Environmental taxes with heterogeneous consumers: an application to energy consumption in France Helmuth Cremer IDEI and GREMAQ, University of Toulouse and Institut universitaire de France Toulouse, France Firouz Gahvari University of Illinois at Urbana-Champaign Champaign, IL 61820, USA Norbert Ladoux IDEI and GREMAQ, University of Toulouse Toulouse, France March 2001 Abstract This paper constructs a model with four groups of households who have prefer- ences over labor supply, consumption of polluting (energy related) and non-polluting (non-energy) goods and emissions. It quantiÞes the model for the French economy and computes its optimal tax equilibria under ten different tax regimes. When preferences are such that polluting good taxes have only an externality-correcting role, we Þnd: (i) environmental taxes result in very modest redistribution from higher- to lower-ability persons; (ii) environmental taxation does not entail a “double dividend”; (iii) in the second-best, the optimal environmental levy is larger than the Pigouvian tax. Secondly, when preferences imply that polluting good taxes embed optimal tax objectives in addi- tion to the externality-correcting role, we show: (iv) polluting goods must be subsidized rather than taxed; (v) this subsidy turns into a tax as the elasticity of substitution between polluting and non-polluting goods increases, but that it continues to remain below the marginal social damage of emissions; (vi) levying a tax on the polluting good equal to its marginal social damage is welfare reducing in that it induces a redistribution from the poor to the rich. 1 Introduction A number of authors have recently studied the optimal tax design problem with exter- nalities, and the structure of environmental taxes, in light of modern optimal tax theory a ` la Mirrlees (1971). This theory allows for heterogeneity among individuals and jus- tiÞes the use of distortionary taxes on the basis of informational asymmetries between tax authorities and taxpayers. A hallmark of this literature is its inclusion of nonlin- ear tax instruments. [See, among others, Kaplow (1996), Mayeres and Proost (1997), Pirttil¨ and Tuomala (1997), and Cremer, Gahvari and Ladoux (1998).] These studies a are exclusively theoretical. The empirical studies of environmental taxes, on the other hand, have remained squarely in the Ramsey tradition. As such, they typically assume identical consumers and allow for linear tax instruments only; see, e.g., Bovenberg and Goulder (1996). Most recently, Mayeres and Proost (2001) have introduced consumer heterogeneity and distributional aims. However, that paper remains within the Ramsey tradition considering only linear tax instruments. The linearity assumption is problematic from a policy perspective. It may severely undermine the role that income taxation can play in offsetting the possible “regressive bias” of environmental taxes. Poterba (1991) estimates that the expenditure shares of such polluting goods as gasoline, fuel oil, natural gas and electricity decrease at all income deciles as income increases. This suggests that environmental taxes may entail undesired redistributive consequences. The question facing the policy makers is thus to determine how serious this problem is and how, i.e. through what tax instruments, it can best be offset. When income taxes are artiÞcially restricted to be linear, it is the linearity restriction that may be behind the apparent redistributive role that emerges for other non-environmental tax instruments. Implementing policies determined on the basis of such restrictions, may then harm rather than help income distribution. The purpose of this study is to examine the efficiency and redistributive power of polluting good taxes in different tax environments, paying particular attention to tax 1 systems that include nonlinear income taxes. This is important. The usefulness of environmental taxes must be evaluated in relation to other tax instruments that the government has at its disposal. Restricting income taxes to be linear, as is often done, has no basis in theoretic or policy grounds. The feasibility of a particular tax instru- ment is ultimately determined by the type of information that is available to the tax administration. To the extent that incomes are publicly observable, they can be taxed nonlinearly (and not just linearly). Consequently, there are no informational grounds for restricting income taxes to be linear. Moreover, as a policy matter, governments of almost all countries do employ graduated income tax schedules. These considerations call for an examination of environmental taxes in presence of nonlinear income taxes. For the purpose of comparisons, we also consider settings with linear income taxes as well as Þrst-best differential lump-sum taxes. A second important feature of our study is our explicit recognition of the two poten- tial roles of polluting good taxes: externality-correcting and optimal tax considerations. That taxation of polluting goods must not be based solely on environmental grounds is often ignored in the discussions of this issue. This is a serious omission that may result in misdirected policy recommendations. Whereas the negative externality properties of polluting goods call for their taxation, their “necessity” attribute calls for their subsi- dization. Whether polluting goods should be taxed or subsidized thus depends, in the absence of explicit emission taxes, on which effect dominates the other. Of course, the (non-environmental) efficiency costs of such taxes or subsidies also play a role here. The paper also attempts to partially Þll another gap in the literature, independently of environmental issues. This concerns numerical calculations of optimal general in- come tax schedules when household types are Þnite. With the notable exception of Saez (2000), we are unaware of any such studies which are calibrated for a “real” econ- omy. The current paper solves numerically a general income tax problem with two dimensions of heterogeneity. The model is calibrated for France. It enables us to ex- 2 amine the directions in which the incentive compatability constraints of various income groups bind. We model an economy consisting of four groups of individuals who differ in earning abilities and may differ in tastes as well.1 They have preferences over labor supply, two categories of consumer goods, “non-polluting” and “polluting”, and total level of emissions in the atmosphere (a negative consumption externality). Emissions result when people consume polluting goods. We identify the consumer types, specify which goods are polluting and which ones are not, and then derive the parameter values of the consumers’ utility functions (assumed to be nested CES in labor supply and goods, and in polluting and non-polluting goods). We carry out these tasks using a mix of calibration and estimation methods as dictated by the limitations of the data available. All the data come from the “Institut National de la Statistique et des Etudes Economiques” (INSEE), France. The four groups are identiÞed as “managerial staff”, “intermediate-salaried employ- ees”, “white-collar workers” and “blue-collar workers”. The data covers 117 consump- tion goods which we aggregate into: non-energy consumption representing non-polluting goods, and energy-related consumption representing polluting goods. The data enables us to determine the groups’ earning abilities, their labor supplies and their net-of- tax-wages. We estimate the values of the elasticities of substitution between labor supply and consumption goods, and between polluting and non-polluting goods, using annual data on labor supply and consumption of different goods (energy-related and non-energy) in France for the years 1970—97. We derive the other parameter values of the utility function (except for emissions) by calibrating the model for the French econ- omy. We base the calculation of the emissions parameter on the assumption that the social damage of a ton of carbon emissions is 850 French francs. This reßects the 1990 recommendation of a carbon tax of this magnitude by the “Groupe Interminist´riel sur e 1 Ideally, one would like to allow for more types. The limitations of the data does not allow this, however. 3 l’Effet de Serre”–a French Government Commission set up to undertake an economics study of the greenhouse effect. We specify ten different tax regimes and compute the consumption levels, labor supplies, and utilities of the four groups, as well as the supporting optimal taxes, un- der each. A system of uniform lump-sum taxes constitutes our “benchmark”. Three tax regimes are built around a linear income tax system. In one, no (differential) con- sumption taxes are levied. In the other two, the polluting good is taxed once at the “Pigouvian” rate and once optimally. Four tax regimes are formed around a general income tax schedule. In one, no (differential) commodity taxes accompany the income tax. The next two constrain the polluting good tax to be linear. The tax is set once at the Pigouvian rate and once optimally. Finally, we allow for a nonlinear tax on the polluting good. In all these cases, the non-polluting good serves as the numeraire and thus goes untaxed. The last two tax regimes allow for differential lump-sum taxes with and without a tax on the polluting good. All computations are performed twice; once under the assumption that different individual types have identical preferences and once that they have heterogeneous pref- erences. Under the Þrst assumption, the tax on the polluting good will have solely an externality-correcting role.2 Our preference speciÞcation implies that, without emis- sions, commodity taxes would be redundant in this case. Income taxation (whether general or linear) is all that is needed for optimal tax policy. The efficacy and the role of environmental taxes are thus understood best in this case. Under the second scenario, when individuals differ in taste, differential commodity taxes are useful instruments of tax policy. Consequently, the optimal tax on the polluting good will have two com- ponents: one for correcting the pollution and the other for conventional optimal tax objectives. The optimal tax computations help shed light on a host of policy issues. SpeciÞcally, 2 See Cremer et al. (1998). 