empec (11) 16:25-33 Public Good Provision Rules and Income Distribution: Some General Equilibrium Calculations By J. Piggott I and J. Whalley 2 Abstract: A central issue in the analysis of public goods is the relationship between the optimal provision level and the distribution of income. Theoretical research has stressed the conditions under which the optimum is independent of the distribution of income. Here we focus on numerical analysis of more policy-relevant concerns. Specifically, to what extent is a given redistribution of income likely to affect the optimal level of public good supply? And how significant are the welfare costs of not adjusting public good supply when income distribution changes? We use an applied general equilibrium (AGE) model of the Australian economy and public sector to generate numerical estimates of the impacts of redistributive policies on these variables. Results suggest that the traditional separation of allocation and distribution in determining the level of public good supply may be a justifiable empirical simplification, except where very dramatic redistributions are involved. 1 Introduction A central issue in the analysis of public goods is the relationship between the optimal provision level and the distribution of income. Does optimal provision change as the income distribution changes, and if so how? Theoretical research has stressed the conditions under which the optimum is independent of the distribution of income. Samuelson (154) showed that independence is guaranteed when the preferences of all consumers are quasi-linear, a restrictive condition, since quasi-linearity implies zero income elasticity of demand for public goods, a property rejected by empirical evidence. More recently, Bergstrom and Comes (183) have formally derived both the necessary and sufficient conditions for in- We wish to acknowledge finanical support from the ARGS and the Reserve Bank of Australia. 1 John Piggott, School of Economics University of New South Wales Sydney Australia & Research School of Social Sciences, Australian National University GPO Box 4, Canberra 2601 Australia. 2 John Whalley, Department of Economics University of Western Ontario London Canada and National Bureau of Economic Research Cambridge Mass. USA. 0377-7332/1/1/25-33 11 Physica-Verlag, Heidelberg
26 J. Piggott and J. Whalley dependence. While the Bergstrom-Comes conditions admit a broader class of preferences than those allowed by quasi-linearity, they remain highly restrictive) Here we focus on numerical analysis of more policy relevant concerns. Specifically, to what extent is a given redistribution of income likely to affect the optimal level of public good supply? And are the welfare costs of not adjusting the level at which public goods are provided when the income distribution changes significant? These questions are important because many taxes levied to finance public provision also redisu'ibute income. Conventional estimates of the welfare costs of taxes (see St. Hilaire and Whalley (182), and Whalley (188)) typically ignore the links to public goods provision. We employ an applied general equilibrium (AGE) model of the Australian economy and public sector due to Piggott (183) to generate numerical estimates of the impacts of redistributive policies on the desired level of public good provision. We also compare assessments of welfare changes from tax changes with and without adjustment of the public good provision level. The results suggest that separation of allocation and distribution in determining optimal public goods supply may be a justifiable empirical simplification, except where very dramatic redistributions are involved. 2 An Applied General Equilibrium Model for the Analysis of Public Goods-Tax Interactions We use a single-period multi-consumer numerical general equilibrium model of the Australian economy and public sector due to Piggott (183) to make calculations of the effects of changes in the income distribution on the optimal provision of public goods, and hence on the welfare costs of taxes. On the demand side, household groups are identified by income range. Public goods are provided by the government and financed by taxes. Each group generates demand by maximizing a multistage CES utility function (defined over consumption goods and public goods), subject to its budget constraint (defined over private goods only). Household incomes are given by capital and labour income and transfers received, less income taxes paid. Government, foreign and corporate sectors are also identified. All the major taxes in Australia operating in 172-73 (the year to which the model is calibrated) are included in the model. These are the personal income tax, company income tax, rates and land tax, payroll tax, excises, tariffs, sales tax, 3 Bergstrom and Comes show that admissible preferences comprise those that can be represented by a utility function of the form A(G)Xi + Bi(G) for each individual i, where G is the public good and X i the ith individual's consumption of the private good.
