The Inflation Tax is Likely to be Inefficient at any Level

Banaian, King; McClure, J. Harold; Willett, Thomas D. Article The Inflation Tax is Likely to be Inefficient at any Level Kredit und Kapital Provided in Cooperation with: Duncker & Humblot, Berlin Suggested Citation: Banaian, King; McClure, J. Harold; Willett, Thomas D. (1994) : The Inflation Tax is Likely to be Inefficient at any Level, Kredit und Kapital, ISSN 0023-4591, Duncker & Humblot, Berlin, Vol. 27, Iss. 1, pp. 30-42, https://doi.org/10.3790/ccm.27.1.30 This Version is available at: https://hdl.handle.net/10419/293269 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Introduction Standard efficiency models in public finance call for equating the marginal cost of taxation from alternative sources. According to standard calculations from the literature on the optimal inflation tax, the double digit inflations most countries suffered during the 1970s were closer to optimal than the substantially lower rates of the 1980s. Several recent papers have pointed to the prospective loss of national seigniorage as a major cost of adopting a common European currency.1 The elasticity of demand for money with respect to expected inflation is usually estimated to lie in the range from 0.5 to 3.0. This implies an optimal inflation rate from 200 to 33 percent by the standard Bailey (1956) type calculations. We argue in this paper that the previous calculations of the standard optimal inflation tax likely contain a substantial upward bias because they are based on the assumption that inflation is perfectly anticipated.2 This is seldom the case. Moreover, a mounting body of evidence suggests a positive relationship between the rate and * Banaian is Associate Professor of Economics, St. Cloud State University. McClure is Assistant Professor, Villanova University. Willett is Horton Professor of Economics, The Claremont Graduate School, and Associate Director of The Lowe Institute of Political Economy at Claremont McKenna College. 1 See, for example, Dornbusch (1988), Drazen (1989), and Grilli (1989). 2 The qualitative implications of inflation uncertainty for the optimal inflation tax was briefly noted in Logue and Willett (1976), but as far as we know, this issue has not been subsequently analyzed in any detail. Kimbrough (1986) suggests another problem with the optimal inflation tax literature: If money is an intermediate good, then the optimal inflation tax literature suggests that its tax rate should be zero. It would be greater than zero only if money is used in final consumption, as the standard inflation tax literature assumes. See also Faig (1988). We note that both papers do not allow for deadweight costs of labor income taxation, making it optimal to fund all government exependitures through the income tax. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.27.1.30 | Generated on 2023-01-16 13:04:41 The Inflation Tax is Likely to be Inefficient at any Level 31 variability of inflation.3 While causation may run both ways if greater variability increases uncertainty (Devereux [1989] and Cooley and Hansen [1989]), this is likely to lower output because individuals must devote more resources to more frequent contracting, shifting in the supply schedules of risk-averse agents. We show in Section II the sensitivity of revenue-maximizing and optimal rates of inflation to output loss due to inflation uncertainty. In section III, calculations for seven OECD economies show that the output loss from inflation uncertainty need not be large for the optimal rate of inflation to fall to zero. Typically the size of the output loss is less than the few available estimates of it.4 The traditional intuition of many economists that inflation is an inefficient form of taxation may be correct after all. In the concluding section, we explore how open economy and political economy considerations are influenced by this analysis. II. How the Output Effects of Inflation Uncertainty Alter the Calculations of Optimal Inflation A government has two sources of revenue. It may tax labor income at a flat rate of t or it may tax money balances at a rate of 7r. Its revenues are calculated to be (1) R = ty + TT(M/P) where is real income and M/P is real money income. If raising revenue from each source reduces the size of the tax base, we set taxes such that MCt = MCn. There is little doubt that the marginal costs of labor income taxation are substantial. Browning (1987) finds that the size of the marginal welfare costs is impossible to estimate with great precision, but that it will likely exceed 9 percent of additional revenues. Stuart (1984) and Ballard, Shoven, and Whalley (1985) find that the range of costs per dollar of revenue raised lies between 15 and 50 percent. 