4 we shall seek to provide answers to the following questions. (i) To what extent envi- ronmental taxes push the society’s utility frontier upwards? In particular, what is the power of environmental taxes in this respect relative to income tax instruments? (ii) Do the redistributive properties of environmental taxes depend on what income taxes are employed, namely, a linear income tax, a general income tax, or differential lump-sum taxes? (iii) Which groups gain and which lose as result of environmental taxes and to what extent (using a Utilitarian social welfare function)? (iv) Does levying environmen- tal taxes imply a “double dividend”? (v) What is the size of the optimal environmental tax relative to the Pigouvian tax in the second-best? Suppose polluting goods must be taxed for two reasons: externality-correcting and optimal tax considerations: (vi) Would optimal tax considerations alone call for their taxation or subsidization relative to non- polluting goods? (vii) Will externality-correcting-cum-optimal-tax-objectives call for a net tax or a net subsidy on the polluting goods? (viii) How is this affected by the elas- ticity of substitution between polluting and non-polluting goods? (ix) If environmental taxes are set at the Pigouvian rate, rather than optimally, will the society’s welfare necessarily improve? Finally, and independently from environmental issues, the paper sheds light on four questions relating to general income and consumption taxation. (x) What is the relationship between the marginal income tax rate and the elasticity of substitution between leisure and consumption goods? (xi) To what extent, relative to a linear tax system, a general income tax schedule enhances the society’s ability to achieve its optimal tax objectives? (xii) How prevalent is bunching in nonlinear income tax schedules, and will it be affected by the availability of consumption taxes? (xiii) How effective are nonlinear consumption taxes as instruments of optimal tax policy? 2 The model The economy consists of four groups of individuals who differ in earning abilities and may differ in tastes as well. Each person, regardless of his type, is endowed with one unit 5 of time. He has preferences over labor supply, L, and two categories of consumer goods: a “non-polluting” good x, a “polluting good” y, and total level of emissions E in the atmosphere. Emissions are created through the consumption of the polluting good. All consumer goods are produced by a linear technology subject to constant returns to scale in a competitive environment. The producer prices of consumer goods are normalized at one. All consumer types have nested CES preferences in goods and labor supply and in the two categories of consumer goods. They also have identical elasticities of substitu- tion between leisure and non-leisure goods, ρ, and between polluting and non-polluting goods, ω. Differences in tastes, if any, are captured by differences in other parameter values of the posited utility function, i.e. aj and bj .3 Assume further that emissions en- ter the utility function linearly. Denote an individual’s wage by w and his gross income by I = wL. The preferences for a person of type j can then be represented by I j fj = U (x, y, ; θ ) − φE, j = 1, 2, 3, 4, (1) wj where θj reßects the “taste parameter” and4 µ ¶ ρ I ρ−1 I ρ−1 2(ρ−1) U (x, y, j , θj ) = j bQ j ρ j + (1 − b )(1 − j ) ρ , (2) w w ³ ω−1 ω−1 ´ ω ω−1 Qj = aj x ω + (1 − aj )y ω . (3) Next, normalize the population size at one and denote the fraction of people of type j to total population by π j . Total level of emissions is then related to the consumption 3 We impose these restrictions because of the data limitations. Goulder et al. (1999) have used a similar structure for consumer preferences to examine the cost effectiveness of different environmental policies. However, because their model is one of identical consumers, they assume that aj and bj also do not vary across consumers. 4 The role of 1/2 in the exponent of (2) is to ensure that the speciÞed CES utility function is strictly concave. 6 of the polluting good according to 4 X E= πj yj , (4) j=1 Consumers choose their consumption bundles by maximizing (1)—(3) subject to their budget constraints. These may be nonlinear functions as we allow for the possibility that the income tax schedule is nonlinear. We will, however, for the purpose of uniformity in exposition, characterize the consumers’ choices, even when they face a nonlinear constraint, as the solution to an optimization problem in which each person faces a (type speciÞc) linearized budget constraint. To do this, introduce a “virtual income” into each type’s budget constraint. Denote the j-type’s marginal income tax rate by tj j and let wn = w j (1 − tj ). We can then write j’s budget constraint as µ j¶ j j j j I px + qy = M + wn , (5) wj where p and q are the consumer prices of x and y, and M j consists of the individual’s exogenous income plus the income adjustment term (virtual income) needed for lin- j j earizing the budget constraint. Note also that I j = wj Lj so that wn (I j /w j ) = wn Lj . The Þrst-order conditions for a j-type’s optimization problem are 1 − aj ¡ xj ¢ ω1 q j j = , (6) a y p ¡ Ij ¢1 j (1 − bj ) xj /(1 − wj ) ρ wn h i ω−ρ = . (7) 1−ω ρ(1−ω) j bj aj + (1 − aj )(xj /y j ) ω p a 3 Types, goods, and the data In order to compute the optimal tax rates, we have (i) to identify the consumer types, (ii) to specify which goods are polluting and which ones are not, and (iii) to estimate the parameter values of the consumers’ utility functions. We carry out these tasks using a mix of calibration and estimation methods as dictated by the limitations of the data 7 available to us. All the data come from the “Institut National de la Statistique et des Etudes Economiques” (INSEE), France. To identify the types, we use two data sources: “budget des familles” and “Enqute sur l’emploi”. The Þrst are consumption surveys conducted for eight different household types; they are available only for four different years. The second are surveys on em- ployment and wages also classiÞed by household types. These surveys are available on an annual basis starting with 1987. The most recent year for which both data sources are available is 1989. We use this year as the basis for our calibrations. Out of the eight categories, only four report any wage incomes. They are classiÞed as: “manage- rial staff”, “intermediate-salaried employees”, “white-collar workers” and “blue-collar workers”. They constitute the four types of individuals in our model. The data covers 117 consumption goods which we aggregate into: (i) non-energy consumption represent- ing non-polluting goods (x), and (ii) energy-related consumption representing polluting goods (y). The 1989 data also enables us to determine the types’ earning abilities; wj ’s. We can compute these from data provided on gross wage incomes (I j = wj Lj ) and labor supplies (Lj ) using the relationship w j = I j /Lj . Wage incomes for each of the four household types are reported in INSEE (1991b) on an annual basis for the year 1989. Labor supplies are reported on a weekly basis as “weekly working hours” (W W H) in INSEE (1989). Given that a typical individual in France works for 47 weeks in a year, his hourly wage is equal to I j /47W W H j . To translate this to a yearly Þgure, we multiply it by 7× 52 ×18 = 6552 where we have assumed that each person has a total endowment time of 18 hours per day (he must sleep for at least 6 hours). In short, w j is computed according to 6552I j wj = . 47W W H j j Additionally, we may compute, for each type, a “net-of-tax-wage” wn = (1 − tj )wj . This is done on the basis of marginal tax rate, tj , that type j faces. The marginal tax 8 Table 1. Data Summary:1989 (monetary Þgures in 100,000 French francs) (1) (2) (3) (4) Managerial Staff Intermediary Level White Collars Blue Collars π 15.41 % 24.77 % 20.00 % 39.82 % px 2.541108 1.742072 1.279786 1.283748 qy 0.155970 0.134835 0.098048 0.117815 pQ Q = px + qy 2.697078 1.876907 1.377834 1.401563 px/pQ Q 0.942171 0.928161 0.928839 0.915940 qy/pQ Q 0.057829 0.071839 0.071161 0.084060 L 0.296164 0.268980 0.257534 0.276849 w 7.254181 4.407053 3.004338 2.760310 t 0.288000 0.192000 0.144000 0.096000 wn 5.164977 3.560899 2.571714 2.495320 M 1.167396 0.919096 0.715530 0.710735 rates for 1989 are from the French official tax publications (Ministere de l’Economie et j des Finances, 1989). Note also that from the Þgures for wn and Lj , we can calculate the value of the j-type’s virtual income, M j , through equation (5). Table 1 provides a summary of the 1989 data. Next, we must estimate of the parameter values of the utility function. The limited (to only four years) time series data on consumption of different types preclude us from estimating the parameters ρ, ω, bj and aj directly from Þrst-order conditions (6)—(7). For this, we have to use another data source. This data, given in INSEE (1998), is annual but macro; i.e. aggregated over all household types. The data covers the years 1970— 1997 and is given both at 1980 constant prices as well as current prices. For estimation purposes, we thus proceed as if equations (6)—(7) apply to a “representative” household. This allows us to estimate ω and ρ from our aggregate data. Upon estimating ω and ρ, we calibrate aj and bj from the 1989 disaggregated data. The calibrations are performed once assuming these parameters differ across individuals (heterogeneous tastes) and once assuming they are the same (identical tastes). 