Public Good Provision Rules and Income Distribution 27 motor vehicles taxes, and a variety of less important government financing instruments. Subsidies and tranfers are included, although with less detail specified. All are represented in ad valorem form, and for all but the personal income tax, average and marginal rates are assumed equal. On the expenditures side, government authorites undertake both real expenditure and transfers, at levels given by data for 172-73. Real expenditures reflect such goods and services as national defence, education, health and social services, maintenance of highways, policy and fire protection, judicial services, and capital expenditures such as highway construction. Capital and labour services along with commodity purchases, are treated as inputs into the production of these goods. Their composition reflects cost minimization. This structure is thus similar to other applied general equilibrium tax models (described in Shoven and Whalley (184)), but differs in that it incorporates public goods as arguments of individual utility functions. The model thus captures both the deadweight losses of taxes and consumer surplus benefits from public goods provision. The public good enters as an argument in household preference functions, with the level of provision determined by a chosen government provision rule. For computational simplicity, public good provision in the benchmark to which the model is calibrated is assumed to be based on the Samuelsonian rule that 2MRS = MRT. In the presence of distorting taxes, this is not an "optimal" provision rule, just as the taxes themselves are not typically "optimal". The Samuelson provision rule implies Pareto optimal allocation in the absence of distortionary taxes, but with distortionary taxes an over or under supply of public goods may be involved (see Atkinson and Stern (174)). However, it is the provision level which would be recommended from a cost benefit analysis which sought to determine the optimal provision level of some public good, on the assumption that the marginal social cost of $1 of revenue was $1. This last assumption is almost universally made in such analysis. The quantity of the public good provided by the government and consumed by all households is the same. Once specified in this way, the model is parameterized using a micro consistent equilibrium data set for Australia for 172-73. Both taxes and transactions of all economic agents specified in the model appear in these data. A units convention is used to separate these equilibrium data observations in value terms into separate price and quantity data. With functional forms chosen for production and demand (utility) functions, parameters are then generated which will reproduce the observed benchmark equilibrium under an unchanged tax and expenditure regime (see Mansur and Whalley (184)). In calibrating the model, we assume a benchmark equilibrium in which the sum of the marginal rates of substitution across households equals the ratio of producer prices for private and public goods in the presence of the 172-73 Australian tax/subsidy system. The government budget is balanced in the data set;
28 J. Piggott and J. Whalley and household preferences for the public good are inferred by imputing 'private expenditures' on public goods to individual households. The attribution rule employed in making these imputations in the calibration process is important for results because through calibration it determines the household public good preferences. There is little emporical guidance as to the appropriate rule. We use two different attribution rules, both of which are used extensively throughout the literature on empirical studies of public good provision. One involves public good expenditure by households proportional to income; the other uses an equal dollar amount per household. Since the quantity of public goods is common to all households, imputing expenditures in this way implicitly defines the personalized public good prices (MRS) used in calibration (see Foley (170)). We note in passing that under this calibration treatment the MRS for each household group in the benchmark equilibrium is independent of the elasticity of substitution chosen for preference functions. 4 When calibrated, the model reproduces the benchmark data as an equilibrium solution for the model. The parameter values for household utility functions determined in this way satisfy the condition that the sum of the marginal rates of substitution between the public good and a composite of private goods is equal to 4 The parameterization procedure may be illustrated as follows. Assume a CES utility function with a private good X and a public good G as arguments: U (o~a -~ X ~ ~ = +(l-a) o- (7 (l) First order conditions yield 1 (2) The personalized public good price Pc is defined such that the consumer would maximise utility by consuming G, if he faced the price PG and could choose the value of G. Thus 17"IFSxG : - -, ~Y where Px is the price of the private good X, which we may normalize to equal unity, Parameterization follows by imputing additional income to the consumer equal to PGG, and then setting a such that G is chosen when the consumer optimizes. It follows from (2) that this will occur when a is set to satisfy (t 2 co - P~
