The n-dimensional bailey-divisia measure as a General-equilibrium measure of the welfare costs of inflation

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The n-Dimensional Bailey-Divisia Measure as a

General-Equilibrium Measure of the Welfare Costs of

In‡ation

Rubens Penha Cysne

Getulio Vargas Foundation

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Bailey’s Measure as an Approximation

Bailey (1956), Lucas (Econometrica, 2000)

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Bailey’s Measure as an Approximation

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Bailey’s Measure as an Approximation

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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 - being subject to the usual criticisms of originating from a partial-equilibrium analysis.

Where does the money demand come from? Lucas’critique.

Cysne (JMCB, 2009) shows that Bailey’s Measure actually turns out to coincide with the exact general-equilibrium measure (emerging from the Sidrauski model) when preferences are quasi-linear. U(c, m) =g(c+λ(m))rather than U(c, m) = _{1 σ}1 c ϕ(m_c ) 1 σ

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Bailey’s PE Measure as an Exact GE Measure

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Bailey’s PE Measure as an Exact GE Measure

Where does the money demand come from?

Lucas’critique.

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Bailey’s PE Measure as an Exact GE Measure

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Bailey’s PE Measure as an Exact GE Measure

Cysne (JMCB, 2009) shows that Bailey’s Measure actually turns out to coincide with the exact general-equilibrium measure (emerging from the Sidrauski model) when preferences are quasi-linear.

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Bailey’s PE Measure as an Exact GE Measure

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Using One - Rather than n - Dimensional Measures

Teles and Zhou (2005): “technological innovation and changes in regulatory practices in the past two decades have made other monetary aggregates as liquid as M1”.

Attanasio et al. (jpe, 2002, p. 341) report that 58.7% of the households in their sample (originated from the Italian economy) hold, besides the two monetary assets currency and interest-bearing bank 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, hat 35% of all households hold bank deposits and at least one additional interest-bearing asset.

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Using One - Rather than n - Dimensional Measures

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Using One - Rather than n - Dimensional Measures

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Using One - Rather than n - Dimensional Measures

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 0.2 0 0.005 0.01 0.015 0.02 0.025 0.03

nominal interest rate R

B w el far e cos t of inf lat ion B U B M

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Using One - Rather than n - Dimensional Measures

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 0.2 0 0.005 0.01 0.015 0.02 0.025 0.03

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Using One - Rather than n - Dimensional Measures

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.005 0.01 0.015 0.02 0.025 0.03

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Using One - Rather than n - Dimensional Measures

0 0.005 0.01 0.015 0.02 0.025 0.03 w el far e cos t of inf lat ion B U B M

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Cysne and Turchick (JEDC, 2010):

Bias Depends on the elasticity of substitution among the di¤erence monies 0.05 0.1 0.15 0.2 0 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 0.2 0 20 40 60 80 100 120 140 160 180

B 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

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N - Dimensional Measures

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Abstract

This paper shows that Bailey’s multidimensional Divisia Index measure BD(m) =

R

χu d m emerges as an exact

general-equilibrium measure of the welfare costs of in‡ation when preferences are quasilinear.

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The Model

Our model is an extension of Sidrauski’s (1967) to an economy with several monies. Let

m= (m1, . . . , mn) 2 [0,+∞]n

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

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

The …rst monetary asset (m1)is assumed to be real currency, in which

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The Model

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

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

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

max c>0, m 0 Z +∞ 0 e ρt_U₍_{c, m}₎_dt _(Pn S) subject to ˙b+1 _m=1 h c+ (r π)b+ (r (π.1)) m b0 >0 and m0>0 given.

where we write 1 for the vector (1, . . . , 1) 2Rn_{, and ‘’for the canonical}

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The Model

Given a rate of monetary expansion equal to θ, in the steady state we have

π= θ and the usual Euler equations imply:

r = θ+ρ (1)

ui : =r ri =

Umi Uc

, ₈i _{2 f}1, . . . , n_g. (2)

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

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The Model

A n-Dimensional (Sidrauski) General-Equilibrium Measure of the Welfare Costs of In‡ation

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

U(c, m) = 1 1 σ c ϕ G(m) c 1 σ (3) where G is a monetary aggregator function, σ >0, σ₆₌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 ϕj[0,m) is strictly

increasing, ϕ_j[m,+_∞) is constant, ϕ00j(0,m)<0 and

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The Model

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 := fm2 Rn++: G(m) =mg. The consumer is satiated with

monetary balances when G(m) =m. We denote by —m those m which lie

in Cm.

Throughout this paper we shall follow the same methodology set forth by Lucas (2000) and implicitly de…ne the welfare costs of in‡ation w(m) by:

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The Model

Take the partial derivatives of (5) and divide by

ϕ0 G (1+w(m)) 1m to obtain: wi(m) ϕ G ₁₊_w1₍_m₎m ϕ0 G ₁₊_w1₍_m₎m + Gmi (m) wi(m)G 1 1+w(m)m =0. Using (4): ϕ G ₁₊_w1₍_m₎m ϕ0 G ₁₊_w1₍_m₎m = Gmi(m) ui ₁₊_w1₍_m₎m +G 1 1+w(m)m ,

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The Model

A n-Dimensional (Sidrauski) General-Equilibrium Measure of the Welfare Costs of In‡ation This leads to w being represented by the n di¤erential equations:

wi(m) = ui

1 1+w(m)m

with initial condition w(—m) =0. Alternatively, consider a C1 path

χ:[0, 1]! [0,+∞]n (6)

such that χ(0) =—m and χ(1) =m. Then, w(m) can be written as the line integral: w(m) = Z χ u 1 1+w(m)m d m (7)