9 3.1 Estimation of ω Logarithmic transformation of (6), for a representative individual, yields, x q ln = constant + ω ln . (8) y p Equation (8) serves as our estimating equation. Its OLS estimation yields (with the standard errors of the estimates in parentheses) q ln x = y - 2.2653 +0.2035 ln p , (9) (0.0193) (0.1072) R2 = 0.1218; DW = 0.1369. The coefficient of ln(q/p) in (9) is not statistically signiÞcant. However, the very low value of DW statistic in (9) indicates that there is a serious problem of autocorrelation among the residuals. To correct for this, we next consider the OLS estimation of equa- tion (8) with the lagged values of ln(x/y) and ln(q/p) also as regressors. Using OLS again, we obtain ln x = y - 0.0265 +0.2689 ln q p q −0.2236 ln( p )−1 +0.9927 ln( x )−1 y (10) (0.1788) (0.0914) (0.0899) (0.0792) R2 = 0.8869; DW = 1.8481. The DW statistic in (10) is close to 2 and the coefficient of ln(q/p) is statistically signiÞcant. Our estimate of ω, the elasticity of substitution between x (non-energy related goods) and y (energy related goods), is thus 0.2689.5 3.2 Estimation of ρ For the purpose of estimating ρ, we assume that individuals choose their optimal al- locations on the basis of a “two-stage” optimization process. Each person chooses Q, 5 We are not aware of any other econometric studies to estimate this parameter–at least not within the context of dividing goods between polluting and non-polluting and certainly not for the French data. Goulder et al. (1999), citing an earlier study by Cruz and Goulder (1992) depicting the US economy in 1990, use a value of 0.85 for this parameter. 10 interpreted as “aggregate expenditure on consumer goods”, and L to maximize (2) in the Þrst stage and then allocates Q between consumption of x and of y in the second stage. The Þrst-order condition for the second-stage problem is then given, as previ- ously, by equation (6). As to the Þrst-stage, one can write the budget constraint of a j-type as pQ Q = wn L + M j , j (11) where pQ is the “price” of Q. This yields the Þrst-order condition µ ¶1 j 1 − bj Qj ρ wn = . (12) bj 1 − Lj pQ Assuming a representative individual, logarithmic transformation of (12) yields, Q wn ln = constant + ρ ln . (13) 1−L pQ Equation (13) is estimated using data on L, wn , Q and pQ for years 1970-1997 from INSEE r´sultats (1998). As far as L is concerned, the reported data are on annual e working hours (AW H). With our previous assumption that the yearly endowment of time (normalized to be one) is 6552 hours, we thus calculate L as AW H/6552. Turning to wn , we have data on total number of wage earners (T W E), their wage incomes (T W I), and their wage income taxes (T W T ). This allows us to calculate wn on a yearly basis as (T W I − T W T )/T W E wn = . AW H/6552 Turning to Q, we have data on the consumption of all households and not just the wage earners as desired. To generate the latter series, we assume that wage earners’ share of total consumption during 1979—1997 has remained the same as in the year 1989 (for which we have the Þgures). Finally, we estimate pQ from 1979—1997 data on consumptions at current and at constant prices. 11 Table 2. Calibrations: heterogeneous tastes (ρ = 0.7927, ω = 0.2689) (1) (2) (3) (4) Managerial Staff Intermediary Level White Collars Blue Collars a 0.999988 0.999971 0.999972 0.999945 b 0.501877 0.467265 0.446461 0.466293 The OLS estimation of (13) yields (with the standard errors of the estimates in parentheses) Q ln 1−L = 3.4706 +0.7927 ln wQ p n (14) (0.0189) (0.0815) R2 = 0.7843; DW = 1.0901. The low value of the DW statistic indicates that the residuals are serially correlated. We nevertheless do not attempt to correct for this. This is because our 0.7927 estimate of ρ is very much within the range of its estimates in the literature. Stern (1976), in his classic study of an optimal linear income tax system, suggests a value of 0.4 on the basis of estimates for married males in the US. Wales and Woodland (1979) give the estimates of 0.83 and 0.91 (depending on the estimation method) based on PSID data. Goulder et al. (1999) use a value of 0.96. More recently, Bourguignon (1999) observes that the existing estimates for the wage elasticity of labor supply are anywhere between 0.1 and 0.5. These values can be shown to correspond to a range of estimates for ρ equal to 0.58 to 1.06; see the Appendix. 3.3 Calibration of aj and bj Given the estimates of δ and ρ, one can then compute aj and bj , for j = 1, 2, 3, 4, on the basis of 1989 INSEE data. We calculate the values of aj ’s and bj ’s numerically as the solution to the non-linear system of equations (6)—(7) using GAUSS. They are presented in Table 2 (and in Table 6 in the Appendix for different values ρ and ω). Note that 12 Table 3. Calibrations: idetical tastes (ρ = 0.7927, ω = 0.2689) a 0.999970 b 0.468704 whereas the values of bj depend on both ρ and ω, the values of aj are independent of ρ (but depend on ω). Finally, we recompute the values of aj and bj on the assumption that they do not differ across types; i.e. all individual types have identical tastes. This is again done on the basis of equations (6)—(7) using the data aggregated over the four types and weighted in proportion to their size. These numbers are reported in Table 3 (and in Table 7 in the Appendix for different values ρ and ω). 3.4 Calulation of φ The starting point for calculation of φ, the coefficient of emissions in the utility function, is a 1990 recommendation of the “Groupe Interminist´riel sur l’effet de Serre”. This e was a French Government Commission set up to undertake an economics study of the greenhouse effect. The recommendation called for a carbon tax of 850 French Francs per ton of emitted carbon. We assume that 850 French francs measures the social damage of a ton of carbon emissions. Next, we calculate the carbon content of a unit of the polluting good (energy-related consumption goods).6 This provides an estimate of the social damage of a unit of emissions, i.e. φ/µ where µ is the shadow cost of public funds (the Lagrange multiplier associated with the government’s budget constraint). To arrive at an estimate of φ, we calculate µ by solving our optimal tax problem in the Þrst best without the externality. This gives us a value for φ for each set of parameter values for ρ, ω, aj and bj , j = 1, 2, 3, 4. 6 We do this based on the carbon content of oil, coal, natural gas and electricity, and by calculating their share in energy-related consumption goods. 13 4 Tax policies The usefulness of environmental taxes must be evaluated in relation to other tax instru- ments that the government has at its disposal. Of particular interest is the structure of the accompanying income taxes, e.g., linear or nonlinear. The feasibility of a particular tax instrument is ultimately determined by the type of information that is available to the tax administration. Public observability of individual incomes typically allows the government to impose nonlinear income taxes. Nevertheless, the income tax liter- ature has traditionally paid a great deal of attention to the study of the linear income taxation. We will consider both income tax instruments. 4.1 The linear income tax The procedure for determining the optimal tax policy when the income tax is linear, is to determine the values of the tax parameters that maximize a social welfare function deÞned in terms of the individuals’ indirect utility functions. For this purpose, we Þrst determine the j-type’s demand functions for nonpolluting and polluting goods, and his labor supply function, from equations (5)—(7). We have xj = x(p, q, wn , M j ; θ j ); yj = y(p, q, wn , M j ; θj ); Lj = L(p, q, wn , M j ; θj ). j j j (15) Note that the demand and supply functions for different consumer types will be of j different functional forms, when written as functions of p, q, wn and M j , whenever aj and bj differ across types. Finally, using (15), we can derive the j-type’s indirect utility j function: v(p, q, wn , M j ; θj ). Turning to social welfare, we adopt a utilitarian outlook.7 The government’s problem can be speciÞed as one of choosing its tax instruments in order to maximize 4 X 4 X j π v(p, q, wn , M j ; θj ) − j φ π j y(p, q, wn , M j ; θj ), j (16) j=1 j=1 7 This may easily be generalized by considering an “iso-elastic” function with a varying “inequality aversion index”. 14 subject to its revenue constraint 4 X £ ¤ ¯ π j (p − 1)xj + (q − 1)y j + tw j Lj − T ≥ R, (17) j=1 where t is the tax rate and T is the lump-sum tax element of the linear income tax ¯ schedule, and R is the government’s external revenue requirement. Note also that in the absence of any other exogenous income, T = −M j . The full array of tax instruments in the government’s optimization problem are: p − 1, q − 1, t and T . However, because the demand functions for goods, and the labor j supply function, are all homogeneous of degree zero in p, q, wn and M j , we can, without any loss of generality, set one of the commodity tax rates at zero (one of the consumer prices at one). We will choose the nonpolluting good to be the one whose tax rate is set at zero. That is, we shall set p = 1 everywhere. Different tax policies are then identiÞed through imposition of different constraints on these instruments and thus on the problem (16)—(17). We consider four tax policies. The Þrst is one of a uniform lump sum tax (U LST ). This serves as our “benchmark” for evaluating other tax regimes. To derive the equi- librium under this policy, we have to impose the additional constraints that q = 1 and t = 0 on problem (16)—(17). Consequently, T will be the only available tax instrument. Next, we consider the possibility of levying a linear income tax absent any commodity taxes (LIT ACT ). To Þnd the equilibrium under LIT ACT , we impose the constraint q = 1 on problem (16)—(17). The feasible tax instruments are now only t and T . Third, we consider a linear income tax accompanied by a tax on the polluting good equal to its marginal damage (LINP DT ). This requires the constraints q − 1 = φ/µ, where µ is the Lagrangian multiplier associated with the government’s budget constraint (17).8 Again, the optimizing tax instruments are t and T . Finally, we consider a tax regime consisting 8 This deÞnition of the “Pigouvian tax” is that of Cremer et al. (1998). Bovenberg and van der Ploeg (1994), and others, deÞne this term differently. We also calculate the value of the Pigouvian tax based on their deÞnition. This is discussed in more detail at the end of Section 5. 