t he household in reducing t ransact ion cost s, wherever it appears.

Bassole and Pessoa (1999) elsewhere t reat ed t his model in det ail. T he most

int erest ing feat ure is t hat it displays t he phenomenon of t he increase of t he

banking sect or for providing t he public goods and services, which are subst it ut es

for money.

Households

T he choice problem of t he household is t he following

max Z₁

0

e¡ ½tu(c1t; l(c1t; m1t; c21t))dt; (4)

subject t o

:

at = rtat+ wt + Ât ¡ c1t¡ ptc2t¡ g(c1t; m2t; c22t) ¡ (¼t + rt)mt; (5)

mt ´

Mt

P1t

; and pt ´

P2t

P1t

; (6)

at ´ kt+ mt; (7)

mt ´ m1t + m2t ; (8)

c2t ´ c21t + c22t: (9)

in which

l-... leisure;

g-... t ransact ion-cost funct ion;

M -... nominal per capita money st ock;

P1-... nominal price of t he …rst good (which could be consumed or accumulat ed

as capit al);

P2-... nominal price of t he banking service;

kt-... per capita capit al st ock;

r -... real int erest rat e or remunerat ion of capit al services;

¼-... in‡at ion rat e;

w-... remunerat ion of t he labor services;

c1-... ‡ow of consumpt ion good;

m1-... services of t he real monet ary st ock allocat ed for saving t ime;

m2-... services of t he real monet ary st ock allocat ed for saving t ransact ion cost ;

c21-... ‡ow of banking services employed for saving t ime;

c22-... ‡ow of banking services employed for saving t ransact ion cost s;

T his is quit e a general model9. For example, if it is supposed t hat leisure depends only on t he quant ity of money and if t here is no banking sect or and

t ransact ion cost s, we are back t o Sidrauski model. On t he ot her hand, if it

is assumed t hat t he inst ant ut ility depends only on consumpt ion and t hat t he

banking sect or does not exist , t hen we are back t o McCallum and Goodfriend

model. Finally, if it is supposed t hat leisure depends only on money and banking

services and t hat t here are no t ransact ion cost s, t he model becomes a simple

two-sect or model which could rat ionalize t he idea of a banking two-sect or. It is possible

t o imagine any combinat ion of t hese t hree models. T he exist ence problem is

not t he main concern of t his paper. It is supposed t hat t he solut ion exist s

and is well-behaved. If leisure and t he t ransact ion funct ion do not depend on

t he consumpt ion ‡ow, it is easy t o see t hat it is possible t o suppose t hat bot h

funct ions are st rict ly concave and, consequent ly, t hat exist ence and uniqueness

is guarant eed10_.

Fir st-Order Conditions

Let ¹_t represent s t he cost at e variable associat ed wit h t he rest rict ion (5), which is obviously t he shadow price of t he capit al good. T he maximizat ion

problem of t he household is a st andard one. T he cont rol variables are11_{: c} 1,

9_{T he st andard assumpt ions are: u}

i > 0, l1< 0, li > 0, g1> 0, gi < 0; ui i < 0, jui jj > 0,

g11 > 0, gi i < 0, jgi jj > 0 and condit ions t hat est ablishes t hat money and banking services

are subst it ut es: l23< 0 and g23< 0. 10_{Evident ly, ruling out monet ary bubbles.}

m1, c21, m2 and c22. It follows t he …rst -order condit ions

u1+ u2l1 = ¹ (1 + g1); (10)

u2l2 = ¹ (¼+ r ); (11)

u2l3 = ¹ p; (12)

¡ g2 = ¼+ r ; (13)

¡ g3 = p: (14)

For t he household st at e variable (asset s), t he Euler equat ion follows

:

¹

¹ = ½¡ r :

Fir ms

T his economy is a twosect or economy. T he …rst sect or, applying a …rst

-order degree homogenous product ion funct ion and employing capit al and labor,

produces a good which could be consumed or accumulat ed as capit al. T he

sec-ond sect or, applying an equivalent t echnology, produces a service called banking

services, which could be acquired by t he household in t he market . It is assumed

t hat t he fact ors market clears cont inuously; fact ors are perfect ly mobile across

sect ors and are supplied ineslat icly. Under t hese condit ions t he equilibrium

supplies funct ions (one for each sect or)

y1= y1(p; k) and y2= y2(p; k);

in which

yi-... per capita product ion of t he i -esimo good.

From t he inclinat ion of t he possibilit ies product ion front ier it is known t hat

y11+ py21= 0; (15)

and from t he social marginal impact of capit al it is known t hat

d

dk(y1+ py2) = y12+ py22= f

0

1(k1(p)) = pf20(k2(p)) = r (16)

in which

fi-... i -esimo sect or product per worker;

ki-... i -esimo sect or capit al per worker rat io.

Gover nment

T he role of t he Government in t his economy is t o print money, which is

a st andard assumpt ion in t his lit erat ure. Evident ly, if it has been assumed

t hat t here has been government consumpt ion which should be …nanced by

dis-t ordis-t ed dis-t axes, dis-t he calculadis-t ion of dis-t he in‡adis-t ion impacdis-t under welfare would had

quant it at ively t his e¤ect is not very large. Under t he monet ary regime, t he

government t ransference t o t he public is t he seigniorage which is equal t o t he

in‡at ionary t ax plus t he increase in t he real quant ity of money. T hat is

 = m + ¼m::

Shor t Run Equilibrium and Dynamics

T he market for banking services clears cont inuously, which means t hat it s

relat ive price (p) adjust s t o accomplish t his equilibrium. Due t o Walras’ law,

t his equilibrium condit ion, plus t he equilibrium in t he money market , implies

t he equilibrium of t he goods market . T he condit ion for t he equilibrium in t he

banking services market

y2(p; k) ¡ c2= 0;

along wit h t he equat ions (8), (9), (10)-(14), det ermine c1, m1, c21, m2, c22, p, c2

and ¼as funct ion of t he st at e variable k, t he cost at e variable ¹ and t he st at e-like

variable m. T his est ablishes t he moment ary equilibrium for t his economy.

T he dynamic is given by t he following equat ions

:

k = y1(p; k) ¡ c1¡ g(c1; m2; c22); (17) :

¹ = ¹ (½¡ f₁0(k1(p)); (18) :

in which

¾´

:

M M.

By looking at t his syst em of equat ions, t he peculiar role played by t he

pa-ramet er ¾is highlight ed. Alt hough it displaces t he equilibrium posit ion, it does

not direct ly change any …rst -order condit ion. T his property will be essent ial

lat er.

