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A note on the welfare cost of money creation
Journal of Monetary Economics 2 (1976) 121-124. 0 North-Holland Publishing Company
A NOTE ON THE WELFARE COST OF MONEY CREATION Alvin L. MARTY Cify University, New York, NY. U.S.A. University of Manchester, Manchester, England
It is well known that an optimal tax mix equalizes the ratios of the marginal change in the deadweight loss to the marginal change in the tax revenue. In the case of a tax on real cash balances, it has been widely overlooked that a simple relationship (which depends only on the elasticity of demand for real balances) exists between the marginal deadweight loss and the marginal tax revenue. Since this relationship avoids the necessity to evaluate a particular integral and holds for any general demand function, it is computationally convenient and has the added advantage of focusing on the economic essence of the marginal concept.’ Let: M
= the nominal money supply,
h/M
= p, the growth rate of the nominal money supply,
P
= the price level,
P/P
= n, the actual and expect bdrate of price change,
Y
= aggregate real income,
Y/Y
= 1.,the growth rate of aggregate real income (assumed constant),
r
= the real rate (assumed constant),
i
c the money rate of interest,
Iv1
= the elasticity of demand for real balances with respect to i,
N,,
= the elasticity of demand for real balances with respect to p.
Thcconccpt of the ratio of the marginal increment to deadweight loss to the increment to revenue is found, for example, in Tower (1971), Cathcart (1973). Marty (l973), and Frcnkel (1975). The average ratio of welfare loss to revenue was first derived for the Cagan function (which involves evaluating a particular integral). and then this average ratio was used to compute the marginal ratio. This is mathcmati *ally cumbersome and economically misleading evaluating a particular integral and holds for since the marginal ratio can be derived withv any general demand function. For an excc lon see Barre ( 1972). who does use a general demand function but restricts his results to the case of a stationary economy.
A. L. Marty, Welfare cost of money creation
122
Under the assumption that the per capita income elasticity of demand for real balances is unity, it is well known that we can write M!P m C-E Y
44 L)
(0
where Y is aggregate real income; and since p = i- r + 1, m = e(p), the welfare loss FV(taken as a ratio to income) is
s”, vm) dx-JI@)P+[r-~1[~(8)_~(p)],
(2)
where #(a) is the stock of real balances held as a zero p. Then the marginal increment to the welfare loss is aW y&
=
ly(p)[il-r-p]
=
-it/v(p)
=
-if/l’(i),
where i is the money rate equal to p - il+r, at the giYlenp. Since the revenue G = p+(p), the marginal increment to revenue is ap =
P* b9 + *id
where i again is the money rate equal to p -r+ II. We note that the elasticity of demand for real balances with respect p:oi is NiE
--
i@(i)
4(i 1’
and the elasticity with respect to p is Np =
dw)P --. fP(i )
(6)
It follows directly, dividing eq. (3) by eq. (4), thst
21n Barrow (1972, ey. (15)) dWMG is given as --
XP 1+N,,
rp+r* ( rp 1 ’
where rp is the expected rate of price change and Nrp is the elasticity of demand with respect to rp. This reduces in the case of the stationary economy to - NJ( 1+ NJ since N,, = N,,.
