The n-dimensional Bailey-Divisia Measure as a General-Equilibrium Measure of the Welfare Costs of In ation

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1 The n-dimensional Bailey-Divisia Measure as a General-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/ / 31

2 Bailey s Measure as an Approximation Bailey (1956), Lucas (Econometrica, 2000) Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

3 Bailey s Measure as an Approximation Bailey (1956), Lucas (Econometrica, 2000) Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

4 Bailey s Measure as an Approximation Bailey (1956), Lucas (Econometrica, 2000) Simonsen and Cysne (JMCB, 2001): w = 1 e A < s < A < B Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

5 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)) raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

6 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. raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

7 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? raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

8 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. raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

9 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. Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

10 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 1 σ c ϕ( m c ) 1 σ Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

11 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. raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

12 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). raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

13 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. raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

14 Using One - Rather than n - Dimensional Measures Suggestion by Lucas (Econometrica, 2000) raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

15 Using One - Rather than n - Dimensional Measures Suggestion by Lucas (Econometrica, 2000) Cysne (JMCB, 2003): A Divisia Index as a Welfare Measure raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

16 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: raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

17 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: welfare cost of inflation B U B M nominal interest rate R B raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

18 Cysne and Turchick (JEDC, 2010): Bias Depends on the elasticity of substitution among the di erence monies ξ= ξ= bias ξ=0.05 ξ=0.2 ξ=0.5 ξ=0.1 ξ= ξ=1 ξ=2 ξ=20 ξ=10 ξ= nominal interest rate R B Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

19 N - Dimensional Measures Cysne and Turchick (MD, forthcoming): w = 1 e A < s < A < B < w Di erence between Measures in the n-dimensional Case Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

20 Abstract This paper shows that Bailey s multidimensional Divisia Index measure B D (m) = R u dm emerges as an exact χ general-equilibrium measure of the welfare costs of in ation when preferences are quasilinear. Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

21 The Model Our model is an extension of Sidrauski s (1967) to an economy with several monies. Let m = (m 1,..., m n ) 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 m i yields a nominal interest rate of r i, and r : = (r 1,..., r n ) 2 R n + The rst monetary asset (m 1 ) is assumed to be real currency, in which case r 1 = 0. Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

22 The Model We shall write the vector of opportunity costs (relatively to holding bonds) as: u : = (u 1,..., u n ) := (r, r r 2,..., r r n ) 2 R n + Given a concave utility function U(c, m), the maximization problem of the representative consumer reads: subject to Z + max c>0, m0 0 e ρt U(c, m)dt (P n S ) ḃ + 1 _ m = 1 h c + (r π)b + (r (π.1)) m b 0 > 0 and m 0 > 0 given. where we write 1 for the vector (1,..., 1) 2 R n, and for the canonical inner product of R n. Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

23 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) u i : = r r i = U m i U c, 8i 2 f1,..., ng. (2) In equilibrium c = 1 and the n equations given by (2) determine the demand for the n monetary assets in the economy. raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

24 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 c ϕ 1 σ G (m) 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 ϕ 0. (ϕ/ϕ 0 ) (0+) = 0. Assumption ϕ. There exists an m 2 (0, + ] such that ϕ j [0,m) is strictly increasing, ϕ j [m,+ ) is constant, ϕ 00 j (0,m) < 0 and ϕ 0 (m ) = 0. c Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

25 The Model A n-dimensional (Sidrauski) General-Equilibrium Measure of the Welfare Costs of In ation Equations (2) now imply: u i (m) = ϕ 0 (G (m)) ϕ(g (m)) G (m) ϕ 0 (G (m)) G m i (m), 8i 2 f1,..., ng. (4) Let C m := fm 2 R n ++ : G (m) = mg. The consumer is satiated with monetary balances when G (m) = m. We denote by m those m which lie in C m. 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: U(1 + w(m), G (m)) = U(1, G ( m)) (5) raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

26 The Model A n-dimensional (Sidrauski) General-Equilibrium Measure of the Welfare Costs of In ation Take the partial derivatives of (5) and divide by ϕ 0 G (1 + w(m)) 1 m to obtain: 1 ϕ G 1+w (m) m w i (m) + G ϕ G w (m) m mi (m) 1 w i (m) G 1 + w(m) m = 0. Using (4): 1 ϕ G 1+w (m) m = ϕ G w (m) m G mi (m) + G 1 u i 1+w (m) m w(m) m, raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

27 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: 1 w i (m) = u i 1 + 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 the line integral: Z 1 w(m) = u 1 + w(m) m dm (7) χ Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

28 The Model The Bailey-Divisia Measure as a General-Equilibrium Measure Note that, w(m) = Z χ 1 u 1 + w(m) m dm the measure of the welfare costs of in ation which emerges from Sidrauski s general-equilibrium model, is not equal to the Bailey-Divisia (B D ) measure, de ned by: B D (m) = u i (m), B D ( m) = 0 (8) or, alternatively, when expressed as a line integral, and considering the path χ de ned above: B D (m) = Z χ u dm (9) Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