A very import ant case t hat will be dealt wit h lat er is t he sit uat ion in which

t he t echnology is t he same across sect ors. If t his is t rue, alt hough from t he

demand point of view t he two goods are dist inct , from t he supply point of view

t hey are equal. Under t his condit ion, t he economy works as if it was an

one-sect or economy, which means t hat t he relat ive price of t he banking service is

const ant and t hat t he int erest rat e is det ermined as usual by

r = f0(k):

It follows in t his sit uat ion, from t his last equat ion and (18), evaluat ed in

t he st at ionary-st at e, t hat t he long-run capit al st ock is …xed and independent of

¾. T hat is, aft er an alt erat ion of t he increase rat e of t he nominal quant ity of

money, t he economy will not present any dynamics. T he following variables

-t he con-t rol variable, -t he s-t a-t e-like variable, and -t he cos-t a-t e variable - jump, and

a new long-run equilibrium is immediat ely at t ained. Only under t his sit uat ion

3

I m pact On W elfar e

In t his represent at ive agent economy, welfare is equal t o t he int ert emporal ut ility

of t he household, expression (4). T hen, it is possible t o direct ly calculat e t he

impact upon welfare of a marginal increase of ¾.

dW d¾ =

Z ₁

0

e¡ ½t d

d¾u(c1; l(c1; m1; c21))dt =

Z ₁

0

e¡ ½t ·

(u1+ u2l1)

dc1

d¾+ u2l2 dm1

d¾ + u2l3 dc21

d¾ ¸

dt: (20)

Subst it ut ing in t his last equat ion t he …rst -order condit ions (10)-(12), it

fol-lows t hat

dW d¾ =

Z ₁

0

e¡ ½t¹ ·

(1 + g1)

dc1

d¾+ (¼+ r ) dm1

d¾ + p dc21

d¾ ¸

dt:

From t he equilibrium in t he market for goods, it is known t hat

Z₁

0

e¡ ½t¹ d d¾

³

y1(p; k) ¡ c1¡ g(c1; m2; c22) :

¡ k ´

dt = 0

and for t he banking services market

Z ₁

0

e¡ ½t_{¹ p}d

which could respect ively be writ t en as

Z ₁

0

e¡ ½t¹ 0 @y11

dp d¾+ y12

dk

d¾¡ (1 + g1) dc1

d¾¡ g2 dm2

d¾ ¡ g3 dc22

d¾¡ dk: d¾

1 A dt = 0

(21)

and

Z₁

0

e¡ ½t¹ p µ

y21

dp d¾+ y22

dk d¾¡

dc2

d¾ ¶

dt = 0: (22)

Int egrat ing by part s t he last t erm in (21) and recalling t hat capit al does not

jump and it is bounded, it follows t hat

Z ₁

0

e¡ ½t¹ d dt

dk d¾dt = ¡

Z ₁

0

e¡ ½t¹ (¡ ½+

:

¹ ¹ )

dk

d¾dt: (23)

Subst it ut ing (23) in (21), adding t he result and (22) t o (20) it is left

dW d¾ =

Z ₁

0

e¡ ½t¹ f ·

(1 + g1)

dc1

d¾+ (¼+ r ) dm1

d¾ + p dc21 d¾ ¸ + Ã y11 dp d¾+ y12

dk

d¾¡ (1 + g1) dc1

d¾¡ g2 dm2

d¾ ¡ g3 dc22

d¾+ (¡ ½+

: ¹ ¹) dk d¾ ! + p µ y21 dp d¾+ y22

dk d¾¡ dc2 d¾ ¶ gdt:

Aft er recalling (13), (14), (15), (16), and (18), every t erm which is not

mult iplied by dmi

d ¾ cancels out . It remains

dW d¾ =

Z1 0

e¡ ½t¹ (¼+ r )dm

T his canceling out expresses t hat besides money, t he ot hers choice variables

present a social bene…t and a social cost , which by t he choice mechanism are

equal, alt hough welfare t heorems are not sat is…ed for monet ary models12_{. In}

ot hers words, t his is a welfare maximizing economy rest rict ed t o t he fact t hat t he

household is consuming less monet ary services t han t he social opt imum. T hat

is, a Social Planer who could not avoid in‡at ion, and who could not induce t he

households t o increase t heir money holdings, would have done no bet t er t han

t he market . Consequent ly, because money has bene…t but does not have cost ,

t here is not t his kind of canceling out ; t he amount expressed by (24) remains.

It is import ant t o not e t hat t here was not any supposit ion about t he speci…c

value of ¾ in deriving t his result , which means t hat expression (24) applies t o

every value for ¾. T hen, regardless of it s value, a furt her increase (or decrease)

produces t hat canceling out , if it is t aken int o considerat ion t hat t he

decision-makers redo t heir opt imum calculat ions13_{. T his result , which is t he one we are}

int erest ed in14_{, is amazingly general. It st at es t hat t he marginal impact of ¾on}

welfare is t he present value, in unit s of ut ilit ies, of t he marginal impact of ¾ on

t he money demand. T he speci…c adjust ment which t akes place in succession a

alt erat ion on ¾ does not mat t er; t he money demand re‡ect s it s. T his result is

a dynamic version of Bailey’s rule.

In deriving (24) t here was not made any hypot hesis respect t o t he variable

12_{T his derivat ion resembles Samuelson’s envelop t heorem; however, it is not quit e t he same.}

I n deriving t he envelop t heorem for a rest rict maximum, t he rest rict ion faced by t he decision maker is added t o t he indirect ut ilit y funct ion. D i¤erent ly, in order t o derive (24), t he rest ric-t ion seen by ric-t he social planer, which are ric-t he physical balance equaric-t ion for ric-t he goods produced by t he economy, was added t o t he indirect welfare funct ion.

¾. T hat is t o say, ¾ could be any exogenous variable. As an example, if it had

been supposed t hat t here was a purchase t ax for any good, following t he same

rout e which leads us t o (24), it would sent us t o

dW d¾j¿= 0=

Z ₁

0

e¡ ½t¹ (¼+ r )dm d¿dt

in which ¿ is t he t ax rat e. T he import ant dist inct ion is t hat t his derivat ive

would apply in t he neighborhood of t he t ax rat e close t o zero; in cont rast , due

t o t he part icular role played by t he paramet er ¾ in monet ary models15_{, (24) is}

a global result .