A. L. Marty,
Welfare cost o~~money creation
123
Clearly no integration is needed; all anyone needs to do is choose a particular p which given r and I. determines both I7 and i, and evaluate the relevant elasticities. 3 The formula is even more simple if the revenue is defined as [(M/P)/ Y]*i = r,li, [Phelps (1973), Aurenheimer (1974)] rather than as (hi/P)/ Y. It would take us here too far afield to discuss the rationale for this new definition of the revenue. We note, however, that since Q/P = M/P(l7 + A) = M/P[i-
r+ A],
the two definitions are equivalent when r = I, since the base for a zero money rate of interest is the same as that for a zero rate of monetary expansion. The reader can easily show as an exercise that when either I’ = i. or the revenue is defined as nt 9i, that ZW
-=-
?G
Ni
IaNi’
(8)
since NP = Ni. Again all that is needed to calculate the relevant elasticities; no integration at all is required. Finally, we note that the economic interpretation of the two formulas does differ. When u/P is taken as the revenue we assume that the authorities produce a welfare loss only on positive rates of monetary expansion, so &V/~-G= Ni/(l ON,,) becomes infinite when N,, = 1, which is, on this definition, the point of maximum revenue. Alternatively, when the revenue is defined as (M/P)I’, we impute a welfare loss whenever i > 0 since the authorities are assumed to get revenue from Fpigniorage which CZI accrue at p < 0 if r > II. The revenue is then maximized and aw/i?G becomes infinite when Ni = 1, which occurs at a lower p (whenever r > A) than in the previous case. The reason for this is that the base for measuring the revenue is i = 0, which is lower along the demand schedule than the point at which p = 0 whenever r > ;I. At any given p, 2w/EG will then be higher when the revenue is taken as (M/P)I. When r = 2. the two interpretations are mathematically and economically equivalent. Vhc above rcvcals an interestingproperty of the demand function for real balances: it is the ratio of the marginal change in the deadweight loss to the marginal change in consumers’ surplus.
References Aurenhcimer, L., 1974, The honest government’s guide to inflationary finance, Journal of Political Economy 80, no. 5,598-606. Barre, R., 1972, Inflationary finance and the welfare cost of inflation, Journal of Political Economy, September. Cathcart, C., 1974, Monetary dynamics, growth and the efficiency of inflationary finance, Journal of Money, Credit and Banking, May.
124
A. L. Marty, Welfare cost of money creation
:Frenkel, J., 1975, Inflationary expectations and some dynamic aspects of the welfare cost, in: Inflation in the world economy (Manchester University Press, Manchester). Marty, A.L., 1973, Growth, satiety and the tax revenue from money creation, Journal of Political Economy 81, September, 1136-l 152. Tower, E., 1971, More on the welfare cost of inflationary finance, Journal of Money, Credit and Banking, November. Phelps, E.S., 1973, Inflation in a theory of public finance, Swedish Economic Journal 75.67-82. :
A NOTE ON THE WELFARE COST OF MONEY CREATION Alvin L. MARTY Cify University, New York, NY. U.S.A. University of Manchester, Manchester, England
It is well known that an optimal tax mix equalizes the ratios of the marginal change in the deadweight loss to the marginal change in the tax revenue. In the case of a tax on real cash balances, it has been widely overlooked that a simple relationship (which depends only on the elasticity of demand for real balances) exists between the marginal deadweight loss and the marginal tax revenue. Since this relationship avoids the necessity to evaluate a particular integral and holds for any general demand function, it is computationally convenient and has the added advantage of focusing on the economic essence of the marginal concept.’ Let: M
= the nominal money supply,
h/M
= p, the growth rate of the nominal money supply,
P
= the price level,
P/P
= n, the actual and expect bdrate of price change,
Y
= aggregate real income,
Y/Y
= 1.,the growth rate of aggregate real income (assumed constant),
r
= the real rate (assumed constant),
i
c the money rate of interest,
Iv1
= the elasticity of demand for real balances with respect to i,
N,,
= the elasticity of demand for real balances with respect to p.
Thcconccpt of the ratio of the marginal increment to deadweight loss to the increment to revenue is found, for example, in Tower (1971), Cathcart (1973). Marty (l973), and Frcnkel (1975). The average ratio of welfare loss to revenue was first derived for the Cagan function (which involves evaluating a particular integral). and then this average ratio was used to compute the marginal ratio. This is mathcmati *ally cumbersome and economically misleading evaluating a particular integral and holds for since the marginal ratio can be derived withv any general demand function. For an excc lon see Barre ( 1972). who does use a general demand function but restricts his results to the case of a stationary economy.