3 See Frohman, Laney and Willett (1981) and Holland (1984) for surveys of the evidence. The point has been debated; for example, Fischer (1981; 1984) argues that inflation and uncertainty are jointly caused by supply shocks. Frohman, Laney and Willett find that for the U.S. the effect persisted even in a sample from which the OPEC shocks were removed, though the size of the effect was reduced. 4 Estimates of this "Friedman effect" are given in Mullinean (1980), Levi and Makin (1980), Evans (1983), and Holland (1986). OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.27.1.30 | Generated on 2023-01-16 13:04:41 32 King Banaian, J. Harold McClure and Thomas D. Willett The welfare costs of inflation are usually depicted as the area under the money demand curve in excess of some (assumed constant) real rate of interest. Using a Cagan money demand function of the form M = A • ya • epi it is usually found that the marginal cost of inflationary finance is equal to 77/(1 - 77), where 77 is the interest semi-elasticity of demand. It is from this calculation that the optimal inflation rates of 30 - 200 percent come. These considerations are based on the traditional Fisherian effect that anticipated inflation raises nominal interest rates and lowers money demand while leaving income and output basically unaffected.5 However, if inflation cannot be perfectly anticipated, the associated reduction in output can be substantial. Since output is an element in the money demand function specified above, output loss will lower the base on which the inflation tax is imposed. If uncertainty is unrelated to average inflation, then incorporation of uncertainty would lower the absolute level of resources obtainable, but would not alter the marginal conditions of optimal inflation rates. Higher inflation on average raises uncertainty and reduces the marginal revenue gains from inflation. Consider the total derivative of the money demand function with respect to inflation: , , dM dM di dM dy di dM dy d 7r di d 7r dy d ir dn dy dir The Fisher effect i = r + TT gives di/dn = 1 if real interest rates are not affected by fully anticipated inflation.6 A vertical long-run Phillips curve renders dy/dir = 0 and the traditional evaluation can follow, such as in Bordo and Stuart (1986). 5 See for example Bailey (1956), Phelps (1973), and Tower (1976). For a recent survey and references to the literature see McClure and Willett (1988). Within the paradigm of the Lucas supply curve, it would be inflation uncertainty relative to price uncertainty that matters. For an interesting treatment of the inflation tax within a cash-in-advance model that also finds an upward-sloping supply curve, see Cooley and Hansen (1989). 6 Reasons why this might not be true include tax wedges between nominal and real interest rates as discussed by Darby (1974) or Feldstein (1976) and the effect of inflation uncertainty directly on money demand as discussed by Klein (1975). We assume this to simplify the analysis; if transactions costs from higher inflation raises real interest rates, the reduction in both revenue-maximizing and optimal inflation rates is reinforced. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.27.1.30 | Generated on 2023-01-16 13:04:41 The Inflation Tax is Likely to be Inefficient at any Level 33 The Friedman hypothesis is that higher inflation lowers output, so that dy/dn = f < 0. To see this expand this derivative by the chain rule dy_ = daL dy dir dir dan The first term in the product represents our claim that higher inflation leads to higher inflation uncertainty; the second term is the Friedman effect from uncertainty to output. The second term is negative. Froyen and Waud (1987) found that higher and more variable inflation increased uncertainty in Canada, the U.K. and the U.S. in the 1970s, though not for West Germany. Welch (1989) provides similar evidence for Brazil. Froyen and Waud also demonstrate that for Canada and the U.K., increases in aggregate price uncertainty had negative output effects. Lower output reduces revenue gained from the income tax, lowering the benefits to government of higher inflation. This is obtained by taking the derivative of the revenue equation (1) with respect to inflation dR dM (4) —— = M + 71" • —— + t • f dir dir Employing equations (2) and (3) we can solve for the revenue-maximiz- ing rate of inflation to be 1 + t(f/y)(y/M) (5) *** = -(/*+ <*(//*)) This includes the traditional result of -1/(3 as a special case where / = 0. Following Bailey (1956, 100 - 01), the area of the money demand curve that results from a rise in nominal interest rates represents the deadweight loss of inflationary finance. This measurement W* requires expansion as well to include the output reduction from the uncertainty effect. Suppose the real rate of interest is .01. Let output equal 1000, and the values of the parameters A, a and (3 equal 1, 1 and -1 respectively. Individuals thus currently desire to hold 990 in real balances. The government desires to raise real revenue by increasing the money supply by 2 % per