Public Good Provision Rules and Income Distribution 2 the corresponding ratio of producer good prices. The level of provision of the public good is not optimal, since distortionary taxes are present. The elasticity values used in the model are discussed in Piggott (183). An especially important set of parameters are elasticities of substitution between public and private goods in household preferences. We use a value of 0.5 which implies an uncompensated elasticity of demand with respect to the 'personal price' of public goods of -0.5, if the public good expenditure shares are small. The empirical evidence relevant to appraising the realism of this assumption is sparse. The major econometric evidence derives frommedian voter models, in which levels of provision of public goods in local jurisdictions are assumed to be determined by the preferences of the median voter (ranked by income). If preferences are identical over a large number of jurisdictions demand parameters for public goods can be estimated. The 0.5 value is in the neighbourhood of the midpoint of the range of estimates generated by these studies. 5 3 Model Results and Interpretation We have used the model described above to estimate the impacts on aggregate welfare and optimal public good provision of alternative lump sum taxes which change the income distribution. The welfare calculations yield estimates of the social costs of taxes, since existing distorting taxes m'e replaced by non-distortory lump sum taxes. Four types of lump sum tax scheme are investigated: regressive, poll, proportional and progressive. The lump sum tax structures are determined according to Ti = ~Yf (1) where/3 takes value of-1,0,1 and 2 to represent regressive, poll, proportional and progressive taxes respectively. Aggregate tax revenues (implied by the value of a) and the level of public good provision (where applicable) are both determined endogenously in the model as part of the equilibrium solution. The benckmark incomes of household groups in the model and implied burden profiles for each tax type are reported in Table 1. Variations in the distribu- 5 In their study of 826 municipalities in the U.S. with 160 populations between 10,000 and 150,000 located in 10 states, Bergstrom and Goodman (173) estimate tax share (price) elasticities. For general municipal expenditures (excluding education and welfare), elasticities by state range from -0.01 to -0.05, for expenditures on police from -0.13 to -0.76, and for expenditures on parks and recreation from +0.25 to -0.81. Pommerehne and Schneider (178) estimate tax share elasticities in their study of 110 Swiss municipalities at -0.17 to -0.72 depending on the classification of municipalities and estimating equation.
30 J. Piggott and J. Whalley tion of income across these replacement taxes are ctearly dramatic. Table 2 reports results from simulations in which all distortionary taxes are removed, with alternative lump sum taxes with widely varying redistributive impacts used to finance the required level of public good provision, transfers and other public sector outlays. Table I. Alternative lump sum tax replacements used in model calculations $A 172-3 Consumer type Average gross Tax burden household income in Benchmark Regressive* Poll Data set Proportional Progressive Poor Rich [ 2552 2424 2480 78 187 2 373 3552 2480 1156 402 3 4142 335 2480 1280 43 4 44 3540 2480 152 704 5 5817 3012 2480 178 73 6 6808 2573 2480 2t04 1333 7 734 2208 2480 2452 1810 8 888 14 2480 2778 2323 10171 1722 2480 3143 274 t0 11537 1518 2480 3565 3827 11 13267 t320 2480 4100 506t 12 21800 804 2480 6737 13664 The tax burdens implied by equation (1) exceed income in the regressive case for the three poorest households. For these households, a tax burden equal to 5 percent of benchmark income is assumed. Since a distortionary tax system is being eliminated in these simulations, substantial aggregate welfare gains result. These are measured as sums of equi- valent variations, 6 and are reported for each case, with G both remaining constant and being allowed to vary to satisfy the Samuelson provision rule. A prominent feature of the results is the relatively small incremental change to aggregate welfare from adjusting the level at which public goods are provided to move to optimal levels along with the tax replacement. The incremental welfare change never exceeds 12 percent of the total welfare variation, and for most model parameterizations and tax changes is less than 1 percent. In this model, at least, the incremental welfare variation is insignificant unless the elasticity of substitution between private and public goods is low. 6 The equivalent variation is given by C(Ui, Po, Go ) - C(Uo, Po, Go ) where C is the consumer's cost function corresponding to his utility function, U i is the level of utility in the z ~h social state (i = 0,1); Po is the initial price vector, and G-o the initial provision level of G.