15 of a linear income tax in which the polluting good is taxed optimally (LIN ODT ). No additional constraints need be imposed on problem (16)—(17); the feasible tax instru- ments are t, T and q. 4.2 The general income tax Next, we consider four other tax regimes formed around a general income tax (where we continue with our normalization rule of setting the tax on the nonpolluting good to be zero; i.e. p = 1). The main complication that arises when one allows for a general income tax is that (in contrast to a linear income tax) one can no longer count on the individuals’ incentive compatibility constraints to be satisÞed automatically. To ensure that the desired equilibrium satisÞes these constraints, one has to impose them on the government’s optimization problem directly. We employ two different procedures depending on the feasibility of nonlinear com- modity tax instruments. 4.2.1 Linear commodity taxes j Denote M j + wn Lj ≡ cj . From equations (5) and (6), determine the demand functions for xj and yj as xj = x(p, q, cj ; θ j ) and y j = y (p, q, cj ; θj ). Next, derive cj and I j as the ˆ ˆ solution to the following problem for the government. Maximize 4 X X 4 ¡ Ij ¢ π U x(p, q, c ; θ ), y (p, q, c ; θ ), j ; θj − φ j ˆ j j ˆ j j π j y (p, q, cj ; θj ), ˆ (18) w j=1 j=1 with respect to cj and I j , subject to the resource constraint 4 X ¡ ¢ ¯ π j I j − cj + (p − 1)xj + (q − 1)y j ≥ R, (19) j=1 the incentive compatibility constraints, for j 6= k; j, k = 1, 2, 3, 4, ¡ Ij ¢ ¡ Ik ¢ U x(p, q, cj ; θj ), y (p, q, cj ; θj ), j ; θ j ≥ U x(p, q, ck ; θj ), y (p, q, ck ; θ j ), j ; θj , ˆ ˆ ˆ ˆ (20) w w 16 and an additional constraint that q = p = 1. Having determined cj and I j , and thus xj and y j , we can then determine tj , the j-type’s marginal income tax rate required to implement these allocations, from (7). Moreove, if implementation is to be carried out through a menue of linear income tax schedules (possibly truncated), we can calculate the required lump-sum tax to be levied on the j-type, T j (= −M j ), from (5). The Þrst case we examine using this procedure, is when no commodity taxes accom- pany the general income tax (GIT ACT ). This is achieved by imposing the constraint q = 1 on problem (18)—(20). The next two tax regimes we examine, complement a general income tax with a tax on the polluting good. One sets this tax at a Pigouvian level (GIT P DT ) and the other chooses it optimally (GIT LDT ). They are found by following exactly the same procedure as above except that in the former q is set equal to 1 + φ/µ (instead of 1), and in the latter q is chosen optimally. 4.2.2 Nonlinear commodity taxes The tax policies considered thus far, have stipulated a tax rate on the polluting good (including zero) which must be the same for all individuals regardless of their type. We next investigate the signiÞcance of differentiating this tax amongst the individual types (i.e. levying a nonlinear tax on the polluting good). Whether or not the government can impose nonlinear taxes (on the polluting good or any other good) would of course depend on the structure of public information in the economy. If consumption levels are known at an individual level (i.e. who buys how much), nonlinear commodity taxes are feasible. On the other hand, if the available public information is only on aggregate sales (anonymous transactions), we can only levy linear commodity taxes. While the latter possibility is more realistic for the majority of goods, there exist real examples where individual consumption levels of a polluting good are observable (e.g. electricity). The problem of nonlinear taxation of such goods is thus a relevant policy consideration. Consequently, we will also study a tax regime in which polluting goods may be taxed nonlinearly (GIT N DT ). 17 The availability of both a general income and a general commodity tax allows us to derive the optimal allocations directly. This requires Þnding the solution to the following government problem. Maximize 4 X X4 Ij j π j U (xj , yj , j ;θ ) −φ πj yj , (21) w j=1 j=1 with respect to xj , y j and I j , subject to the resource constraint 4 X ¯ π j (I j − xj − y j ) ≥ R, (22) j=1 and the self-selection constraints Ij j Ik U (xj , y j , ; θ ) ≥ U (xk , y k , j ; θ j ), j 6= k; j, k = 1, 2, 3, 4. (23) wj w Having determined the optimal allocations (xj , y j , I j ), one can calculate the (marginal) tax rate on the polluting good, for the j-type, from equation (6). This is given by (1/aj − 1)(xj /y j )1/ω − 1. Then, T j and tj are determined from equations (5) and (7). 4.3 First best and welfare For comparison purposes, we will also calculate two tax regimes in which differential lump-sum taxation is feasible. They differ in their tax treatment of the polluting good. In one, the polluting good goes tax free (F BADT ). This is found by dropping the self- selection constraints (20) in problem (18)—(20) and adding the constraint that p = q = 1. In the other tax regime, the polluting good is taxed optimally. This is of course, the Þrst-best allocations (F B). This is attained by dropping the self-selection constraints (23) in problem (21)—(23). Finally, to conduct welfare comparisons, we report equivalent variation, EV , of a change in policy from the “benchmark allocation” B to one of the tax “alternatives” discussed. Thus, for each type j = 1, 2, 3, 4, we calculate an EV j from the following relationship v(pB , qB , wn,B , TB + EVij ) = v(pi , qi , wn,i , Tij ), j j j 18 where subscript B denotes the benchmark (U LST ) and subscript i refers to one of the tax options: LIT ACT , LIN P DT , LIN ODT , GIT ACT , GIT P DT , GIT LDT , GIT N DT , F BADT , and F B. 5 Optimal taxes with identical tastes In this section, we compute the solutions under our various tax schemes assuming that the four types have identical tastes. The efficacy and the role of environmental taxes are understood best in this case. The reason for this is that the tax on the polluting good here will have solely an externality-correcting role. Our speciÞcation of preferences in (2)—(3) implies that without emissions, commodity taxes are redundant. Income taxation (whether general or linear) is all that is needed for optimal tax policy. [See Atkinson and Stiglitz (1976) and Deaton (1979)]. The results are reported in Table 4. To examine the robustness of our results, we additionally calculate the tax solutions for a number of other values of ρ and ω around their estimated values. These solutions (corresponding to ρ = 0.5, 0.99 and ω = 0.1, 0.5, 0.99) are reported in Tables 8—12 in the Appendix. The general pattern of the results and the lessons that emerge do not appear to depend on the values of ρ and ω. For the sake of brevity, we limit our discussions below to the case where ρ = .79 and ω = 0.2689. However, when relevant, we will also mention the changes that occur as either ρ or ω changes. We begin with tax policies that do not include environmental taxes thus leaving pollution “uncorrected”. This allows us to isolate the impact of environmental taxes when we introduce them. First is the benchmark case of a uniform tax on all types (U LST ). This requires that everyone pays a tax equal to 36,764 French francs to pay for government expenditures. At the other extreme, we have the case with differential lump-sum taxes but no environmental taxes (F BADT ). This is characterized by a lump-sum tax of 494,986 and 128,684 francs on types 1 and 2 and a positive grant of 90,163 and 134,042 francs on types 3 and 4. It is clear that 19 the uniform taxation of all types leaves us way off the “ideal” redistributive tax. Next, consider a linear income tax. This gets the economy closer to the F BADT by increasing the tax payments of types 1 and 2 to 57,751 and 40,214 while reducing the taxes of types 3 and 4 to 31,027 and 29,375 francs. [A j-type household’s total tax payments, from all sources, is denoted by T P j in the Tables.] The tax schedule that achieves this consists of a rate of 17.6% coupled with a lump-sum tax of 7,312 francs. Note that the lump-sum element here is a tax and not a positive grant. This suggests that our linear tax system is in fact regressive.9 A general income tax (GIT ACT ) further improves the redistributive power of the tax system. The tax payments of types 1 and 2 are now increased to 106,698 and 43,429 while the taxes of types 3 and 4 are reduced to 16,984 francs each. Types 3 and 4 end up paying the same taxes because the tax equilibrium here calls for pooling these two types together giving them an identical income and consumption bundle (but of course different labor supply levels). This is implemented by marginal tax rates of 0.6, 21.4, 22.8 and 9.1 percent on types 1 to 4. If we use a menu of linear tax schedules for implementation, these marginal income tax rates will have to be accompanied by lump- sum taxes of 104,818 , 3,626, -11,919 and 4,874 francs. Note also that the highest-wage person’s marginal income tax rate is positive (though very small). This is in keeping with Cremer et al.’s (1998) result who showed that unless the polluting good is taxed optimally, the allocation of the “top” individuals must be distorted. We now turn to the tax systems that include environmental taxes. Begin with the implications of introducing an environmental tax when we have differential lump-sum taxes. The F B equilibrium is supported by a lump-sum tax of 494,495 and 127,981 francs on types 1 and 2 and a positive grant of 91,087 and 135,022 francs on types 3 and 4. Additionally, there will be a 10% tax on the polluting good. This latter tax raises 493, 704, 923 and 979 francs from types 1 to 4. Note that the lower wage persons will 9 The average tax rates for types 1 to 4 are: 20.11%, 21.47%, 22.98% and 23.38%. 