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The Model

The Bailey-Divisia Measure as a General-Equilibrium Measure

Note that, w(m) = Z χ u 1 1+w(m)m d m

the measure of the welfare costs of in‡ation which emerges from Sidrauski’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 the path χ de…ned above:

BD(m) =

Z

χ

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The Model

Our main purpose here is showing that BD =w when preferences, rather

than de…ned by the general form (3), can be expressed by the quasi-linear form:

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

where c and G are as de…ned before. f : [0,+∞]_!R+ and

g :[0,+∞]_!R are twice-di¤erentiable functions such that f0 >0, f00 <0, g0 >0 and g00 0. Any U in the class of functions represented by (10) is concave in (c, m), since it is given by the composition of concave and increasing functions. This makes e gtU concave with respect to

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The Model

In this particular case, equations (2) give, ₈i _{2 f}1, . . . , n_g:

ui =f0(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)

Proposition

Let an economy be described as above. Then

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The Model

Proof.

The demonstration follows basically the same steps as in Cysne (2009). Since g0(.) >0, we can write, from (12):

w(m) =f (G(—m)) f (G(m)) (14)

Taking the derivative with respect to mi :

wi(m) = f0(G(m))Gmi(m), 8i 2 f1, . . . , ng (15) By using (11) in (15):

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The Model

Proof.

Normalize w(m)by making w(—m) =0, which is equivalent to writing BD(χ(0)) =0. Now use the de…nition of BD in (9) to obtain the main

result:

w(m) = Z

χ

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The Model

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

Remark

Note, by using the parameterized path (6), that (17) can also be written as: w(m):= Z 1 0 d d λw(χ(λ))d λ= Z 1 0 [ u(χ(λ)) rχ(λ)]d λ (18)

In (18), λ stands for a real parameter taking values in [0, 1], χ(λ)for the

vector of monetary aggregrates and _rχ(λ) for the vector of derivatives of

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Applications

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Applications

Example 1: Area Under the Inverse Demand for Monetary Base

Here we assume that the spreads ui,8i 2 f2, . . . , ngare competitively

determined by a costless banking system operating under constant and non-remunerated reserve requirements. In this case ui :=r ri =kir , ki

standing for the reserve requirements on deposit i .Let z stand for the monetary base, k := (k2, . . . , kn), m( 1):= (m2, . . . , mn) 2 [0,+∞]n 1.

With ui =kir for all i (17) becomes:

BD(m) =w(m) =

Z

χ

r(dm1+k d m( 1)) (19)

But in such an economy the monetary base (equal to currency plus non-interest-bearing reserves deposited in the Central Bank) reads

z :=m1+ k m. Since k is a vector of constants, dz =dm1+k d m( 1),

in which case (19) can be written as:

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Applications

Example 1: Area Under the Inverse Demand for Monetary Base

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

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Applications

Example 2: Constant Spreads

Let λ=r in (18) and assume that all banking spreads ui other than that

on currency (i _{2 f}2, . . . , n_g) remain constant. This would be the case for instance if m2, m3, ...mn were issued by the government and their interest

payments increased pari-passu with the interest rate on bonds, the only nonmonetary 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 the role of parameter λ by a rede…nition of domain) under in the following way:

BD(m) =

Z r 0

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Applications

Example 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’s unidimensional formula is the correct one to be used. The remaining term ∑n

i=2¯uim0_i(x)reminds us that one should also take into consideration the

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

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Applications

Example 3

Take, in (10), g as the identity function, f de…ned by f(x) = x1 σ_{1 σ}1, where σ>0 and σ6=1, f (x) =ln x when σ=1 and

G(m1, m2) =m1µm 1 µ

2 , with 0<µ<1. From (11) we have

ui(m) =G(m) σGxi(m). So, from (9), BD(m) = Z χ G(m) σ (Gx1(m)dm1+Gx2(m)dm2) = Z G(m) G(—m) G σ_dG ₌ 1 1 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) = 1 1 σ[m¯ µ(1 σ) 1 m¯ (1 µ)(1 σ) 2 m µ(1 σ) 1 m (1 µ)(1 σ) 2 ]

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Applications

Example 3 When σ =1 : BD(m) =ln G(—m) G(m) =µ ln ¯ m1 m1 + (1 µ)lnm¯2 m2 (21) In this case the welfare costs of in‡ation are given by a weighted average which measures how relatively distant the representative agent is from satiation with respect to each monetary asset.

Finally, note that both (20) and (21) can also be directly obtained from (14).

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Conclusions

Bailey’s and Lucas’s one-dimensional formulas do not take into consideration an important fact common to most economies

nowadays: the presence of several types of money other than currency or demand deposits.

The present work has tackled this issue and extended Cysne’s (2009) result by showing that in economies with several monies a

Divisia-index version of Bailey’s original measure can also be regarded as an exact general-equilibrium measure of the welfare costs of in‡ation. Three applications following from this result have been presented.

The intuition is the same as before, now applied to a higher dimension: in the absence of wealth e¤ects, the consumers’surplus (here, the multidimensional consumers’surplus de…ned by the Bailey-Divisia measure), rather than an approximation, provides an

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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 of In‡ation as a General-Equilibrium Measure”. Journal of Money, Credit and Banking, March/2009, 41(2-3), p.451-459.

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References

Cysne, Rubens P., and David Turchick. (2007) “An Ordering of Measures of the Welfare Cost of In‡ation in Economies

with Interest-Bearing Deposits.”EPGE Working Paper 659, available at http://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 a Monetary Economy". American Economic Review; 57; 534-544.