Public Good Provision Rules and Income Distribution 31 ~ 0 0 ~ ~ 0 0 ~ C~ n ~ 0 0 0 0 0 0 0 ~::~ > 0 ~ ~2 ~: b ~ e.0 m ~ r-- D- o M < O 2 o ~ s O r-- ~" e,i t/3 > (D e~._> O ~_~ b ~ e4 [.-
32 J. Piggott and J. Whalley Secondly, the value of the change in G required to satisfy the Samue!son condition after redistribution varies significantly with the elasticity of substitution and attribution rule assumed. The welfare gain from adjusting G varies with the size of the required change for given paramenter values, but decreases as the elasticity of substitution rises. This is because the greater is the substitutability between private and public goods in consumption, the less costly is the non-optimality in the level of G to the consumer. 4 Conclusion This paper reports numerical simulation results on the relationship between the income distribution and optimal public good supply in a numerical general equilibrium model with many consumers. Previous work has focussed on establishing theoretical conditions for the independence of income distribution and optimal public goods provision. Here we use a numerical general equilibrium model based on Australian data to investigate how strong these interactions are empirically. Results suggest that the optimal level of public good provision is not greatly affected by changes in the income distribution unless the redistribution is dramatic. Of equal importance for policymakers, our results also suggest that the incremental welfare effect from allowing the level of public good supply to vary to meet changed optimality conditions when taxes change is not great. This result is of some comfort to policymakers, who until now could only defend the separation of redistributive and allocative decisions by assuming very restrictive preferences, such as the Gorman (153) "polar" form. References Atkinson A B, Stem N H (174). "Pigou, Taxation and Public Goods". Review of Economic Studies, 41, January: 11-I28 Bergstrom T C, Comes R (183). "Independence of Allocative Efficiency from Distribution in the Theory of Public Goods". Econometrica, 51, 6:1753-1766 Bergstrom T C, Goodman R P (173). "Private Demands for Public Goods". American Economic Review, 63:280-26 Foley D (170). "Lindahl's Solution and the Core of an Economy with Public Goods". Econometrica, 38: 1, 66-72 Gorman W M (153). "Community Preference Fields". Econometrica, 21:63-80
Public Good Provision Rules and income Distribution 33 Mansur A, Whalley J (t84). "Numerical Specification of Applied General Equilibrium Models" in Scarf, H E, Shoven J (eds) Applied General Equilibrium Analysis. Cambridge University Press, New York Piggott J (183) "A Walrasian Model of the Australian Economy and Public Sector: Specification Procedures and Data Set Construction". Working Papers in Economic and Econometrics, No. 86, Australian National University Pommerehne W W, Schneider F S (178) "Fiscal Illusion, Political Institutions, and Local Public Spending"., Kyklos, 31:381408 Samuelson P A (154) "The Pure Theory of Public Expenditure". Review of Economics and Statistics, 36:387-38 Shoven J, Whalley J (184) "Applied General Equilibrium Models of Taxation and International Trade". Journal of Economic Literature, September: 1007-1051 St Hiiaire F, Whalley J (182) "Recent studies of Efficiency and Distributional Impacts of Taxes", in Thirsk W R, Whalley J (eds), Tax Policy Options in the 180s, Canadian Tax Foundation Whalley J, (188) "Lessons from General Equilibrium Models", in Aaron H J, Galper H and Pechman J A (eds), Uneasy Compromise: Problems of a Hybrid Income-Consumption Tax, Brookings Institution, Washington D.C., 15-58