20 Insert Table 4 here. 21 pay higher environmental taxes. This reßects the fact that at the Þrst-best utilitarian solution, lower wage-earners will consume more of all goods, including the polluting good, than the higher wage-earners and thus pay higher taxes too.10 The introduction of environmental taxes in a Þrst-best environment results in a redistribution of total tax payments by different households. A comparison of F BADT and F B reveals that total tax payments of types 1 and 2 are increased by two and one francs while the tax payments of types 3 and 4 are reduced by one franc each. These changes are very modest indeed. The welfare implications of the environmental tax can be determined by considering the changes in the EV terms for different types (as we move from F BADT to F B). Types 1—4 gain 29, 18, 11 and 9 francs. Obviously, we have a Pareto improving environmental tax, albeit, a modest one.11 Next, consider introducing an environmental tax into second-best settings. The in- teresting point to note now is that our preference speciÞcation implies that LIT P DT = LIT ODT and GIT P DT = GIT LDT = GIT N DT . That is, the optimal tax on the polluting good is equal to the Pigouvian tax (and is linear). Start with the case when the income tax instrument is linear. The polluting good should be taxed at a rate of 9.4%. This allows the income tax rate to be cut from 17.6% to 17.1%. On the other hand, the lump-sum tax element of the linear income tax is increased from 7,312 to 7,365 francs. The introduction of the environmental tax thus allows the government to cut other distortionary taxes in the economy. The introduction of an environmental tax into a linear income tax system, has very modest redistributive implications. Types 1 and 2 pay eleven and one francs less and types 3 and 4 pay three and four francs more in total taxes. In welfare terms, EV 10 This result corresponds to the general property that with a utilitarian, or any concave, social welfare function, Þrst-best allocations require the higher wage persons to work more but to receive no more (in after tax pay) than the lower-wage persons. See Stiglitz (1987). 11 The introduction of an additional instrument (which is not redundant) pushes the utility frontier upwards. However, when the optimum is characterized by the maximum of a particular social welfare function (utilitarian or otherwise), this does not necessarily imply a Pareto improvement. This point should be borne in mind later on also when we discuss other tax policies that are not Pareto improving. 22 Þgures in going from LIT ACT to LIT ODT indicate gains of 27, 15, 7 and 7 francs for types 1 to 4. While, per household, these are clearly modest gains, it is interesting to note that the introduction of the environmental tax makes all household types better off. The tax is Pareto improving. Now consider the introduction of an environmental tax into a general income tax framework. The optimal environmental tax is 9.7%. As with the linear income tax case, introduction of an environmental tax allows “other” distortionary taxes (now consisting of the marginal income tax rates on all types) to be cut. We observe that the marginal tax rate of types 2 to 4 are reduced from to 21.4% to 21.0%, from 22.8% to 22.4% and from 9.1% to 8.6%. Additionally, the marginal tax rate of type 1 goes to zero. This is as expected, and reßects the famous no distortion at the top result. Note also the tax equilibrium continues to be one of pooling types 3 and 4. Turning to redistributive implications, the changes in total tax payments continue to be very modest. Payments of types 1 and 2 are reduced by ten and one francs while those of types 3 and 4 are increased by three francs each. To gauge the welfare implications of these changes, consider what happens to the various EV terms as we go from GIT ACT to GIT P DT . They indicate gains of 26, 15, 9 and 8 francs for types 1 to 4. The environmental tax is thus Pareto improving in this setting as well. We also note that the gains for each household type is very similar to the gains under a linear income tax. It appears that the welfare gains due to environmental taxes do not depend on whether the government employs a linear or a general income tax to achieve its optimal tax objectives. Examining Tables 8—12 indicate that the nature of our results is robust to the vari- ations in the values of ρ and ω. Two points are worth mentioning. First, as the value of ρ increases, the optimal marginal income tax rate decreases (with linear as well as gen- eral income tax schedules). This is intuitive enough. A higher elasticity of substitution between leisure and goods imply a higher efficiency cost of taxation. This in turn calls 23 for the optimal tax rate to be lower. Note also that when we have a linear tax schedule, the lump-sum tax element also increases. The tax remains positive even for the lowest value of ρ (=0.5) thus implying a regressive income tax system. Second, the optimal environmental tax is not sensitive to the variations in ω. This is not surprising. With identical tastes, the role of the environmental tax is solely externality correcting. This role is insensitive to the variations in ω. We close this section by making two Þnal observations. The Þrst observation con- cerns the recent controversy over the “double-dividend” hypothesis. According to this hypothesis, environmental taxes are more welfare enhancing in second-best environ- ments. The “argument” is that there will be two sources of beneÞts in the second-best. One is, as in the Þrst-best, the welfare improvement due to the imposition of Pigouvian taxes. The second, and the purportedly “additional” source, is due to the reduction in the existing distortionary taxes (that the “new revenues” make possible). There are a number of different interpretations of this claim; see Goulder (1995) for a survey. One simple and direct way to examine the validity of the double dividend hypothesis is by comparing the welfare gains that we have computed for the Þrst- and the second-best settings. The gains in going from F BADT to F B were 29, 18, 11 and 9 francs for types 1—4. The corresponding gains were 26, 15, 9 and 8 francs in going from GIT ACT to GIT P DT and 27, 15, 7 and 7 francs in going from LIT ACT to LIT ODT . Each type thus gains more when the environmental tax is introduced in the Þrst-best than when it is introduced in either of the two second-best settings considered. Evidently, not only is there no double dividend, there is even less of a dividend in the second-best! Note also that this result is robust in that it holds in all the tax solutions derived under the different values of ρ and ω we have considered. Our second observation relates to the concept of “the Pigouvian tax”. Our discus- sion of the Pigouvian tax and its equality to the optimal environmental tax, given our speciÞcation of preferences, is based on Cremer et al.’s (1998) deÞnition of the Pigouvian 24 tax. According to this deÞnition, a tax is called Pigouvian if it is equal to the marginal social damage of pollution, as measured by φ/µ. Bovenberg and van der Ploeg (1994), Bovenberg and de Mooij (1994), Kaplow (1996), Fullerton (1997) and others deÞne the Pigouvian tax differently. Their deÞnition is based on the Samuelson’s rule for optimal provision of public goods. They term a tax Pigouvian if it is equal to the sum of the private dollar costs of the environmental damage per unit of the polluting good across P j j j all households. In our notation, their Pigouvian tax is j π φ/α , where α is the j-type’s private marginal utility of income. To see how the optimal environmental tax compares with this conception of the Pigouvian tax, we have also calculated the values P for this alternative deÞnition. This is shown in our tables by τ ≡ φ j π j /αj . Note that whereas the optimal environmental tax is calculated to be 9.4%, under LIT ACT , τ is equal to 9.1%. That is, the optimal environmental tax is larger than the Pigouvian tax. This Þnding may appear surprising in light of Bovenberg and de Mooij’s (1994) result that the optimal environmental tax must be lower than the Pigouvian tax. It does not contradict their claim, however. The point is that their result holds if preferences satisfy certain separability assumptions and that labor supply functions are upward sloping. Our preference speciÞcation does satisfy their separability assumptions, but it does not guarantee an upward-sloping labor supply function. Indeed, under LIT ACT , the tax equilibrium calls for all types to be on the backward-bending part of their labor supply functions. This in turn results in a marginal cost of public funds which is less than one causing the optimal environmental tax to be larger than τ . The importance of our Þnding is that it occurs under an empirically relevant optimal tax scheme. It thus indicates that, as a policy prescription, one may not be able to rely on Bovenberg and de Mooij’s result. This result is also robust and holds in all the tax solutions derived under the different values of ρ and ω we have considered. Note also that the optimal environmental tax continues to be larger than τ under a general income tax (9.7% versus 25 9.4%).12 6 Optimal taxes with heterogeneous tastes We now turn to the case when individuals have heterogeneous tastes. Under this circum- stance, differential commodity taxes are useful instruments of tax policy. Consequently, the optimal tax on the polluting good will have two components: one for correcting the pollution and the other for conventional optimal tax objectives. Failure to understand this point may result in wrong policy recommendations. The