To go furt her, t he long-run capit al st ock should not be sensit ive t o t he

in‡at ion rat e. As it was seen, it is necessary t o assume t hat t he t echnology is

t he same among sect ors; ot herwise, t he concept of a st able demand funct ion is

meaningless. In such a sit uat ion, it is possible t o int egrat e (24) t o get16

½dW d¾= ¹

¤_R¤dm¤

d¾:

T his means t hat in unit s of capit al17_{, it follows t hat}

½ ¹¤

dW d¾ = R

¤dm¤

d¾:

15_{Generally, a paramet er displaces some …rst -order condit ions. T hat is not t he case}

regard-ing t o ¾.

16_{T he ‘* ’ is t o remember t hat from t his point on t he result s refer t o a st at ionary st at e}

capit al st ock t hat does not change wit h in‡at ion.

Int egrat ing t his last equat ion on ¾ we are back t o (3)

½¢ WU nit s of Capi t al =

Z _¾

¡ ½

R¤dm

¤

d¾0d¾ 0_{= ¡}

Z _m¤

( ¡ ½) m¤_{( ¾)}

R¤(m)dm: (25)

T he equat ion (25) is t he main result of t he paper. For a very general class

of monet ary models, t he area under t he inverse money demand funct ion is t he

accurat e measurement of t he impact of in‡at ion under welfare. Said di¤erent ly:

“ T his conclusion, t hat t he area under t he observed demand curve

for real cash balances during an in‡at ion measures t he welfare cost s

of t he reduct ion of t hese balances, applies r egar dless of t he

par-t icular manner in which par-t hese cospar-t s a¤ecpar-t real income and leisure.”

(Bailey, (1956), pg.102, emphasis added.)

Usually, t he welfare cost of in‡at ion is measured in unit s of consumpt ion

goods and not in unit s of capit al goods. Why, in general is Bailey’s rules valid

when welfare is measured in unit s of capit al but not in unit s of consumpt ion

goods?

Looking at (11) or (13), it is clear t hat t he relat ive price of money in unit s

of capit al is t he nominal int erest rat e. On t he ot her hand, it follows from (10)

t hat t he relat ive price of t he consumpt ion good in unit s of capit al is 1 + g1.

T hen, in t erms of consumpt ion good, it is possible t o rewrit e (25) as

¡ ½¢ WU nit s of Cons. G oods=

Z _m¤

( ¡ ½) m¤_{( ¾)}

R(m) 1 + g1(c(m); m)

dm · Z _m¤

( ¡ ½) m¤_{( ¾)}

t he equality occurring if g1 = 0. If t his last condit ion applies, Bailey’s rule is

exact for welfare in unit s of consumpt ion good. It means t hat for t he st

an-dard Sidrauski model, t he McCallum-Goodfriend model (if t he t ransact ion cost

funct ion does not depend on consumpt ion) and for t he two-sect or model wit h

a banking sect or o¤ering subst it ut es for money (if t he t echnology is t he same

across sect or), Bailey’s rule is exact . T he area under t he inverse demand

func-t ion overesfunc-t imafunc-t es func-t he welfare cosfunc-t of in‡afunc-t ion in unifunc-t s of consumpfunc-t ion goods

if t he t ransact ion cost funct ion is sensit ive t o t he amount which has been

con-sumed.

In t he last paragraph it was seen t hat Bailey’s rule overest imat es t he

wel-fare cost of in‡at ion measured in unit s of consumpt ion goods if g1> 0, for t he

McCallum-Goodfriend model. Once Bailey’s rule is not exact in t his cont ext

t he quest ion remains: is t here anot her int erpret at ion for t he in‡at ion impact

on welfare? A possible rout e is t o calculat e welfare in t erms of t he

consump-t ion demand insconsump-t ead of consump-t he money demand. Iconsump-t is useful consump-t o work on a simpli…ed

version of t he general model of t he second Sect ion t o accomplish t his. If it

is supposed t hat t he moment ary ut ility funct ion depends only on

consump-t ion, and consump-t haconsump-t consump-t here is only one secconsump-t or, consump-t he second Secconsump-t ion model consump-t urns inconsump-t o

t he McCallum-Goodfriend model. T he unique depart ure of t his model from t he

st andard Ramsey-Cass-K oopmans model is t he t ransact ion cost g(c; m), which

subt ract resources from t he household budget const raint . As we know,

expres-sion (25) applies t o t his economy. It is allowed t o st art from t here. From t he

follows t hat

f (k¤) = c¤+ g(c¤; m¤)

which means t hat

(1 + g1)

dc¤

d¾= ¡ g2 dm¤

d¾ = (¼+ r ) dm¤

d¾: (27)

T he last equality follows from (13). Subst it ut ing (27) in (26), it follows t hat

¡ ½¢ WU nit s of Cons. Goods= ¢ c¤:

As expect ed, t he welfare cost of in‡at ion for t he McCallum-Goodfriend

frame-work18 _{is t he reduct ion of consumpt ion which t akes place due t o t he increase}

of t he in‡at ion rat e. It is clear now why t he welfare cost of in‡at ion measured

in unit s of consumpt ion is lower t han in unit s of income, for t he

McCallum-Goodfriend model; t o produce one unit of consumpt ion good it is necessary

1 + g1unit s of income.

Summing up and remembering t hat r¤_{= ½, it is possible t o writ e}

r¤¢ WU n it s of A sset s= ¡

Z _m¤

( ¡ ½) m¤_{( ¾)}

R(m)dm: (28)

18_{T his result is valid if t his model augment ed wit h banking services in t he t ransact ion}

funct ion is considered. T his is t rue because t he t erm ¡ dc2in t he budget const raint is canceled

It is import ant t o emphasize here t hat t he marginal t ransformat ion rat e between

money and asset s, for t he household, is t he nominal int erest rat e19_{. If t he capit al}

relat ive price in unit s of asset s is const ant , Bailey’s rule applies in unit s of

capit al; if t he consumpt ion relat ive price in unit s of asset s is const ant , Bailey’s

rule applies in unit s of consumpt ion goods.

4

M oney D em and and G r ow t h

T he model t hat was discussed in Sect ion 2 does not present growt h. It is known

t hat at …rst approximat ion t he income elast icity of money demand is roughly

one20_{. It would be int erest ing t o know how t he result t hat we have so far got t en}

would change, or not , in a model which exhibit s a st at ionary solut ion when

t here is a long-run t rend in income. T he st andard Sidrauski model augment ed

wit h exogenous t echnological progress present s t his property of const ancy in t he

long-run of t he income money velocity21_{. On t he ot her hand it will be possible}

t o compare t he result in t his paper wit h Lucas (1997), which is qualit at ively

di¤erent22_{. It will be shown t hat t he int roduct ion of a t rend in income does not}

change t he result t hat Bailey’s rules is exact in t he st andard Sidrauski model.