29 The Model The Bailey-Divisia Measure as a General-Equilibrium Measure Our main purpose here is showing that B D = 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 f 0 > 0, f 00 < 0, g 0 > 0 and g 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 gt U concave with respect to (b, ḃ, m, _ m), which makes the Euler equations su cient for an optimum. Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

30 The Model The Bailey-Divisia Measure as a General-Equilibrium Measure In this particular case, equations (2) give, 8i 2 f1,..., ng: u i = f 0 (G (m))g mi (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 B D (m) = w(m) (13) raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

31 The Model The Bailey-Divisia Measure as a General-Equilibrium Measure 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 m i : By using (11) in (15): w i (m) = f 0 (G (m))g mi (m), 8i 2 f1,..., ng (15) w i (m) = u i (m), 8i 2 f1,..., ng (16) raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

32 The Model The Bailey-Divisia Measure as a General-Equilibrium Measure Proof. Normalize w(m) by making w( m) = 0, which is equivalent to writing B D (χ (0)) = 0. Now use the de nition of B D in (9) to obtain the main result: Z w(m) = u dm = B D (m) (17) χ raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

33 The Model The Bailey-Divisia Measure as a General-Equilibrium Measure 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 u i, 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 dλ w (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 the vector of monetary aggregrates and rχ(λ) for the vector of derivatives of χ with respect to λ. raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

34 Applications Examples raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

35 Applications Example 1: Area Under the Inverse Demand for Monetary Base Here we assume that the spreads u i, 8i 2 f2,..., ng are competitively determined by a costless banking system operating under constant and non-remunerated reserve requirements. In this case u i := r r i = k i r, k i standing for the reserve requirements on deposit i.let z stand for the monetary base, k := (k 2,..., k n ), m ( 1) := (m 2,..., m n ) 2 [0, + ] n 1. With u i = k i r for all i (17) becomes: Z B D (m) = w(m) = r(dm 1 + k dm ( 1) ) (19) rdz But in such an economy the monetary base (equal to currency plus non-interest-bearing reserves deposited in the Central Bank) reads z := m 1 + k m. Since k is a vector of constants, dz = dm 1 + k dm ( 1), in which case (19) can be written as: B D (m) = w(m) = Z m1 +km ( 1) m 1 +k m ( 1) Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31 χ

36 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 M 1, as used, for instance, by Lucas (2000)). Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

37 Applications Example 2: Constant Spreads Let λ = r in (18) and assume that all banking spreads u i other than that on currency (i 2 f2,..., ng) remain constant. This would be the case for instance if m 2, m 3,...m n 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, B D (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: B D (m) = Z r 0 [xm 0 1(x) + n i=2 ūi m 0 i (x)]dx Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

38 Applications Example 2: An Important Conclusion B D (m) = Z r 0 [xm 0 1(x) + n i=2 ūi m 0 i (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 ū i mi 0 (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. Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

39 Applications Example 3 Take, in (10), g as the identity function, f de ned by f (x) = x 1 σ 1 1 σ, where σ > 0 and σ 6= 1, f (x) = ln x when σ = 1 and G (m 1, m 2 ) = m µ 1 m1 µ 2, with 0 < µ < 1. From (11) we have u i (m) = G (m) σ G xi (m). So, from (9), B D (m) = = Z G (m) σ (G x1 (m) dm 1 + G x2 (m) dm 2 ) χ Z G (m) G ( m) G σ dg = 1 G ( m) 1 σ G (m) 1 σ (20) 1 σ The de nition of G is only used in the nal step to generate: B D (m) = 1 σ) [ mµ(1 1 σ 1 m (1 µ)(1 σ) µ(1 σ) (1 µ)(1 σ) 2 m 1 m 2 ] Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

40 Applications Example 3 When σ = 1 : B D (m) = ln G ( m) G (m) = µ ln m 1 m 1 + (1 µ) ln m 2 m 2 (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). raduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

41 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 exact measure of the deadweight loss stemming from taxation. Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

42 References Bailey, Martin J. (1956) "Welfare Cost of In ationary Finance", Journal of Political Economy 64, Cysne, Rubens Penha. (2003). "Divisia Index, In ation and Welfare". Journal of Money, Credit and Banking, The Ohio State University, v. 35, n. 2, 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 Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31

43 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 Lucas, R. E. Jr. (2000) "In ation and Welfare". Econometrica 68, 62, Sidrauski, M. (1967). "Rational Choice and Patterns of Growth in a Monetary Economy". American Economic Review; 57; Graduate School of Economics (EPGE/FGV) Advances in Macroeconomics 05/27/ / 31