results are reported in Table 5 (for ρ = 0.7927 and ω = 0.2689), as well as Tables 13—17 in the Appendix (corresponding to ρ = 0.5, 0.99 and ω = 0.1, 0.5, 0.99). We again limit our discussion to the case reported in Table 5 while noting any changes in the results in the footnotes. We start with tax policies that do not include a tax on the polluting good. These tax policies are thus suboptimal in two ways. They do not use commodity taxes that can enhance welfare, and they also leave pollution “uncorrected”. These tax structures appear to be quantitatively very much like those that resulted under the assumption of identical tastes. The benchmark case of a uniform tax on all types now requires everyone to pay a tax equal to 36,584 French francs. The case with differential lump-sum taxes (F BADT ) requires a lump-sum tax of 481,445 and 128,438 francs on types 1 and 2 and a positive grant of 85,462 and 131,461 francs on types 3 and 4. The linear income tax schedule consists of a marginal tax rate of 18.3% and a lump-sum tax of 5,724 francs. Again, this is a regressive tax.13 It raises 61485, 39,834, 29,385 and 28,539 francs from types 1 to 4. Finally, the general income tax is supported with marginal income tax rates of 0.4, 22.6, 17.8 and 11.3 percent on types 1 to 4. It raises 122,813, 40,691, 12665 and 12,665 francs from them. Types 3 and 4 are again pooled ending up with the same before- and after-tax incomes in equilibrium.14 12 Cremer et al. (2000) have shown that with a general income tax, the optimal environmental tax may be larger, equal to, or smaller than τ . 13 As ρ decreases, the lump-sum tax element turns into a rebate, making the tax progressive. 14 Their consumption bundles though differ. Identical incomes do not imply identical consumption 26 Now consider taxing the polluting good starting with the case when we have dif- ferential lump-sum taxes. The optimal Pigouvian tax is again 10%. To study the redistributive implications in going from F BADT to F B, we examine the changes in total tax payments and welfare of different groups. Total tax payments of types 1 and 2 are increased by 22 and 6 francs while the tax payments of types 3 and 4 are reduced by 2 and 11. Considering the EV terms, we note that the environmental tax is Pareto improving resulting in gains of 9, 14, 14 and 19 for types 1—4. We next turn to second-best settings, beginning with the linear income tax. Out- comes under LIT P DT and LIT ODT are now markedly different. If we tax the polluting good simply for “corrective” purposes (LIT P DT ), we will tax it by 9.4% (equal to mar- ginal social damage of pollution). The optimal tax on this good, on the other hand, is a subsidy of 4.9%. It is apparent that optimal tax objectives calls for a subsidy on pollut- ing goods (relative to non-polluting goods). Even when adjusted because of pollution consideration, we still want to subsidize these goods.15 Clearly, tax recommendations based on Pigouvian considerations alone, can be very misleading. Note that LIT P DT is supported by a marginal tax rate of 17.9% and a lump-sum tax of 5,759 francs, while LIT ODT is supported by a marginal tax rate of 18.6% and a lump-sum tax of 5,704 francs. That is, the marginal income tax rate decreases when we go to LIT P DT , but it increases if we go to LIT ODT . Taxing the polluting good optimally in this case increases the other distortionary tax in the economy rather than reducing it!16 Taxation of the polluting good, also implies redistribution among types. In going from LIT ACT to LIT P DT , we reduce the total tax payment of type 1 by 253 francs while increasing the tax payments of types 2—4 by 6, zero and 94 francs. On the other hand, in going from LIT ACT to LIT ODT , we increase the total tax payment of type 1 patterns when tastes differ. 15 This result does not hold at high values of ω. 16 This result does not hold at high values of ω. 27 Insert Table 5 here. 28 by 138 francs while reducing the tax payments of types 2—4 by 2, zero and 52 francs. This suggests that a Pigouvian tax on the polluting good results in an income redistribution from less well-off to more well-off. An optimal tax on the polluting good, on the other hand, brings about the desired redistribution from the rich to the poor. The redistributive implications of LIT P DT and LIT ODT are best highlighted by contrasting the welfare implications of the two tax schemes. The EV values indicate that, unlike the homogeneous taste case, the introduction of a tax on the polluting good (whether Pigouvian or optimal) is no longer Pareto improving. More interestingly, the two policies will have opposite redistributive implications. Levying a Pigouvian tax beneÞts the wealthy and hurts the poor. On the other hand, if the polluting good is taxed optimally, the rich would lose and the poor would gain.17 According to the EV Þgures, in moving from LIT ACT to LIT P DT , types 1—3 gain 408, 6 and 11 francs while type 4 (the poor) loses 94 francs. In moving from LIT ACT to LIT ODT , types 1—3 lose 171, 12 and 12 francs while type 4 gains 45 francs. That “environmental taxes” result in a higher degree of redistribution when tastes are heterogeneous should not be surprising. These taxes now reßect an optimal tax objective in addition to their externality correcting objective. Next consider the introduction of an environmental tax into a general income tax framework. There are three different tax policies: GIT P DT, GIT LDT and GIT N DT . In case of GIT P DT , we should levy a tax of 9.7% on the polluting good. This allows a cut in the marginal income tax rates for all types. Households of types 1—4 now face marginal income tax rates equal to zero (down from 0.4), 22.2 (down from 22.6), 17.3 (down from 17.8) and 10.8 (down from 11.3) percent. The total tax payments (income and consumption) for types 1 and 2 are cut by 305 and 17 francs while types 3 and 4 see their taxes raised by 15 and 120 francs. Clearly, levying a tax on the polluting good equal to its marginal social damage, induces redistribution from the poor to the rich. 17 This result does not hold at high values of ω. 29 This is precisely what we observed in the case with the linear income tax. In welfare terms, the EV values indicate that in going from GIT ACT to GIT P DT , types 1—3 gain by 320, 27 and 3 francs while type 4 (the poor) loses by 113 francs. Turning to GIT LDT , the optimal tax on the polluting good is a subsidy of 5.1%.18 This is in line with what we observed in the linear income tax case. This subsidy results in an increase in the marginal income tax rates for all types. Types 1—4 now face marginal income tax rates equal to 0.7 (up from 0.4), 22.8 (up from 22.6), 18.1 (up from 17.8) and 11.6 (up from 11.3) percent. GIT LDT also results in a redistribution from the rich to the poor in marked contrast to GIT P DT .19 In going to GIT LDT (from GIT ACT ), types 1 and 2 pay 169 and 8 francs more in total taxes, while types 3 and 4 pay 9 and 66 francs less. In terms of welfare, the EV values indicate that the move entails losses of 190, 23 and 8 francs for types 1—3. On the other hand, type 4 gains by 55 francs. Finally, consider going from GIT ACT to GIT NDT so that the government levies nonlinear taxes on the polluting good. The tax will be 9.3% for type 1. This is equal to the marginal social damage of emissions so that the tax is wholly Pigouvian. This should not be surprising. With the ability to levy nonlinear commodity taxes, the no distortion at the top result applies. That is, in the absence of pollution, type 1 must face a zero marginal tax rate on income and on consumption goods. In the presence of pollution, the top individuals must face only a Pigouvian tax just for the purpose of correcting the pollution. Turning to types 2—4, type 2 will now face a subsidy of 14.3%, type 3 a tax of 11.5% and type 4 a subsidy of 4.0%. As far as marginal income tax rates are concerned, type 1 faces a zero rate (down from 0.4), type 2 a tax rate of 23.2 (up from 22.6), type 3 a tax rate of 17.4 (down from 17.8) and type 4 a tax rate of 11.5 (up from 11.3) percent. Note that types 1 and 3 face a lower rate, and types 2 and 18 This will turn to a tax at high values of ω. 19 At high values of ω, GIT LDT , as with GIT P DT , also implies a redistribution from the poor to the rich! 30 4 a higher rate. Imposition of optimal nonlinear taxes on the polluting good does not induce a universal cut in other distortionary taxes. Like GIT LDT , and in contrast to GIT P DT , levying an optimal nonlinear tax on the polluting good results in a redistribution from the rich to the poor. The extent of the redistribution, however, is much more signiÞcant. In going from GIT ACT to GIT N DT , type 1 pays 1,394 francs more in total taxes, while types 2—4, pay 41, 103 and 463 francs less. In terms of welfare, this translates into EV changes of 1,413 francs loss for type 1 and of 17, 76 and 463 francs gains for types 2—4. Evidently, the nonlinear tax on the polluting good, and nonlinear commodity taxation in general, is a powerful redistributive mechanism. The strong redistributive power of the nonlinear tax on the polluting good can best be measured by directly comparing GIT LDT and GIT N DT (which differ only in that the former levies a linear tax on the polluting good and the latter a nonlinear tax). If we were to go from GIT LDT to GIT N DT , type 1 will pay 1,225 francs more in total taxes, while types 2—4, pay 49, 94 and 397 francs less. In terms of welfare, this translates into EV changes of 1,223 francs loss for type 1 and of 40, 84 and 408 francs gains for types 2—4. It appears that switching from a linear tax on the polluting good to a nonlinear tax, induces redistributive changes that dwarf those brought about by the introduction of the linear pollution tax in the Þrst place. One further aspect of the nonlinear taxation of the polluting good that signiÞes the redistributive power of these taxes is worth mentioning. The