Household

19_{I t follows direct ly from t he privat e budget rest rict ion (5).}

20_{I n fact , it is known t hat t his elast icit y is lower t han one (see for example L ucas (1997),}

…gure 1).

21_{T he models which cont emplat e a banking sect or generally do not present a}

long-run-growt h st at ionary solut ion. T he exogenous t echnological change cont inuously reduces t he relat ive price of t he banking services, if t he in‡at ion rat e does not present a t rend, and, consequent ly, t he long-run money demand present s a lessening t endency. I t seems t o me t hat A iyagari et ali i (1998) did not not ice t his fact (see t heir discussion on pg.1289).

T he household solves

max Z ₁

0

e¡ ½t(u(

»

c;m))» 1¡ ®_{¡ 1}

1 ¡ ® dt

subject t o

d»a dt =

»

w +»ar ¡ »c ¡ (¼+ r )m +» »Â;

in which u(¢; ¢) is …rst -order degreehomogenous. It is possiblet o writ e23_u(»_{c;m) =}» »

c' (m

c), in which t he variables wit hout ‘» ’ are det rended ones. Let »

¹ be t he

shadow price of»a and ¹ ´ »¹ e®gt _{t he det rended price; t hen t he …rst -order}

con-dit ions for t he cont rol variables follows

(c' (m c))

¡ ®h_{' (}m

c) ¡ m

c'

0₍m

c) i

= ¹ ;

(c' (m c))

¡ ®_' 0₍m

c) = ¹ (¼+ r )

and for t he st at e variable

:

¹ = ¹ (½+ ®g ¡ r ):

Fir ms

T he …rst -order condit ions for t he …rms lead t o

w ´

»

w

egt = f (k) ¡ kf

0_{(k); (29)}

r = f0_{(k): (30)}

General Equilibrium and Dynamics

Remembering t hat t he Government det rended t ransfer sat is…es

 = m + (¼+ g)m;: (31)

and subst it ut ing (29)-(31) int o t he det rended household budget rest rict ion, we

are left wit h t he following dynamic syst em

:

k = f (k) ¡ c ¡ gk; (32)

:

m = m(¾¡ (¼+ g));

:

¹ = ¹ (½+ ®g ¡ f0(k)):

Welfare

T he impact on Welfare of t he in‡at ion could be calculat ed from

dW d¾ =

Z ₁

0

e¡ ½t_(u(»_c;m))» ¡ ®

" u1(

»

c;m)» d

»

c d¾+ u2(

»

c;m)» d

» m d¾ # dt; = Z ₁ 0

e¡ ( ½¡ g( 1¡ ®) ) t¹ ·

dc

d¾+ (¼+ r ) dm

d¾ ¸

dt

follows aft er t he subst it ut ion of t he …rst -order condit ion for consumpt ion and

money services. Aft er di¤erent iat ing (32) against ®, and redoing t he st eps of

t he last Sect ion, it follows t hat

dW d¾ =

Z₁

0

e¡ ( ½¡ g( 1¡ ®) ) t¹ ·

(¼+ r )dm

d¾+ (® ¡ 1) dk d¾

¸ dt:

Recalling t hat in t he st at ionary-st at e t he capit al st ock does not vary wit h

¾; it is possible t o solve t he int egral

(½¡ g(1 ¡ ®))dW d¾ = (r

¤_{¡ g)}dW

d¾ = ¹

¤_R¤dm¤

d¾;

which means t hat

(r¤¡ g)¢ WU n it s of G oods= ¡

Z _m¤_{( ¡ ( ½+ ®g) )}

m¤_{( ¾)}

R(m)dm: (33)

Comparing (33) wit h (28), t he di¤erences in t he measurement of t he welfare

cost of in‡at ion when t echnological exogenous progress t akes place are twofold.

First ly, t o calculat e t he impact of in‡at ion under welfare, t he area under t he

inverse money demand funct ion should be divided by t he int erest rat e net of

t he growt h rat e. T herefore, t he presence of unbounded growt h st rengt hens or

weakens t he case against monet ary …nance whet her t he int ert emporal elast icity

(33), m¤(¾), is higher under growt h24; 25, st rengt hened t he case against in‡at ion …nance in t his cont ext26_{. T he next Sect ion shows t hat t he validity of (1) and (3)}

are not an art ifact of t he Sidrauski model or t he McCallum-Goodfriend version

of it .

5

A C ash-i n-A dvance Economy

From t he point of view of get t ing a deeper acquaint ance of t he monet ary

phe-nomenon, t he models t hat were invest igat ed unt il t he last Sect ion belong t o t he

family of Sidrauski models. T he next cat egory of monet ary models in

increas-ing order of underst andincreas-ing of t he monet ary phenomenon are t he cash-in-advance

models. T he aim of t his Sect ion is t o demonst rat e t hat t he result s which were

derived for t he Sidrauski-type models are valid t o t his family of monet ary

mod-els. The same rout e will be followed: for a very general cash-in-advance model,

which could encompass many models as a part icular case, (1) and (3) will be

est ablished.

T he drawback of t he st andard27 _{cash-in-advance model is t he const ancy}

in income velocity. T he manner which has been suggest ed t o cope wit h t his

24_{T he nominal int erest rat e, when t he increase rat e of t he nominal quant it y of money is ¾,}

is ¾+ ½+ ®g, which is higher t han ¾+ ½ whenever g > 0. On t he ot her hand, t he inverse money demand funct ions are iqual, once it is recalled t hat t he marginal condit ions t hat bring t hem about are equal.

25_{I n t his growing economy, Friedman’s rule requires a de‡at ion rat e equal t o ½+ ®g, which}

is higher t han t he usual ½.

26_{T he quali…cat ions on t he measure of t he impact of in‡at ion under welfare when growt h}

t akes place was quit e an import ant issue in t he sixt ies and sevent ies. See Tower (1971), M art y (1973) and (1976), Cat hcart (1974), Tat om (1976), and Chappell (1981). H owever, t hese works address t his issue under a diverse set of hypot heses, and, consequent ly, are not appropriat e for comparison wit h t his paper.

limit at ion is t o add goods t hat can be purchased by credit28. As put fort h by Gillman (1993), it is possible t o consider a cont inuum of goods, which, from

t he preference point of view possesses symmet ric roles, alt hough not from t he

t ransact ion t echnology point of view. Under t his formulat ion, every good can

be purchased by money or credit . T he dist inct ion is t hat t here is a credit cost

at t ached t o each good which varies across goods, in such a way t hat as in‡at ion

increases, t he range of goods which are credit goods increases. If it is considered

t hat t hese credit services are o¤ered by a sect or of t he economy which employes

product ion fact ors in order t o produce it , we are in t he Aiyagari, Braun, and

Eckest ein (1998) framework.

