the n-dimensional bailey-divisia measure as a general ... · rubens penha cysne getulio vargas...

The n-Dimensional Bailey-Divisia Measure as aGeneral-Equilibrium Measure of the Welfare Costs of

In�ation

Rubens Penha Cysne

Getulio Vargas Foundation

May 2011

Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/2011 1 / 31

Bailey�s Measure as an Approximation

Bailey (1956), Lucas (Econometrica, 2000)

Simonsen and Cysne (JMCB, 2001): w = 1� e�A < s < A < B


Bailey�s PE Measure as an Exact GE Measure

Relevance: Bailey�s formula remains widely used in the profession(e.g., Lucas (Econometrica, 2000), Mulligan and Sala-i-Martin (JPE,2000), Attanasio et al. (JPE, 2002) Ireland (AER, 2009))

The use of Bailey�s measure is usual followed by a disclaimer - beingsubject to the usual criticisms of originating from a partial-equilibriumanalysis.

Where does the money demand come from?

Lucas�critique.

Cysne (JMCB, 2009) shows that Bailey�s Measure actually turns outto coincide with the exact general-equilibrium measure (emergingfrom the Sidrauski model) when preferences are quasi-linear.

U(c ,m) = g(c + λ(m)) rather than U(c ,m) = 11�σ

�cϕ(mc )

�1�σ


Using One - Rather than n - Dimensional Measures

Teles and Zhou (2005): �technological innovation and changes inregulatory practices in the past two decades have made othermonetary aggregates as liquid as M1�.

Attanasio et al. (jpe, 2002, p. 341) report that 58.7% of thehouseholds in their sample (originated from the Italian economy)hold, besides the two monetary assets currency and interest-bearingbank deposits, at least one other interest-bearing non-monetary asset(e.g., bonds).

Regarding the United States, Mulligan and Sala-i-Martin (2000, p.962) report, following the 1989 Survey of Consumer Finances, hat35% of all households hold bank deposits and at least one additionalinterest-bearing asset.



Suggestion by Lucas (Econometrica, 2000)

Cysne (JMCB, 2003): A Divisia Index as a Welfare MeasureCysne and Turchick ( JEDC, 2010): On the Error of Using One-Rather than n-Dimensional Measures:

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.005

0.01

0.015

0.02

0.025

0.03

nominal interest rate RB

wel

fare

cos

t of i

nfla

tion

BUBM



Suggestion by Lucas (Econometrica, 2000)Cysne (JMCB, 2003): A Divisia Index as a Welfare Measure

Cysne and Turchick ( JEDC, 2010): On the Error of Using One-Rather than n-Dimensional Measures:

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.005

0.01

0.015

0.02

0.025

0.03


wel

fare

cos

t of i

nfla

tion

BUBM



Suggestion by Lucas (Econometrica, 2000)Cysne (JMCB, 2003): A Divisia Index as a Welfare MeasureCysne and Turchick ( JEDC, 2010): On the Error of Using One-Rather than n-Dimensional Measures:

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.005

0.01

0.015

0.02

0.025

0.03


wel

fare

cos

t of i

nfla

tion

BUBM


Cysne and Turchick (JEDC, 2010):

Bias Depends on the elasticity of substitution among the di¤erencemonies

0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

bias

0.05 0.1 0.15 0.20

20

40

60

80

100

120

140

160

180


ξ=1

ξ=0.5

ξ=0.02

ξ=0.05

ξ=0.1

ξ=0.2

ξ=5

ξ=10

ξ=20

ξ=50

ξ=1

ξ=2


N - Dimensional Measures

Cysne and Turchick (MD, forthcoming): w = 1� e�A < s < A < B < w

Di¤erence between Measures in the n-Dimensional CaseGraduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/2011 7 / 31

Abstract

This paper shows that Bailey�s multidimensional Divisia Indexmeasure BD (m) = �

Rχ u � dm emerges as an exact

general-equilibrium measure of the welfare costs of in�ation whenpreferences are quasilinear.