tax solutions under GIT ADT, GIT P DT and GIT LDT all involved pooling types 3 and 4 together offering them the same before- and after-tax incomes.20 On the other hand, GIT N DT allows a separation in the before- and after-tax incomes offered these groups. We end our discussion by making two observations regarding the implications of 20 Types 3 and 4 consume different amounts of polluting and non-polluting goods despite having the same before-tax incomes. This is due to their different tastes. In the previous identical tastes case, these types consumed identical bundles. Note also that this property does not hold for higher values of ω. 31 changing ρ and ω. The former affects the income tax rate and the latter the environ- mental tax rate. An increase in ρ lowers the optimal marginal income tax rate (with linear as well as nonlinear income tax schedules), just as it did when tastes were iden- tical. In the present case, however, the lump-sum element of the linear income tax will eventually turn into a subsidy. That is, the linear income tax schedule will become progressive. Second, observe that the optimal environmental levy will turn from a net subsidy at low values of ω to a tax at high values of ω. The reason is that as ω increases the efficiency cost of a potential tax differential between polluting and non-polluting goods increases. This reduces the optimal rate of subsidy on polluting goods due to redistributive considerations.21 7 Conclusion This paper has constructed a model with four different groups of households who have preferences over labor supply, consumption of polluting (energy related) and non- polluting (non-energy) goods and emissions. It has quantiÞed the model for the French economy and has computed its optimal tax equilibria under ten different tax regimes. In doing so, it has been able to shed light on a number of questions concerning the properties of optimal environmental taxes. In particular, it has shown that: (i) En- vironmental taxes are welfare improving but the changes are very modest; they add minimally to gains due to income tax instruments. (ii) The welfare gains due to envi- ronmental taxes remain low regardless of the income tax instrument employed (linear, general or differential lump-sum). (iii) Environmental taxes allow for further redistribu- tion from higher-ability to lower-ability persons and can make all types better-off. (iv) Environmental taxation does not entail a “double dividend”; they result in higher wel- fare gains in the Þrst-best than in the second-best. (v) In the second-best, the optimal environmental levy is larger than the Pigouvian tax. When polluting goods must be 21 In the identical tastes case, when the environmental tax reßects only Pigouvian considerations, its optimal value was invariant to changes in ω. 32 taxed for both externality-correcting and optimal tax considerations: (vi) they should be subsidized relative to non-polluting goods because of their redistributive properties. (vii) Externality-correcting-cum-optimal-tax-objectives call for a net subsidy on pollut- ing goods. (viii) This net subsidy will turn into a net tax as the elasticity of substitution between polluting and non-polluting goods increase. (ix) Levying a tax on the pollut- ing good equal to its marginal social damage is welfare reducing in that it induces a redistribution from the poor to the rich. (x) The optimal marginal income tax rate decreases (with linear as well as with general income tax schedules) as the elasticity of substitution between leisure and consumption goods increase. (xi) Graduating marginal income tax rates (i.e. using a general income tax schedule instead of a linear tax sys- tem), enhances the society’s ability to achieve its optimal tax objectives considerably. (xii) When using a general income tax alone, or in combination with linear commodity taxes, we found that the two lowest ability groups must be bunched together and of- fered the same before- and after-tax incomes. However, the ability to use a nonlinear consumption tax (on the polluting good) eradicates the bunching property and allows a separation in the before- and after-tax incomes offered these groups. (xiii) Nonlinear commodity taxation is a powerful redistributive mechanism. These Þndings are, of course, not meant to be the last word on such important policy question as the taxation of energy. The research can be extended in a number of directions. One may examine different preference structures, other social welfare functions, a more disaggregated set of goods, and different parameter values including the assumed marginal social damage of emissions. In particular, it would be enlightening to compute the optimal tax structures for a model consisting of a greater number of types than four. This needs more extensive data; maybe one can Þnd it for other countries. 33 References Atkinson, A.B. and J.E. Stiglitz (1976) “The design of tax structure: direct versus indirect taxation,” Journal of Public Economics, 6, 55—75. Bourguignon, F. (1999) “Redistribution and labor-supply incentives,” mimeo. Bovenberg, A.L. and F. van der Ploeg (1994) “Environmental policy, public Þnance and the labor market in a second-best world,” Journal of Public Economics, 55, 349—390. Bovenberg, A.L. and R.A. de Mooij (1994) “Environmental Levies and Distortionary Taxation,” American Economic Review, 84, 1085—1089. Bovenberg, A.L. and L. Goulder (1996) “Optimal environmental taxation in the pres- ence of other taxes: general equilibrium analyses,” American Economic Review, 86, 985—1000. Cremer, H., Gahvari, F. and N. Ladoux (1998) “Externalities and optimal taxation,” Journal of Public Economics, 70, 343—364. Cremer, H., Gahvari, F. and N. Ladoux (2000) “Second-best environmental levies and the structure of preferences,” mimeo. Cruz, M. and L.H. Goulder (1992) “An intertemporal general equilibrium mode for an- alyzing U.S. energy and environmental policies: data documentation,” Unpublished manuscript, Stanford University. Deaton, A. (1979) “Optimally uniform commodity taxes,” Economics Letters, 2, 357—361. Fullerton, D. (1997) “Environmental Levies and Distortionary Taxation: Comment,” American Economic Review, 87, 245—251. Goulder, L.H., Parry, I.W.H., Williams III, R.C. and D. Burtraw (1999) “The cost effectiveness of alternative instruments for environmental protection in a second-best setting,” Journal of Public Economics, 72, 329—360. Goulder, L.H. (1995) “Environmental taxation and double dividend: A reader’s guide,” International Tax ans Public Finance, 2, 157—183. INSEE (1989) “Emplois-revenus: enquˆte sur l’emploi de 1989, r´sultats d´taill´s,” S´rie e e e e e INSEE R´sultats, 28—29, ISBN 2-11-065325-6. e INSEE (1991a) “Consommation modes de vie: le budget des m´nages en 1989,” S´rie e e Consommation Modes de Vie, 116—117, ISBN 2-11-065922-X. INSEE (1991b) “Emplois-revenus: les salaires dans l’industrie, le commerce et les ser- vices en 1987—1989,” S´rie INSEE R´sultats, 367—369, ISBN 2-11-06-244-1. e e INSEE (1998) “Comptes et indicateurs ´conomiques: rapport sur les comptes de la e nation 1997,” S´rie INSEE R´sultats, 165—167, ISBN 2-11-0667748-6. e e 34 Kaplow, L. (1996) “The optimal supply of public goods and the distortionary cost of taxation,” National Tax Journal, 49, 513—533. Mayeres, I. and S. Proost (1997) “Optimal tax and public investment rules for congestion type of externalities,” Scandinavian Journal of Economics, 99, 261—279. Mayeres, I. and S. Proost (2001) “Marginal tax reform, externalities and income distri- bution,” Journal of Public Economics, 79, 343—363. Minist`re de l’Economie et des Finances (1989) “Publication N 2041 S,” France. e Pirttil¨, J. and M. Tuomala (1997) “Income tax, commodity tax and environmental a policy,” International Tax and Public Finance, 4, 379—393. Poterba, J.M. (1991) “Tax policy to combat global warming: on designing a carbon tax,” in Global Warming: Economic Policy Responses, ed. Dornbusch, R. and J.M. Poterba. Cambridge, Massachusetts: The MIT Press. Saez, E. (2000) “Optimal income transfer programs: intensive versus extensive labor supply responses,” mimeo. Stern, N.H. (1976) “On the speciÞcation of models of optimum income taxation,” Jour- nal of Public Economics, 6, 123—162. Stiglitz, J.E. (1987) “Pareto efficient and optimal taxation and the new new welfare economics,” in Handbook of Public Economics, Vol 2, ed. Auerbach, A. and M. Feld- stein. Amsterdam: North-Holland, 991—1042. Wales, T.J. and A.D. Woodland (1979) “Labour supply and progressive taxes,” Review of Economic Studies, 46, 83—95. 