T he model t hat will be st udy in t his Sect ion is a generalizat ion of t heir

model in one direct ion: t he aggregat or funct ion, which de…nes t he consumpt ion

good and t he invest ment good, present s elast icity of subst it ut ion across types

of goods larger t han zero. T here are two main reasons for t his choice. First ly,

it is int ended t o work in a more general set up, which can deliver ot her models

as a part icular case. Secondly, t he sit uat ion in which t he elast icity across types

of goods is higher t han zero produces anot her impact of in‡at ion under welfare.

Due t o t he symmet ric role played by t he goods in preference, t he household

prefers t o smoot he consumpt ion across types. Notwit hst anding t his, among t he

goods acquired as credit goods, t he relat ive price - t he credit cost relat ive t o

t he nominal int erest rat e - varies in such a way t hat following an increase in

relat ively rich descript ion of a monet ary economy under cert ainty. Following

an increase in in‡at ion, t he range of cash goods decreases, t he consumpt ion

pro…le of t he household twist s, t he banking sect or absorbs product ion fact ors t o

o¤er t ransact ion services, and t he accumulat ion of capit al is hindered. However,

it will be shown t hat (1) represent s t he marginal impact under welfare of t he

in‡at ion. Moreover, if it is supposed t hat capit al accumulat ion is not a¤ect ed

by in‡at ion, Bailey’s rules is again valid.

5.1

T he M odel

T here is a cont inuum of goods index by z 2 [0; 1]. T hey are ident ical goods from

t he supply point of view, which means t hat t he producer price Pt is t he same,

regardless of t he type. T here is anot her sect or in t his economy, t he banking

sect or, which produces a service. Each good could be acquired as cash good

or credit good. In t he …rst case, t he household pays Pt, but has t o have it as

cash, which means t hat t he cost it faces is (1+ Rt)Pt, in which R is t he nominal

int erest rat e. When buying a good as credit good t he household pays Pt t o

t he good’s producer plus t he int ermediat ion services cost . Following A iyagari et

alii, it is supposed t hat t o acquire a unit of good of any quality as credit good,

it is necessary t o buy R(z) unit s of banking services, which cost qsR(z) in unit s

of goods. Consequent ly, t he e¤ect ive cost of a credit good t o t he household

is Pt(1 + qsR (z)). It is supposed t hat t he product ion funct ion for goods and

t ransact ion services are t he same, which means t hat it is possible t o normalize

which nt is t he per capita supply of labor services. Moreover, t he t ransact ion

services cost funct ion is increase in t he index z and R(0) = 0. At any moment

t here is a cut -o¤ index, zt, such t hat any good whose index is lower t han t he

cut -o¤ is bought as credit good, and t he ot hers are bought as cash.

Household Choice

T he household solves

max

1

X

t = 0

¯tu(ct; 1 ¡ nt) (34)

in which

ct =

µ Z ₁

0

c

µ ¡ 1 µ

t (z) dz

¶ µ µ ¡ 1

(35)

is an aggregat or funct ion t hat de…nes t he unit of consumpt ion.

T he household faces two sort s of rest rict ions. One is t he cash-in-advance

and t he ot her is t he budget const raint . Before going t o t he good market , it is

possible t o go t o t he credit market , in order t o t ake cash. T his operat ion is

cost less. Let Mt; Bt and Xt be, respect ively, t he nominal quant ity of money, of

bonds in t he household port folio, and t he nominal value of government t ransfer.

T he cash-in-advance rest rict ion is

Mt+ Xt

Pt

+ Bt Pt

¡ Bt + 1 Pt(1 + Rt)

¸ 1 Pt

Z ₁

zt

Pt(z)(ct(z) + it(z))dz: (36)

going t o t he goods market in t he inst ant t, and t he right side is t he nominal

cost of cash goods. T he budget const raint is

Mt+ Xt

Pt

+ Bt Pt

+ wtnt+ rtkt ¸

1 Pt

Z ₁

0

Pt(z)(ct(z) + it(z))dz +

Mt + 1

Pt

+ Bt + 1 Pt(1 + Rt)

:

(37)

T he movement equat ion for capit al is

kt + 1= it + (1 ¡ ±)kt; (38)

in which, it is an aggregat or funct ion t hat de…nes t he invest ment good

it =

µ Z ₁

0

i

µ ¡ 1 µ

t (z) dz

¶ µ µ ¡ 1

: (39)

Taking t he limit µ ! 0 t his model deliver Aiyagari et alii model; t he limit

µ ! 1 reproduces Gillman model if an economy wit hout capit al is considered.

If t he cut -o¤ index, zt; is …xed and if t here are neit her banking services nor

t ransact ion services, t he model reproduces Lucas and St okey’s (1983) economy,

and if t here are no credit goods, t he model generat es St ockman’s (1981) model.

Addit ionally, if capit al is cost less credit good, Lucas’s (1980) model is obt ained.

For t his const ant -subst it ut ion-elast icity aggregat or is known t hat

µ ct(z)

ct

¶¡ 1 µ

= µ

it(z)

it

¶¡ 1 µ

= 1 + R(z) Qt

if z · zt (40)

and µ

ct(z)

ct

¶_¡ 1 µ

= µ

it(z)

it

¶_¡ 1 µ

= 1 + Rt Qt

if z > zt; (41)

in which

Qt ´ Pt(1 + ¿t) ´ Pt

· Z _zt

0

(1 + R (z))1¡ µdz + (1 ¡ zt)(1 + Rt)1¡ µ

¸ 1 1 ¡ µ

(42)

is t he e¤ect ive price index faced by t he household.

Let ¯t¸t¹t, ¯t¸t, and ¯t¸tqt be t he Langranger mult ipliers of (36), (37),

and (38). T he …rst -order condit ions for t he ‡ows variables, consumpt ion and

invest ment , are

u1(ct; 1 ¡ nt)c

1 µ

t c ¡ 1

µ

t (z) = ¸t(1 + ¹t)

Pt(z)

Pt

and i1µ

t i ¡ 1

µ

t (z) = (1 + ¹t)

Pt(z)

Pt

if z > zt; (43)

and u1(ct; 1 ¡ nt)c

1 µ

t c ¡ 1

µ

t (z) = ¸t(1 + R (z))

Pt(z)

Pt and i 1 µ t i ¡ 1 µ

t (z) = (1 + R (z))

Pt(z)

Pt

if z · zt: (44)

T he …rst -order condit ions for t he labor supply and t he cut -o¤ index are

and

1 + ¹_t = 1 + R(zt):

T his last condit ion st at es t hat t he relat ive price of money in unit s of bonds

is equal t o t he credit cost of t he cut -o¤ good. T his relat ive price should be equal

t o t he nominal int erest rat e in order t o keep t he Budget rest rict ion bounded;

ot herwise it would be possible t o gain money selling (or buying) cash t he zt

good, and buying (or selling) it as credit good. At each inst ant t he cut -o¤ good

is det ermined wit h t he aim of meet ing t his non-arbit rage condit ion. T hat is

¹_t = Rt = R (zt): (46)

As Gillman (1993) st ressed, (46) is a Baumol-type condit ion which equat es t he

marginal cost of hold money wit h t he marginal t ransact ion cost .