A. L. Marty, Welfare cost of money creation
122
Under the assumption that the per capita income elasticity of demand for real balances is unity, it is well known that we can write M!P m C-E Y
44 L)
(0
where Y is aggregate real income; and since p = i- r + 1, m = e(p), the welfare loss FV(taken as a ratio to income) is
s”, vm) dx-JI@)P+[r-~1[~(8)_~(p)],
(2)
where #(a) is the stock of real balances held as a zero p. Then the marginal increment to the welfare loss is aW y&
=
ly(p)[il-r-p]
=
-it/v(p)
=
-if/l’(i),
where i is the money rate equal to p - il+r, at the giYlenp. Since the revenue G = p+(p), the marginal increment to revenue is ap =
P* b9 + *id
where i again is the money rate equal to p -r+ II. We note that the elasticity of demand for real balances with respect p:oi is NiE
--
i@(i)
4(i 1’
and the elasticity with respect to p is Np =
dw)P --. fP(i )
(6)
It follows directly, dividing eq. (3) by eq. (4), thst
21n Barrow (1972, ey. (15)) dWMG is given as --
XP 1+N,,
rp+r* ( rp 1 ’
where rp is the expected rate of price change and Nrp is the elasticity of demand with respect to rp. This reduces in the case of the stationary economy to - NJ( 1+ NJ since N,, = N,,.
A. L. Marty,
Welfare cost o~~money creation
123
Clearly no integration is needed; all anyone needs to do is choose a particular p which given r and I. determines both I7 and i, and evaluate the relevant elasticities. 3 The formula is even more simple if the revenue is defined as [(M/P)/ Y]*i = r,li, [Phelps (1973), Aurenheimer (1974)] rather than as (hi/P)/ Y. It would take us here too far afield to discuss the rationale for this new definition of the revenue. We note, however, that since Q/P = M/P(l7 + A) = M/P[i-
r+ A],
the two definitions are equivalent when r = I, since the base for a zero money rate of interest is the same as that for a zero rate of monetary expansion. The reader can easily show as an exercise that when either I’ = i. or the revenue is defined as nt 9i, that ZW
-=-
?G
Ni
IaNi’
(8)
since NP = Ni. Again all that is needed to calculate the relevant elasticities; no integration at all is required. Finally, we note that the economic interpretation of the two formulas does differ. When u/P is taken as the revenue we assume that the authorities produce a welfare loss only on positive rates of monetary expansion, so &V/~-G= Ni/(l ON,,) becomes infinite when N,, = 1, which is, on this definition, the point of maximum revenue. Alternatively, when the revenue is defined as (M/P)I’, we impute a welfare loss whenever i > 0 since the authorities are assumed to get revenue from Fpigniorage which CZI accrue at p < 0 if r > II. The revenue is then maximized and aw/i?G becomes infinite when Ni = 1, which occurs at a lower p (whenever r > A) than in the previous case. The reason for this is that the base for measuring the revenue is i = 0, which is lower along the demand schedule than the point at which p = 0 whenever r > ;I. At any given p, 2w/EG will then be higher when the revenue is taken as (M/P)I. When r = 2. the two interpretations are mathematically and economically equivalent. Vhc above rcvcals an interestingproperty of the demand function for real balances: it is the ratio of the marginal change in the deadweight loss to the marginal change in consumers’ surplus.
References Aurenhcimer, L., 1974, The honest government’s guide to inflationary finance, Journal of Political Economy 80, no. 5,598-606. Barre, R., 1972, Inflationary finance and the welfare cost of inflation, Journal of Political Economy, September. Cathcart, C., 1974, Monetary dynamics, growth and the efficiency of inflationary finance, Journal of Money, Credit and Banking, May.
124
A. L. Marty, Welfare cost of money creation
:Frenkel, J., 1975, Inflationary expectations and some dynamic aspects of the welfare cost, in: Inflation in the world economy (Manchester University Press, Manchester). Marty, A.L., 1973, Growth, satiety and the tax revenue from money creation, Journal of Political Economy 81, September, 1136-l 152. Tower, E., 1971, More on the welfare cost of inflationary finance, Journal of Money, Credit and Banking, November. Phelps, E.S., 1973, Inflation in a theory of public finance, Swedish Economic Journal 75.67-82. :