annum. The area under the demand for money function as the quantity of real balances desired declines from M to M* represents the usual welfare loss. This would be computed by taking the definite integral of the money demand function between nominal interest rates of 1 and 3 3 Kredit und Kapital 1/1994 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.27.1.30 | Generated on 2023-01-16 13:04:41 34 King Banaian, J. Harold McClure and Thomas D. Willett irtertist raíes 0.06 t 005 004 0.03 0.02 • 0.01 0 M' Kf M Rbeì money Manees Figure 1 : Money Demand with output effects percent, graphically shown as the triangle A. With our parameters, the loss is 0.392, or about 0.4% of output. The gain would be two percent of the remaining base, or 19.41. The marginal cost per dollar gained is 0.02. But this understates the total decline in the demand for money if output effects occur. A decline in output due to expected inflation adds to the loss by shifting in the money demand curve. As we showed above, this would decrease the revenue base on which the inflation tax would operate, and reduce the revenues generated from an income tax. It also increases the size of the welfare cost by the area of rectangle B in the graph. This loss represents the loss of additional services from money because fewer transactions are being made: higher transactions costs reduce the number of transactions. If / = 0.1, which we argue later is the lower bound of possible Friedman effect parameters, the amount of money demanded will fall by an additional 1.94. This raises the welfare cost of the inflation tax to .4308, thus understating the cost by about Algebraically, the marginal welfare cost of raising the inflation rate by one more point can be viewed as where <\> is the change in welfare resulting from a one-unit change in income, which we assume equals one. To simplify, we assume a social welfare function with output as the sole determinant. The traditional welfare cost of (3iM thus understates the true welfare cost in two ways: (i) the traditional deadweight loss per point of inflation is increased by 10%. dW ——=(/? + a • f/y) • i - M +f •<£ < 0 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.27.1.30 | Generated on 2023-01-16 13:04:41 The Inflation Tax is Likely to be Inefficient at any Level 35 the lowering of money demand as demonstrated above (as given by the expression a(f/y)iM)\ and (ii) the inclusion of output reduction in the consideration of welfare losses (the value /</>). We can now calculate the marginal welfare cost of inflationary finance by taking the ratio of equation (6) to equation (4) _ -dW/dn _ -i((3 + q • f/y) + 4, (f/y) • (y/M) dR/dn 1 + (0 + a • (f/y)) + t(f/y) • (y/M) Note again that the Bailey solution of -i/3/(l + f3) is obtained if / = 0. Using the Fisher relation for the value of i above, we can solve for the optimal inflation rate as m . = MC,(1 + t • (f/y) (y/M)) + <j> • (f/y) (y/M) _ U * -(i + a.//y)(l+MCf) r Note that the optimal inflation rate depends on the marginal tax rate and the velocity of money when the Friedman effect is present. Higher marginal tax rates lower the revenue received from a one percent change in the inflation rate without any change in the welfare costs. Thus higher income tax rates lower the optimal inflation rate.7 Higher velocity will also reduce the optimal inflation rate both by lowering the level of revenues received from seigniorage (note the expression (f/y) (y/M) < 0) and by raising the welfare costs. Countries like Germany with relatively high velocity and income tax rates can be expected to have lower optimal inflation rates than those with lower tax rates and lower velocity like Japan. More surprisingly, countries with low nominal interest rate elasticities will have lower revenue maximizing rates when a Friedman effect is present.8 It is precisely those countries that gain the most from seigniorage according to the Bailey formula. III. How Big an Effect? To illustrate the likely order of magnitude the Friedman effect would have on the revenue-maximizing and optimal inflation rates, we use pub- 7 This point is also made by Bordo and Stuart (1986). In that work they calculate the optimal inflation rate for tax rates between 21 and 70 percent given the marginal welfare burdens of those income tax rates demonstrated by Stuart (1984). Bordo and Stuart do not include discussion of a Friedman effect. 8 You see this by noting that d"RM = 1 + WyHy/M) _}_ wjiere ^he right hand J 6 d0 (0 + a(f/y))2 ffi & side of the inequality represents the comparative static if / = 0. 3* OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.27.1.30 | Generated on 2023-01-16 13:04:41 36 King Banaian, J. Harold McClure and Thomas D. Willett lished estimates of money demand for seven industrial countries from Boughton (1981). He models money demand roughly similar to Cagan (1956), regressing money on inflation, real interest rates, income, and lagged money. His long run inflation elasticities are in the second column of table 1. The next two columns show historical averages of the ratio of GNP to money and the average tax rate for the period 1960 - 86. There were often breaks in the data series; where this occurred, we used only the more recent part of the series. All data are from the 1988 International Financial Statistics Yearbook. Browning (1987) estimates that the marginal welfare burden of the income tax was roughly a minimum 8-10% of revenue.9 Shoven, Ballard, and Whalley (1985) and Fullerton suggest marginal costs in the area of 30 - 50%. We use the rates of 10, 30, and 50 percent as three estimates that likely encompass all plausible values. Note that higher costs would dramatically increase the optimal inflation rate when uncertainty effects are absent. Levi and Makin (1980) and Froyen and Waud (1987) have estimated the size of the Friedman effect. Levi and Makin find that a one percentage point rise in inflation lowers employment by 0.1 to 0.2 percent in the United States. Using the higher estimate and Okun's Law that a one percent rise in unemployment reduces GNP by 3 percent gives a value for f/y = 0.6. If there were a one-to-one relationship between rates of change of employment and output, the lower end of their estimate would give a value for//?/ = 0.1. We considered these along with the values 0.3 and zero. The latter four columns of Table 1 show the revenue-maximizing inflation rates. The actual inflation rate is show for comparison purposes. The reduction in revenue-maximizing rates is most pronounced in the cases we delineated at the end of section II: countries with high velocities and high tax rates lose the most from the presence of uncertainty effects. With a sizable Friedman effect the revenue-maximizing rates decline a minimum of 40% for Italy, the country Boughton shows to have the highest interest elasticity and lowest income tax rate. In two cases, Germany and the United Kingdom, a high Friedman effect leads government to not inflate at all. 9 Browning suggests that the welfare costs cannot be accurately measured. His range of estimates runs from 10 - 200 percent. We are taking his low-end estimate for our computations without comment on its accuracy. Gordon (1976) suggests that 3.3% is too high an estimate, but we find no agreement elsewhere with this assertion. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.27.1.30 | Generated on 2023-01-16 13:04:41 The Inflation Tax is Likely to be Inefficient at any Level 37 Table 1 Revenue-Maximizing Inflation Rates Country Interest Elasticity GNP/ Money Avg. Tax Rate Actual Inflation TTKM if f/y = 0 TTKM if f/y = 1 TTÄM if f/y = .3 TTÄM if f/y = -6 Canada - .30 6.76 .18 6.2 333.3 219.6 105.8 30.0 France -1.66 3.27 .37 7.4 60.2 49.9 32.5 12.1 Germany - .42 6.12 .27 4.4 238.1 160.5 70.0 0.8 Italy -2.99 2.27 .20 11.1 33.4 30.9 26.3 20.3 Japan -1.41 3.24 .11 4.6 70.9 63.9 52.2 39.1 U.K. -1.19 5.21 .33 8.5 84.0 64.2 32.5 - 1.8 U.S.A. -0.76 4.66 .20 5.2 131.6 105.4 68.0 32.4 Dates for averages run 1960 - 1983 except Japan ends 1979, France begins 1972, Germany begins 1968, and Canada begins 1969. Elasticities from Boughton (1981). Velocity and average taxes from IMF Yearbook, 1988. Optimal inflation rates for f/y equal to zero and 0.1 are shown in Table 2. In every case when the Friedman effect coefficient was set to 0.3 or 0.6, we obtained negative optimal rates.10 With a small Friedman effect of 0.1, optimal inflation rates are positive for some countries if the marginal cost of the income tax is high, but these still fall substantially from their perfect certainty levels. Interestingly, the optimal rate for Italy in this case is very close to its actual rate. With lower deadweight costs of the income tax, the optimal rate is negative in all but one case. To view this slightly differently, we display in Table 3 the size of the output loss ratio f/y necessary to make the revenue- and welfare-maxi- mizing rates of inflation equal zero. In five of seven cases, the output loss necessary to remove any revenue gain from the inflation tax is implausibly large. For example, the output loss necessary to remove incentives for inflationary finance in Japan would be 2.8% for a one perio We do not believe that the current empirical evidence should be interpreted as suggesting that negative inflation rates would show less variability and uncertainty than zero rates. Thus, on uncertainty-minimizing grounds, a zero rate of inflation would seem to be the appropriate long-run target and we treat calculations of negative rates as if they were zero. Friedman (1956) argues that a slightly negative rate of inflation would maximize consumer surplus by driving the nominal rate to zero and maxizing the area under the money demand curve. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.27.1.30 | Generated on 2023-01-16 13:04:41