Aft er subst it ut ing (40) and (41) int o (43) and (44), recalling (42) and (46)

it follows t hat

u1(ct; 1 ¡ nt) = ¸t(1 + ¿t) and qt = 1 + ¿t: (47)

T he Euler equat ions for t he capit al st ock and bonds are respect ively

¸t(1 + ¿t) = ¯ ¸t + 1(1 + ¿t + 1)(1 ¡ ± +

rt + 1

1 + ¿t + 1

and,

¸t = ¯ ¸t + 1(1 + Rt + 1)

Pt

Pt + 1

:

5.2

I m pact U nder W el far e

From (34), aft er subst it ut ing t he …rst -order condit ions (43) and (44), recalling

(40) and (41), it follows t hat

dW d¾ =

1

X

t = 0

¯t¸t

· Z zt

0

(1 + R (z))dct(z) d¾ dz +

Z ₁

zt

(1 + Rt)

dct(z)

d¾ dz ¡ wt dnt

d¾ ¸

:

(49)

T he mat erial balance equat ion for t his economy is

f (kt; nt) ¡

Z zt

0

(1 + R(z))(ct(z) + it(z))dz ¡

Z ₁

zt

(ct(z) + it(z))dz = 0;

which means t hat

0 =

1

X

t = 0

¯t¸t

· rt

dkt

d¾+ wt dnt

d¾¡ Z zt

0

(1 + R(z))(dct(z) d¾ +

dit(z)

d¾ )dz ¸ (50) ¡ 1 X

t = 0

¯t¸t

·

R(zt)(ct(zt) + it(zt))

dzt

d¾+ Z₁

zt

(dct(z) d¾ +

dit(z)

d¾ )dz ¸

Adding (50) t o (49) it follows t hat

dW d¾ =

1

X

t = 0

¯t¸t

· Z ₁

zt

Rt

dct(z)

d¾ dz ¡ R(zt)ct(zt) dzt d¾ ¸ (51) + 1 X

t = 0

¯t¸t

· rt

dkt

d¾¡ Z _zt

0

(1 + R(z))dit(z) d¾ dz ¡

Z ₁

zt

dit(z)

d¾ dz ¡ R (zt)it(zt) dzt

d¾ ¸

:

From t he …rst -order condit ion for t he invest ment , it follows t hat

(1 + ¿t)it =

Zzt

0

(1 + R(z))it(z)dz + (1 + Rt)

Z ₁

zt

it(z)dz;

which means t hat

0 =

1

X

t = 0

¯t¸t

· Z zt

0

(1 + R (z))dit(z)

d¾ dz + (1 + Rt) Z ₁

zt

dit(z)

d¾ dz ¸

+

1

X

t = 0

¯t¸t

· dRt

d¾ Z ₁

zt

it(z)dz ¡ it

d(1 + ¿t)

d¾ ¡ (1 + ¿t) dit

d¾ ¸

: (52)

Adding (52) t o (51), recalling t hat

dRt

d¾ Z 1

zt

it(z)dz ¡ it

d(1 + ¿t)

d¾ = 0;

it follows t hat

dW d¾ =

1

X

t = 0

¯t¸t

· Z ₁

zt

Rt

d

d¾(ct(z) + it(z))dz ¡ R(zt)(ct(zt) + it(zt)) dzt d¾ ¸ + 1 X

t = 0

¯t¸t

· rt

dkt

d¾¡ (1 + ¿t) dit

d¾ ¸

: (53)

line in (53) as

1

X

t = 0

¯t¸t

· rt

dkt

d¾¡ (1 + ¿t) µ

dkt + 1

d¾ ¡ (1 ¡ ±) dkt d¾ ¶ ¸ = 1 X

t = 0

£

¯t¸trt+ ¯t¸t(1 + ¿t)(1 ¡ ±) ¡ ¯t ¡ 1¸t ¡ 1(1 + ¿t ¡ 1)

¤ dkt

d¾

= 0; (54)

in which t he …rst equality follows becauset heinit ial capit al st ock is an exogenous

variable, and t he second equality follows from t he…rst -order condit ion for capit al

accumulat ion, equat ion (48). Subst it ut ing (54) int o (53) it remains

dW d¾ =

1

X

t = 0

¯t¸tRt

d d¾

µ Z ₁

zt

(ct(z) + it(z))dz

¶

=

1

X

t = 0

¯t¸tRt

dmD em and t

d¾ : (55)

T he second equality follows …rst ly from (36) and secondly form t he fact t hat

t he cash-in-advance rest rict ion is binding. Equat ion (55) is equivalent t o (1).

Cont inuing along t he same pat h t hat was t aken in t he …rst part of t he paper,

let ’s suppose t hat t he economy present s a long-run capit al st ock t hat does not

vary wit h ¾. Int egrat ing (55), Bailey’s rule follows

(1 ¡ ¯ )¢ WU nit s of A ssest = ¡

Z _m¤

( ¡ ½) m¤_{( ¾)}

R(m)dm:

For t his economy, Bailey’s rule is t he measure, in unit s of asset s, of t he impact

t akes int o considerat ion …rst ly t he diversion of product ion fact ors t o t he banking

sect or and t he reduct ion of labor supply29_{, which result s in t he decrease of}

t he average consumpt ion level, and, secondly, t he increase in t he variability of

consumpt ion across types of consumpt ion goods.

Let ’s suppose t hat labor supply does not change. From (47) and (55), it

follows t hat

(1 ¡ ¯ )dW

d¾U ni t s of C on sum p t ion B asket

= Rt 1 + ¿t

d d¾

Z ₁

zt

ct(z)dz

= dc d¾;

in which t he second equality follows from (35) and from t he fact t hat t he income

y¤= Z ₁

0

c(z)dz + Z z¤

0

R(z)c(z)dz + ±k¤

is const ant under t hese hypot heses. As it was shown for t he McCallum-Goodfriend

model30_{, t he welfare cost of in‡at ion measured in unit s of consumpt ion goods is}

smaller t han in unit s of asset s or income31_{. T he reason is t he same. When}

cal-culat ing welfare in unit s of consumpt ion bundle, t he t ransact ion cost associat ed

wit h t he consumpt ion is not t aken int o considerat ion.