The Model

Our model is an extension of Sidrauski�s (1967) to an economy withseveral monies. Let

m = (m1, . . . ,mn) 2 [0,+∞]n

represent the vector of real quantities of each type of money, as a fractionof nominal GDP. Real output is supposed to be constant and equal toone. Each mi yields a nominal interest rate of ri , and

r : = (r1, . . . , rn) 2 Rn+

The �rst monetary asset (m1) is assumed to be real currency, in whichcase r1 = 0.


The Model

We shall write the vector of opportunity costs (relatively to holding bonds)as:

u : = (u1, . . . , un) := (r , r � r2, . . . , r � rn) 2 Rn+

Given a concave utility function U(c ,m), the maximization problem of therepresentative consumer reads:

maxc>0, m�0

Z +∞

0e�ρtU(c ,m)dt (PnS)

subject to

b+ 1 �_m = 1� h� c + (r � π)b+ (r� (π.1)) �mb0 > 0 and m0 > 0 given.

where we write 1 for the vector (1, . . . , 1) 2 Rn, and ��for the canonicalinner product of Rn.


The Model

Given a rate of monetary expansion equal to θ, in the steady state we haveπ = θ and the usual Euler equations imply:

r = θ + ρ (1)

ui : = r � ri =UmiUc, 8i 2 f1, . . . , ng. (2)

In equilibrium c = 1 and the n equations given by (2) determine thedemand for the n monetary assets in the economy.


The ModelA n-Dimensional (Sidrauski) General-Equilibrium Measure of the Welfare Costs of In�ation

This subsection is based on Cysne and Turchick (2007). Assume for amoment that our representative agent�s utility had the general form:

U(c ,m) =1

1� σ

�cϕ

�G (m)c

��1�σ

(3)

where G is a monetary aggregator function, σ > 0, σ 6= 1. ϕ : R+ ! R+

is a di¤erentiable function satisfying:

Assumption ϕ0. (ϕ/ϕ0) (0+) = 0.

Assumption ϕ. There exists an m 2 (0,+∞] such that ϕ j[0,m) is strictlyincreasing, ϕ j[m,+∞) is constant, ϕ00 j(0,m)< 0 andϕ0 (m�) = 0.



Equations (2) now imply:

ui (m) =ϕ0(G (m))

ϕ(G (m))� G (m) ϕ0(G (m))Gmi (m), 8i 2 f1, . . . , ng. (4)

Let Cm := fm 2 Rn++ : G (m) = mg. The consumer is satiated with

monetary balances when G (m) = m. We denote by�m those m which liein Cm .Throughout this paper we shall follow the same methodology set forth byLucas (2000) and implicitly de�ne the welfare costs of in�ation w(m) by:

U(1+ w(m), G (m)) = U(1,G (�m)) (5)



Take the partial derivatives of (5) and divide by

ϕ0�G�(1+ w(m))�1m

��to obtain:

wi (m)ϕ�G�

11+w (m)m

��ϕ0�G�

11+w (m)m

�� + �Gmi (m)� wi (m)G � 11+ w(m)

m��

= 0.

Using (4):

ϕ�G�

11+w (m)m

��ϕ0�G�

11+w (m)m

�� = Gmi (m)

ui�

11+w (m)m

� + G � 11+ w(m)

m�,



This leads to w being represented by the n di¤erential equations:

wi (m) = �ui�

11+ w(m)

m�

with initial condition w(�m) = 0.Alternatively, consider a C 1 path

χ : [0, 1]! [0,+∞]n (6)

such that χ (0) =�m and χ (1) = m. Then, w(m) can be written as theline integral:

w(m) = �Z

χu�

11+ w(m)

m�� dm (7)


The ModelThe Bailey-Divisia Measure as a General-Equilibrium Measure

Note that,

w(m) = �Z

χu�

11+ w(m)

m�� dm

the measure of the welfare costs of in�ation which emerges fromSidrauski�s general-equilibrium model, is not equal to the Bailey-Divisia(BD ) measure, de�ned by:

BD (m) = �ui (m) , BD (�m) = 0 (8)

or, alternatively, when expressed as a line integral, and considering thepath χ de�ned above:

BD (m) = �Z

χu � dm (9)



Our main purpose here is showing that BD = w when preferences, ratherthan de�ned by the general form (3), can be expressed by the quasi-linearform:

U(c ,m) = g(c + f (G (m))) (10)

where c and G are as de�ned before. f : [0,+∞]! R+ andg : [0,+∞]! R are twice-di¤erentiable functions such that f 0 > 0,f 00 < 0, g 0 > 0 and g 00 � 0. Any U in the class of functions represented by(10) is concave in (c ,m), since it is given by the composition of concaveand increasing functions. This makes e�gtU concave with respect to(b, b,m,_m), which makes the Euler equations su¢ cient for an optimum.



In this particular case, equations (2) give, 8i 2 f1, . . . , ng:

ui = f 0(G (m))Gmi (m), i 2 f1, . . . , ng (11)

Here, (5) and (10) imply:

g(1+ w(m) + f (G (m))) = g(1+ f (G (�m)) (12)

PropositionLet an economy be described as above. Then

BD (m) = w(m) (13)



Proof.The demonstration follows basically the same steps as in Cysne (2009).Since g 0(.) > 0, we can write, from (12):

w (m) = f (G (�m))� f (G (m)) (14)

Taking the derivative with respect to mi :

wi (m) = �f 0(G (m))Gmi (m), 8i 2 f1, . . . , ng (15)

By using (11) in (15):

wi (m) = �ui (m), 8i 2 f1, . . . , ng (16)



Proof.Normalize w(m) by making w(�m) = 0, which is equivalent to writingBD (χ (0)) = 0. Now use the de�nition of BD in (9) to obtain the mainresult:

w(m) = �Z

χu � dm = BD (m) (17)



General-equilibrium considerations remind us that the vector m in (17) is afunction not only of the nominal interest rate r , but also of all remainingspreads ui , i 2 f2, . . . , ng.

RemarkNote, by using the parameterized path (6), that (17) can also be writtenas:

w (m) :=Z 1

0

ddλw (χ (λ)) dλ =

Z 1

0[�u (χ(λ)) � rχ(λ)] dλ (18)

In (18), λ stands for a real parameter taking values in [0, 1], χ(λ) for thevector of monetary aggregrates and rχ(λ) for the vector of derivatives ofχ with respect to λ.


Applications

Examples


ApplicationsExample 1: Area Under the Inverse Demand for Monetary Base

Here we assume that the spreads ui , 8i 2 f2, . . . , ng are competitivelydetermined by a costless banking system operating under constant andnon-remunerated reserve requirements. In this case ui := r � ri = ki r , kistanding for the reserve requirements on deposit i .Let z stand for themonetary base, k := (k2, . . . , kn), m(�1) := (m2, . . . ,mn) 2 [0,+∞]n�1.With ui = ki r for all i (17) becomes:

BD (m) = w(m) = �Z

χr(dm1 + k � dm(�1)) (19)

But in such an economy the monetary base (equal to currency plusnon-interest-bearing reserves deposited in the Central Bank) readsz := m1+ k �m. Since k is a vector of constants, dz = dm1 + k � dm(�1),in which case (19) can be written as:

BD (m) = w(m) = �Z m1+k�m(�1)

m1+k��m(�1)rdz


ApplicationsExample 1: Area Under the Inverse Demand for Monetary Base

The conclusion of this example is that under quasilinear preferences theBailey-Divisia measure leads to an exact general-equilibrium welfaremeasure equal to the area under the inverse demand for monetary base(rather than the inverse demand for M1, as used, for instance, by Lucas(2000)).


