APPENDIX: Effects of Inflation and Control Problem Solution
This Appendix provides a formal demonstration of the analysis contained in section II of the paper. The effects of inflation on total revenue from taxation and inflation and on the effective saving and effective consumption ratios are derived in section 1. The solution to the control problem described in equation (15), its existence and stability, as well as the effects of a change in the structural parameters on the optimal rate of monetary expansion, are demonstrated in section 2.
Cathcart, C. D., “Monetary Dynamics, Growth and the Efficiency of Inflationary Finance,” Journal of Money, Credit, and Banking, Vol. 6 (May 1974), pp. 169-190.
Choudhry, N. N., “Collection Lags, Fiscal Revenue and Inflationary Financing: Empirical Evidence and Analysis, WP/91/41, April 1991, International Monetary Fund.
Dixit, A., “The Optimal Mix of Inflationary Finance and Commodity Taxation with Collection Lags,” Staff Papers, Vol. 38 (September 1991), pp. 643-654.
Driffill, J., Mizon, G., and Ulph, A., “Costs of Inflation,” Discussion Paper No. 293, Center for Economic Policy Research, April 1989.
Frenkel, J. A., “Inflation and the Formation of Expectations,” Journal of Monetary Economics, Vol. 1 (October 1975), pp. 403-421.
Frenkel, J. A., “Some Aspects of the Welfare Cost of Inflationary Finance,” in Money and Finance in Economic Growth and Development: Essays in Honor of S. Shaw, edited by R. I. McKinnon, New York, Marcel Dekker, 1976.
Marty, A., “Growth and the Welfare Cost of Inflationary Finance,” Journal of Political Economy, Vol. 75 (February 1967), pp. 71-76.
Marty, A., “Growth, Satiety and Tax Revenue from Money Creation,” Journal of Political Economy, Vol. 81 (September/October 1973), pp. 1136-52.
Tanzi, V., “Inflation, Real Tax Revenue, and the Case for Inflationary Finance: Theory with an Application to Argentina,” Staff Papers, Vol. 25 (September 1978), pp. 417-451.
The author wishes to particularly thank Abbas Mirakhor, Vicente Galbis and Peter Wickham and also Bijan Agehlvi and Vito Tanzi for helpful suggestions. Thanks too are due to G. Hettiarachchi and Marco Lari for providing excellent research assistance. All remaining errors are the responsibility of the author.
The insight on collection lags and its importance for government finance in developing countries was developed by Tanzi (1977). This analysis has been elaborated further by others, including Tanzi (1978), and most recently by Choudhry (1990, 1991); the latter analyzes and provides evidence on collection lags, the erosion of real fiscal revenue by inflation, and the scope of inflationary finance for a large number of developing countries.
Dixit (1990) has argued that the optimal rates of consumption taxes should also be reconsidered in the light of collection lags. When this is done, the erosion of fiscal revenue from inflation can be recouped by changing the rates on these taxes. While indexation of taxes shifts the burden of fiscal erosion away from the inflation tax, Dixit’s analysis ignores the social costs and economic distortions stemming from inflation, as evidenced by the experience of many countries.
Aghevli (1977) made a case in favor of moderate rates of inflation when countries resort to inflationary finance in pursuit of their growth objectives.
The model described below retains the salient features of the model used by Aghevli (1977). Henceforth, all variables are expressed in real per capita terms, unless otherwise specified.
The assumption of a constant private saving ratio facilitates the analysis to focus on the use of the inflation tax.
This simplification avoids the complications associated with the adjustment of price expectations. The divergence between actual and anticipated inflation, as has been frequently observed, leads not only to the transfer of real resources to the government but to other costs associated with the distribution of income and wealth. For the dynamic aspects of the welfare costs of inflation, when expectations adjust with a lag to the actual price developments, see Cathcart (1974) and Frenkel (1975, 1976). Also, for a survey of the recent theoretical and empirical work on the costs of inflation, see Driffill, et al (1989).
The rate μse is derived from the properties of ɸ(μ), and λ(μ) in the Appendix, section 1, which also formally demonstrates other statements made later in the text.
This measure of wc, which is the area under the demand curve for real balances, is based on the assumption that the marginal utility of consumption goods is constant. For a detailed analysis of the welfare cost of inflation and the derivation of the standard expression, see Bailey (1956) and Marty (1967, 1973). Also for the associated adjustment costs pertaining to price expectations, see Cathcart (1974) and Frenkel (1975 and 1976).
This definition, which is also used by Aghevli (1975), is based on the assumption that real balances are a substitute for consumption goods.
The solution of the control problem of equation (15) is derived in the Appendix, section 2. If the buoyancy of fiscal revenue is greater than unity (β1>1), the relationship determining the optimal rate of monetary expansion is given by:
The local stability requires the conditions:
A high value of α implies that the elasticity of inflation of real balances is greater for a given expected rate of inflation. Thus, an increase in the expected rate of inflation would lower the desired real balance holdings more than if the value of α were small. Hence, with a smaller value of α, inflation revenue would be higher at a relatively low rate of inflation.
This implies that, for a given expected inflation elasticity of real balances, the higher the value of this coefficient, the lower will be the optimal rate of monetary expansion whether or not there is fiscal erosion.
The Golden rule prescribes that in the steady state:
f′(k) = n + δ.
Using the steady-state expression for
Therefore, given n and σ, and since se(n) = s(1-βo) + nαo, the lower the rate of private saving, the higher is the social discount rate.
The choice of δ = 0.05 may be considered an admissible value. This can be seen by rewriting the p = 0 equation in (27) as:
Also, using the
Equating these two relationships, we have:
rearranging, we can write:
Given the values for the other parameters, as shown in Table 1, and for the values of: μ = 0.012; α = 2.0 and β = 0.35 and p = 1.41, which are obtained from Table 2, it is seen that for s = 0.05, the value of δ is 0.056.
While there is still debate in the literature on the choice of an appropriate rate of discount, a rate of 3 percent provides some room for use of inflationary finance.
The steady-state values of output and consumption of goods were 84.9 and 68.2, respectively, when μ = n.
Whether or not there is fiscal erosion, the conventional measure of the welfare cost of inflation, which is based on the assumption that output is fixed, does not reflect the “true” cost of resources. The measure,
A paper on the optimal mix of income taxation and inflationary finance by Abbas Mirakhor and this author will be forthcoming shortly.
Analytically, the increased burden of interest payment, which has the same effect on the fiscal deficit as the erosion of fiscal revenue, is likely to further limit the use of he inflation tax.
The second derivative, r″(μ) = ɸ″(μ) + λ″(μ) = β2ɸ(μ) - (2/μ - α)αλ(μ), is
If the extent of fiscal erosion is sufficiently large, it is possible that a positive inflation rate at which total revenue is maximum does not exist, implying μr < n. Such a solution is not considered here.
See footnote one on page 11.
Notice that the saddle point exists as long as μ* ≤ μr.