29_{I n t he models of t he …rst part of t his paper it was supposed t hat t he labor supply was}

inelast ic.

30_{See discussion in t he t hird sect ion.}

31_{A iyagari et ali i (1998) found t hat bot h are equal (see page. 1290). In fact , t he demand}

6

C om p ensat e I ncom e

In t his paper, t he t ot al impact of in‡at ion under welfare has been de…ned as t he

int egrat ion of t he marginal impact , in unit s of asset s. T his is a direct measure of

t he variat ion in welfare in unit s of asset s or income, and, as was seen, provides a

general t heoret ical foundat ion for Bailey’s rule. T he compensat e income which

should be given t o t he household, in order t o keep it indi¤erent t o t he sit uat ion

in t he presence of in‡at ion as compared t o an init ial posit ion wit hout in‡at ion,

is anot her measure of t he welfare cost of in‡at ion. T his concept seems more

nat ural when t he researcher is considering a speci…c model, which could be

calibrat ed t o a real economy t o deliver numerical calculat ions. Bailey’s rule

is more appropriat e when t he researcher has only an empirical est imat ion of

t he money demand funct ion. To est ablish t he link between t hose two di¤erent

de…nit ions of t he impact of in‡at ion under welfare, let ’s solve t he dual problem

of t he general model of t he second Sect ion. For t he st at ionary-st at e, it follows

t hat32

min yPr ivat e = c1+ c2+ g(c1; m2; c22) + (¾+ ½)m + Â

subject t o : u(c1; l (c1; m1; c21)) = const .

Aft er adding t o t he derivat ive of t he income against ¾ t he derivat ive of t he

rest rict ion, recalling t he …rst -order condit ions and t he government rest rict ion,

it follows t hat

dy

d¾= ¡ (¾+ ½) dm

d¾;

in which

y ´ c1+ c2+ g(c1; m2; c22):

T hen, t he income t hat should be given t o t he household t o compensat e it for

t he harm of in‡at ion is

¢ y = Z ¹

m ( ¡ ½)

¹

m ( ¾)

R(m)d¹ m;¹

in which t he bar over t he money demand is t o remind us t hat t his is t he

compen-sat e demand. T his is t he social income t hat should be given t o t he household.

T he variat ion of income t hat t he household observes is

¢ yPr ivat e₌

Z ¹

m ( ¡ ½)

¹

m ( ¾)

R(m)d¹ m + (¾+ ½)(¹ m(¾) ¡ m(¾)):¹

T he following t hought experiment helps t o underst and t he dist inct ion

be-tween t hese two concept s of income. Suppose an economy, in which t he increase

rat e of t he nominal quant ity of money is ¾. Suddenly, a st ock of mineral

re-sources, valued at 1

r¢ y, is discovered. Wit h t his addit ional income, t he money

level of ut ility as it is possible under Friedman rule, would be at t ained. T he

addit ional quant ity of money could be provided by t he economy wit hout cost .

For t he cash-in-advance model t he dual problem is

min y = Z ₁

0

c(z)dz + Z z¤

0

R (z)c(z)dz ¡ n

subject t o33

u( µ Z ₁

0

cµ ¡ 1µ _{(z) dz}

¶ µ µ ¡ 1

; 1 ¡ n) = const .

It is st raight forward t o show t hat

dy dR = ¡ R

dm¹

dR, in which

¹

m = Z z

0

c(z)dz:

Becausem(¾) ¸ m(¾) and¹ m(¡ ½) = m(¡ ½); it is not possible t o compare¹ t he areas under t he two inverse money demand funct ions. T hey should be

quan-t iquan-t aquan-t ively very close, buquan-t whenever income e¤ecquan-t is presenquan-t , iquan-t is noquan-t possible quan-t o

compare t hem34_{. T he ot her common employed measure is t he consumpt ion}

which leaves t he household in t he same ut ility level. For t he st andard Sidrauski

model, t his measure overst at es t he welfare cost of in‡at ion because it does not

consider t hat t he decision-maker will increase her money demand if her

con-sumpt ion level is augment ed. Applying it t o t he McCallum-Goodfriend model

33_{T he wage rat e was normalized t o one.}

34_{I t is a microeconomic t ext -book result t hat t he consumer surplus is a perfect measure of}

and t he cash-in-advance model invest igat ed in t he last Sect ion, t his measure

underest imat es because it does not consider t he increase in t he t ransact ion cost

due t o t he addit ional quant ity of consumpt ion good.

T he general conclusion of t he paper is t hat t he money demand caries wit h it

a lot of informat ion. But what is meant by ‘money’ ? T he next Sect ion argues

t hat t he relevant concept of money for t his subject - t he impact under welfare

of in‡at ion - is t he narrow monet ary aggregat e, t he monet ary base.

7

A M odel wi t h I nside M oney

At t his point t he message of t his paper should be very clear. Abst ract ing from

impact s of in‡at ion under long-run capit al, Bailey’s rule is t he accurat e

mea-sure of t he reduct ion on welfare caused by a perfect ly predict ed in‡at ion. T his

conclusion is quit e general and does not depend on t he speci…c role played by

money in t his economy or t he speci…c kind of adjust ment faced by t he real

sec-t or in order sec-t o avoid or sec-t o help sec-t he public sec-t o cope wisec-t h in‡asec-t ion. Busec-t whasec-t is

meant exact ly by ‘money demand’ ? What is money? Whenever t he researcher

is st udying t he short -run equilibrium of t he economy, money is t he asset which

possesses t he property of liquidity. It is usually cash out of t he banking sect or

plus demand deposit s. But , t hat is not what is meant by money in t his

cost35 ; 36.

When in‡at ion increases, t he public demand for demand deposit s decreases,

which could be considered a welfare cost of in‡at ion. However, because t his

service - demand deposit - requires capit al and work force t o be supplied, t he

reduct ion in t he public demand for demand deposit is not a cost , from t he

social point of view. What occurs is t hat t he increase of in‡at ion decreases

t he demand-deposit demand, but it increases t he demand for t he ot her bank

services, in such a way t hat t he demand for an aggregat ed bundle of banking

services increases. T hose e¤ect s were t aken int o considerat ion in t he models

st udied in t his paper. Saying it di¤erent ly, t he demand-deposit is just anot her

service which is supplied by t he banking sect or t o help t he public t o cope wit h

in‡at ion. T he variant of t he second Sect ion model sket ched below is int ended

t o clarify t his issue.