ApplicationsExample 2: Constant Spreads

Let λ = r in (18) and assume that all banking spreads ui other than thaton currency (i 2 f2, . . . , ng) remain constant. This would be the case forinstance if m2,m3, ...mn were issued by the government and their interestpayments increased pari-passu with the interest rate on bonds, the onlynonmonetary asset in the economy. In this case, using Remark 1, BD (m)can be written as a function of the nominal interest rate (which takes therole of parameter λ by a rede�nition of domain) under in the followingway:

BD (m) = �Z r

0[xm01(x) +∑n

i=2 uim0i (x)]dx


ApplicationsExample 2: An Important Conclusion

BD (m) = �Z r

0[xm01(x) +∑n

i=2 uim0i (x)]dx

Note above that having all spreads constant does not imply that Bailey�sunidimensional formula is the correct one to be used. The remaining term∑ni=2 uim

0i (x) reminds us that one should also take into consideration the

implications of the changes of the nominal interest rate on the demand forall other monies. Cysne and Turchick (JEDC, 2010) concentrate oncalculating the bias originated by measurements which neglect this fact.


ApplicationsExample 3

Take, in (10), g as the identity function, f de�ned by f (x) = x 1�σ�11�σ ,

where σ > 0 and σ 6= 1, f (x) = ln x when σ = 1 andG (m1,m2) = m

µ1m

1�µ2 , with 0 < µ < 1. From (11) we have

ui (m) = G (m)�σ Gxi (m). So, from (9),

BD (m) = �Z

χG (m)�σ (Gx1 (m) dm1 + Gx2 (m) dm2)

= �Z G (m)

G (�m)G�σdG =

11� σ

�G (�m)1�σ � G (m)1�σ

�(20)

The de�nition of G is only used in the �nal step to generate:

BD (m) =11�σ

[mµ(1�σ)1 m(1�µ)(1�σ)

2 �mµ(1�σ)1 m(1�µ)(1�σ)

2 ]


ApplicationsExample 3

When σ = 1 :

BD (m) = lnG (�m)G (m)

= µ lnm1m1+ (1� µ) ln

m2m2

(21)

In this case the welfare costs of in�ation are given by a weighted averagewhich measures how relatively distant the representative agent is fromsatiation with respect to each monetary asset.Finally, note that both (20) and (21) can also be directly obtained from(14).


Conclusions

Bailey�s and Lucas�s one-dimensional formulas do not take intoconsideration an important fact common to most economiesnowadays: the presence of several types of money other than currencyor demand deposits.

The present work has tackled this issue and extended Cysne�s (2009)result by showing that in economies with several monies aDivisia-index version of Bailey�s original measure can also be regardedas an exact general-equilibrium measure of the welfare costs ofin�ation. Three applications following from this result have beenpresented.

The intuition is the same as before, now applied to a higherdimension: in the absence of wealth e¤ects, the consumers�surplus(here, the multidimensional consumers�surplus de�ned by theBailey-Divisia measure), rather than an approximation, provides anexact measure of the deadweight loss stemming from taxation.


References

Bailey, Martin J. (1956) "Welfare Cost of In�ationary Finance",Journal of Political Economy 64, 93-110.

Cysne, Rubens Penha. (2003). "Divisia Index, In�ation and Welfare".Journal of Money, Credit and Banking, The Ohio State University, v.35, n. 2, 221-238.

Cysne, Rubens P. (2009). �Bailey�s Measure of the Welfare Costs ofIn�ation as a General-Equilibrium Measure�. Journal of Money, Creditand Banking, March/2009, 41(2-3), p.451-459.


References

Cysne, Rubens P., and David Turchick. (2007) �An Ordering ofMeasures of the Welfare Cost of In�ation in Economieswith Interest-Bearing Deposits.�EPGE Working Paper 659, available athttp://virtualbib.fgv.br/dspace/bitstream/handle/10438/396/2248.pdf.

Lucas, R. E. Jr. (2000) "In�ation and Welfare". Econometrica 68, 62 ,247-274.

Sidrauski, M. (1967). "Rational Choice and Patterns of Growth in aMonetary Economy". American Economic Review; 57; 534-544.


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