Household

T here are t hree liquidity inst rument s: cash, demand deposit s and anot her

banking service. T he household solves

max Z ₁

0

e¡ ½tu(c1t; l(m1t; m2t; c2t))dt; (56)

35_{T his concept of money applies t o Friedman’s rule. T he asset whose consumpt ion should}

be pushed t o sat iat ion is t he monet ary base.

36_{D i¤erent ly, L ucas (1981) pg. 44, de…nes money, as far as t he welfare impact of in‡at ion}

is concerned, as any

“ nonint erest -bearing asset s or t o asset s t he int erest on which is rest rict ed t o below-market rat es.”

subject t o

¢

at = rtat+ wt+ ÂH ;t+ {t¡ c1t¡ ptc2t¡ pdtm1t ¡ (¼t+ rt)(m1t + m2t) (57)

in which37

a ´ k + m1+ m2;

m1t-... st ock of cash in household’s port folio;

m2t-... st ock of demand deposit s in household’s port folio;

Â_{H ;t}-... Government t ransfers t o t he household;

{t-... bank’s pro…ts;

pd

t-... demand deposit price.

For simplicity, t he ot her banking services are t reat ed as ‡ow of services and

not as asset s. Because of t he possibility of very low in‡at ion rat es t he banks

charge a fee t o held demand deposit s. It is possible, if in‡at ion is su¢ cient ly

high, t hat t his price could be zero. T he …rst -order condit ions for t his st andard

37_{Not hing would change if t his model had been const ruct ed as general as t he model in t he}

problem is

u1 = ¹ ; (58)

u2l1 = ¹ (¼+ r );

u2l2 = ¹ (¼+ r + pd);

u2l3 = ¹ p; ¢

¹

¹ = ½¡ r: (59)

The Banks

T his is a two-sect or economy. T he real sect or produces a good, which can

be consumed and accumulat ed as capit al. T he second sect or, banks, in t his

Sect ion are mult iproduct …rms. T hey employ capit al and work force t o produce

a service, (called banking services, which help t he household in saving t

rans-act ion t ime) and t o produce anot her liquidity service, named demand deposit .

As usual, it is supposed t hat t he demand deposit s are denominat ed in nominal

unit s; consequent ly, t he income of t he banking in o¤ering t his services is t he

price t hat it could charges plus t he nominal int erest rat es. T herefore, t he per

capita pro…t funct ion for t he banks, in unit s of t he good are

{ = pc2+ (pd+ (¼+ r ) (1 ¡ ³ ))m2¡ (r k2+ w)l2+ ÂB (60)

k2-... capit al-labor rat io in t he banking sect or;

l2-... rat io of t he work force employed by t he banking sect or;

Â_B-... Govern’s t ransfer t o t he Banks.

It is supposed t hat t he issue of a new demand deposit is a cost less act ivity,

as it is t o t he government t o issue base, such t hat t he seigniorage is an income

appropriat ed by t he banking inst it ut ion. T he banks maximize (60), subject t o

t he t echnological rest rict ion38

y2= l2f2(k2) = g(c2; m2): (61)

It st at es t hat t he per capita product ion of t his indust ry can be dist ribut ed across

t he two product s according t o t he t ransformat ion funct ion g. T his funct ion is

concave and …rst -order-degree homogeneous. Let q be t he Lagrange mult iplier

for (61). T he …rst -order condit ions for t he maximizat ion problem for t he banks

are as follows

p = qg1; (62)

pd+ (¼+ r ) (1 ¡ ³ ) = qg2; (63)

r = qf₂0(k2); (64)

w = q(f2¡ k2f20(k2)): (65)

Due t o t he homogeneity of g, it follows from (62) and (63) t hat

pc2+ (pd+ (¼+ r ) (1 ¡ ³ ))m2= qy2; (66)

which means t hat t he t ot al per capita product ion of t he Banks, evaluat ed in

unit s of goods, is equal t o t he product ion of services, priced at p, and t he

product ion of demand deposit s, priced at pd_{+ (¼+ r ) (1 ¡ ³ ). T he price q is t he}

price, in unit s of goods, of a opt imum bundle of t ransact ion services and demand

deposit s. T his is t he relevant price for t he allocat ion decision for t he product ion

fact ors39_{. At each inst ant t he price q det ermines t he relat ive rent ability across}

t he sect ors, and, accordingly, t he allocat ion of fact ors between t he real sect or

and t he banking sect or40_{. Consequent ly, t he sect or’s o¤ers funct ion can be}

writ t en as follows

y1(q; k) and y2(q; k):

Similar t o t he ot her sect ions, propriet ies (15) and (16) are sat is…ed. Given an

amount of banking out put , y2, t he relat ive price between services and demand

deposit s det ermines at which point of t he t ransformat ion funct ion g t he banking

sect or will be posit ioned. On t he ot her hand, equat ion (66) could be seen as an

equilibrium equat ion for t he banking sect or. Tot ally di¤erent iat ing (66) aft er

39_{From (64) and (65) it is possible t o verify it direct ly.}

40_{I t apparent t hat t his economy does not sat i…ces Friedman’s rule for demand deposit . I f}

in‡at ion decrease, t o o¤er t his service t he banks will charge t he fee pd_{, in order t o pay for t he}

subst it ut ing

dy2= q¡ 1(pdc2+ (pd+ (¼+ r ) (1 ¡ ³ ))dm2)

it follows t hat

y2dq = c2dp + m2d(pd+ (¼+ r ) (1 ¡ ³ )): (67)

T his last result will be useful lat er.

General Equilibrium and Welfare

Because t he t ransformat ion front ier for t he Banks is …rst -order-degree

ho-mogeneous t he payment of fact or by it s marginal product ivity is equal t o t he

product ion of liquidity services - pc2+ (pd+ (¼+ r ) (1 ¡ ³ ))m2. Consequent ly,

t he bank’s pro…t is t he bank’s seigniorage - (1 ¡ ³ )m¢2 - plus t he government

t ransfer - Â_B. Aft er subst it ut ing t he liquidity services equilibrium equat ion (66), remembering t hat t he per capita income r k+ w is equal t o t he per capita out

-put - y1+ qy2- and t hat t he t ot al government t ransfer is equal t o t he seigniorage

of t he monet ary base, t he good’s market equilibrium equat ion follows from (57)

¢

k = y1(q; k) ¡ c1: (68)