35 Appendix Computing ρ on the basis of wage elasticities: Rewrite equation (12) as µ ¶ρ µ ¶ ρ b wn Q= (1 − L). (A1) 1−b pQ Substituting for Q from (11) in above and solving for L, we have −ρ 1 − AM wn L= 1−ρ , (A2) 1 + Awn where µ ¶ρ 1−b A≡ pρ−1 . Q (A3) b From (A2), we the derive the elasticity of labor supply as wn ∂L ρM 1−ρ ²LL ≡ = ρ − ρ−1 . L ∂wn wn /A − M wn /A + 1 Substituting for A in terms of L from (A2) in above, we can rewrite the elasticity of labor supply as ρM 1−ρ ²LL = − . (A4) (wn L + M )/(1 − L) − M (wn L + M )/wn (1 − L) + 1 Equation (A4) governs the relationship between ρ and ²LL . Given any value for ²LL , one can compute the corresponding value of ρ for every individual type. Simple calculations give ²LL = 0.1 ²LL = 0.5 ρ1 0.67 1.06 ρ2 0.60 0.95 ρ3 0.56 0.89 ρ4 0.58 0.93 36 Table 6. Calibrations: heterogeneous tastes (1) (2) (3) (4) Managerial Staff Intermediary Level White Collars Blue Collars 6.1. ρ = 0.5, ω = 0.2689 a 0.999988 0.999971 0.999972 0.999945 b 0.705082 0.603996 0.524800 0.549138 6.2. ρ = 0.99, ω = 0.2689 a 0.999988 0.999971 0.999972 0.999945 b 0.428783 0.420778 0.420182 0.438268 6.3. ρ = 0.7927, ω = 0.10 a 1.000000 1.000000 1.000000 1.000000 b 0.503043 0.468617 0.447747 0.467825 6.4 ρ = 0.7927, ω = 0.50 a 0.997322 0.995749 0.995835 0.994034 b 0.499749 0.464573 0.443768 0.463126 6.5 ρ = 0.7927, ω = 0.99 a 0.943852 0.930069 0.930737 0.918016 b 0.491792 0.455777 0.435085 0.453718 Table 7. Calibrations: idetical tastes 7.1. ρ = 0.5, ω = 0.2689 a 0.999970 b 0.590297 7.2. ρ = 0.99, ω = 0.2689 a 0.999970 b 0.427287 7.3. ρ = 0.7927, ω = 0.10 a 1.000000 b 0.469961 7.4. ρ = 0.7927, ω = 0.50 a 0.995689 b 0.465890 7.5. ρ = 0.7927, ω = 0.99 a 0.929604 b 0.457065 37 Table 4. Optimal allocations and supporting taxes when tastes are identical: φ = 0.0259, ρ = 0.7927, ω = 0.2689 (monetary Þgures in 100,000 French francs) ULST LITACT LITPDT LITODT GITACT GITPDT GITLDT GITNDT FBADT FB φ/µ 0.093311 0.094253 0.094249 0.094249 0.096815 0.096812 0.096812 0.096812 0.100005 0.100000 τ 0.101443 0.090502 0.090987 0.090987 0.093826 0.094343 0.094343 0.094343 0.099768 0.100000 q 1 − 1 0.000000 0.000000 0.094249 0.094334 0.000000 0.096812 0.096797 0.096781 0.000000 0.100001 q 2 − 1 0.000000 0.000000 0.094249 0.094334 0.000000 0.096812 0.096797 0.096816 0.000000 0.100001 q 3 − 1 0.000000 0.000000 0.094249 0.094334 0.000000 0.096812 0.096797 0.096807 0.000000 0.100001 q 4 − 1 0.000000 0.000000 0.094249 0.094334 0.000000 0.096812 0.096797 0.096810 0.000000 0.100001 t1 0.000000 0.175626 0.171180 0.171176 0.005522 0.000002 0.000005 0.000005 0.000000 0.000010 t2 0.000000 0.175626 0.171180 0.171176 0.214118 0.209764 0.209764 0.209763 0.000000 0.000005 t3 0.000000 0.175626 0.171180 0.171176 0.228160 0.223884 0.223879 0.223879 0.000000 0.000003 t4 0.000000 0.175626 0.171180 0.171176 0.090846 0.085800 0.085794 0.085793 0.000000 0.000003 T1 0.367637 0.073123 0.073647 0.073647 1.048175 1.054187 1.054177 1.054177 4.949857 4.944946 T2 0.367637 0.073123 0.073647 0.073647 0.036260 0.036602 0.036601 0.036600 1.286843 1.279806 T3 0.367637 0.073123 0.073647 0.073647 -0.119191 -0.119720 -0.119713 -0.119713 -0.901632 -0.910865 T4 0.367637 0.073123 0.073647 0.073647 0.048738 0.049163 0.049172 0.049172 -1.340422 -1.350216 TP1 0.367637 0.577513 0.577404 0.577404 1.066980 1.066878 1.066875 1.066875 4.949857 4.949877 TP2 0.367637 0.402135 0.402117 0.402117 0.434286 0.434280 0.434278 0.434278 1.286843 1.286851 TP3 0.367637 0.310266 0.310296 0.310296 0.159838 0.159867 0.159868 0.159868 -0.901632 -0.901639 TP4 0.367637 0.293750 0.293789 0.293789 0.159838 0.159867 0.159868 0.159868 -1.340422 -1.340432 EV 1 0.000000 -0.231617 -0.231352 -0.231352 -0.698531 -0.698276 -0.698274 -0.698274 -4.582588 -4.582300 EV 2 0.000000 -0.047710 -0.047565 -0.047564 -0.087748 -0.087601 -0.087599 -0.087599 -0.919477 -0.919304 EV 3 0.000000 0.048451 0.048530 0.048530 0.190069 0.190157 0.190156 0.190156 1.269053 1.269157 EV 4 0.000000 0.065720 0.065788 0.065788 0.205986 0.206061 0.206060 0.206060 1.707853 1.707942 1 x 2.459916 2.162836 2.165985 2.165987 2.204086 2.207386 2.207384 2.207383 0.823119 0.822048 x2 1.536740 1.386846 1.388845 1.388847 1.342906 1.344892 1.344893 1.344893 1.187041 1.185635 x3 1.051041 0.980355 0.981751 0.981753 1.002142 1.003610 1.003613 1.003613 1.556997 1.555294 x4 0.963463 0.907279 0.908567 0.908568 1.002142 1.003610 1.003613 1.003613 1.651251 1.649477 y1 0.149680 0.131603 0.128641 0.128638 0.134113 0.131017 0.131017 0.131018 0.050085 0.048754 y2 0.093507 0.084386 0.082485 0.082484 0.081712 0.079825 0.079825 0.079825 0.072228 0.070317 y3 0.063953 0.059652 0.058308 0.058306 0.060978 0.059568 0.059569 0.059568 0.094739 0.092241 y4 0.058624 0.055206 0.053961 0.053960 0.060978 0.059568 0.059569 0.059568 0.100474 0.097827 w1 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 w2 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 w3 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 w4 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 L1 0.410416 0.395903 0.395914 0.395914 0.469409 0.469423 0.469423 0.469423 0.802718 0.802389 L2 0.453338 0.425084 0.425102 0.425102 0.421802 0.421823 0.421823 0.421823 0.577736 0.576985 L3 0.493497 0.449441 0.449468 0.449468 0.407064 0.407093 0.407095 0.407095 0.249673 0.248273 L4 0.503467 0.455106 0.455136 0.455136 0.443051 0.443083 0.443084 0.443084 0.149006 0.147401 21 Table 5. Optimal allocations and supporting taxes when tastes are heterogenous: φ = 0.0259, ρ = 0.7927, ω = 0.2689 (monetary Þgures in 100,000 French francs) ULST LITACT LITPDT LITODT GITACT GITPDT GITLDT GITNDT FBADT FB φ/µ 0.093374 0.094384 0.094379 0.094388 0.097100 0.097094 0.097104 0.097108 0.100005 0.100001 τ 0.100905 0.089592 0.090064 0.089338 0.093186 0.093689 0.092913 0.093122 0.099761 0.100007 q1 − 1 0.000000 0.000000 0.094379 -0.049091 0.000000 0.097094 -0.050745 0.097183 0.000000 0.099999 q2 − 1 0.000000 0.000000 0.094379 -0.049091 0.000000 0.097094 -0.050745 -0.143489 0.000000 0.099999 q3 − 1 0.000000 0.000000 0.094379 -0.049091 0.000000 0.097094 -0.050745 0.115138 0.000000 0.099999 q4 − 1 0.000000 0.000000 0.094379 -0.049091 0.000000 0.097094 -0.050745 -0.039588 0.000000 0.099999 t1 0.000000 0.183398 0.179101 0.185717 0.004237 -0.000036 0.006632 0.000003 0.000000 -0.000185 t2 0.000000 0.183398 0.179101 0.185717 0.225998 0.221896 0.228252 0.232259 0.000000 -0.000136 t3 0.000000 0.183398 0.179101 0.185717 0.178289 0.173260 0.181017 0.174012 0.000000 -0.000113 t4 -0.000000 0.183398 0.179101 0.185717 0.113479 0.108236 0.116324 0.114837 -0.000000 -0.000090 T1 0.365838 0.057235 0.057590 0.057040 1.212649 1.214692 1.211288 1.231555 4.814449 4.811363 T2 0.365838 0.057235 0.057590 0.057040 -0.005824 -0.006128 -0.005717 -0.005611 1.284378 1.277807 T3 0.365838 0.057235 0.057590 0.057040 -0.085113 -0.084751 -0.085315 -0.087549 -0.854621 -0.863159 T4 0.365838 0.057235 0.057590 0.057040 -0.008136 -0.007487 -0.008496 -0.011281 -1.314605 -1.325978 TP1 0.365838 0.614848 0.612315 0.616224 1.228130 1.225081 1.229824 1.242067 4.814449 4.814669 TP2 0.365838 0.398343 0.398395 0.398316 0.406907 0.406743 0.406989 0.406498 1.284378 1.284435 TP3 0.365838 0.293849 0.293852 0.293847 0.126645 0.126804 0.126555 0.125622 -0.854621 -0.854639 TP4 0.365838 0.285386 0.286332 0.284871 0.126645 0.127848 0.125985 0.122020 -1.314605 -1.314717 EV 1 0.000000 -0.273575 -0.270652 -0.275279 -0.861604 -0.858395 -0.863499 -0.875726 -4.449681 -4.449590 EV 2 0.000000 -0.047123 -0.047061 -0.047244 -0.065187 -0.064922 -0.065423 -0.065021 -0.919375 -0.919236 EV 3 0.000000 0.062196 0.062303 0.062080 0.229139 0.229169 0.229060 0.229901 1.219783 1.219920 EV 4 0.000000 0.071383 0.070440 0.071829 0.235925 0.234786 0.236476 0.240562 1.679803 1.679986 1 x 2.651033 2.314690 2.319342 2.312114 2.314519 2.318265 2.312322 2.311690 0.938950 0.938064 x2 1.534044 1.378308 1.380190 1.377247 1.338483 1.340359 1.337396 1.335699 1.185366 1.184027 x3 1.009839 0.940091 0.941397 0.939356 1.001204 1.002933 1.000233 1.002651 1.476948 1.475333 x4 0.951961 0.894542 0.895251 0.894127 0.990147 0.991135 0.989572 0.991700 1.620048 1.618120 y1 0.127020 0.110905 0.108465 0.112291 0.110897 0.108342 0.112354 0.108033 0.044988 0.043809 y2 0.092686 0.083277 0.081392 0.084347 0.080870 0.078991 0.081944 0.084135 0.071619 0.069728 y3 0.060394 0.056223 0.054952 0.056944 0.059878 0.058505 0.060663 0.058232 0.088330 0.086000 y4 0.068199 0.064086 0.062600 0.064929 0.070935 0.069258 0.071894 0.071822 0.116062 0.112990 w1 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 w2 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 w3 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 w4 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 L1 0.433390 0.419130 0.419086 0.419156 0.503647 0.503391 0.503779 0.504783 0.799317 0.799062 L2 0.452132 0.422034 0.422046 0.422030 0.414395 0.414357 0.414411 0.414411 0.576658 0.575938 L3 0.477999 0.429433 0.429446 0.429428 0.395337 0.395509 0.395245 0.394931 0.236543 0.235225 L4 0.502117 0.450679 0.450740 0.450647 0.430288 0.430474 0.430187 0.429496 0.152702 0.150850 28 Table 8. Optimal allocations and supporting taxes when tastes are identical: φ = 0.0263, ρ = 0.50, ω = 0.2689 (monetary Þgures in 100,000 French francs) ULST LITACT LITPDT LITODT GITACT GITPDT GITLDT GITNDT FBADT FB φ/µ 0.093312 0.094356 0.094353 0.094353 0.097443 0.097439 0.097439 0.097438 0.100004 0.100000 τ 0.101724 0.088015 0.088488 0.088489 0.092081 0.092592 0.092592 0.092592 0.099778 0.100000 q 1 − 1 0.000000 0.000000 0.094353 0.094703 0.000000 0.097439 0.097451 0.097440 0.000000 0.100000 q 2 − 1 0.000000 0.000000 0.094353 0.094703 0.000000 0.097439 0.097451 0.097446 0.000000 0.100000 q 3 − 1 0.000000 0.000000 0.094353 0.094703 0.000000 0.097439 0.097451 0.097693 0.000000 0.100000 q 4 − 1 0.000000 0.000000 0.094353 0.094703 0.000000 0.097439 0.097451 0.097487 0.000000 0.100000 t1 0.000000 0.212597 0.208333 0.208325 0.005560 0.000003 -0.000002 -0.000005 0.000000 -0.000000 t2 0.000000 0.212597 0.208333 0.208325 0.261752 0.257639 0.257634 0.257638 0.000000 -0.000000 t3 0.000000 0.212597 0.208333 0.208325 0.284109 0.280117 0.280109 0.280125 0.000000 0.000000 t4 0.000000 0.212597 0.208333 0.208325 0.114559 0.109611 0.109600 0.109633 0.000000 0.000000 T1 0.357961 0.016173 0.016401 0.016391 0.986395 0.992095 0.992103 0.992125 5.000344 4.996405 T2 0.357961 0.016173 0.016401 0.016391 -0.030188 -0.030229 -0.030224 -0.030222 1.284760 1.278271 T3 0.357961 0.016173 0.016401 0.016391 -0.191991 -0.192937 -0.192926 -0.192963 -0.929027 -0.938255 T4 0.357961 0.016173 0.016401 0.016391 0.017829 0.018080 0.018094 0.018047 -1.369211 -1.379148 TP1 0.357961 0.548598 0.548508 0.548515 1.002967 1.002899 1.002894 1.002906 5.000344 5.000397 TP2 0.357961 0.394987 0.394970 0.394971 0.435635 0.435630 0.435628 0.435636 1.284760 1.284783 TP3 0.357961 0.304918 0.304943 0.304941 0.159598 0.159617 0.159619 0.159613 -0.929027 -0.929044 TP4 0.357961 0.287775 0.287808 0.287805 0.159598 0.159617 0.159619 0.159613 -1.369211 -1.369238 EV 1 0.000000 -0.210427 -0.210172 -0.210180 -0.644429 -0.644200 -0.644194 -0.644206 -4.643255 -4.643007 EV 2 0.000000 -0.049749 -0.049612 -0.049614 -0.098565 -0.098431 -0.098429 -0.098437 -0.927428 -0.927281 EV 3 0.000000 0.044196 0.044269 0.044270 0.179447 0.179529 0.179529 0.179531 1.286489 1.286589 EV 4 0.000000 0.062047 0.062107 0.062109 0.196327 0.196398 0.196397 0.196401 1.726696 1.726788 1 x 2.060422 1.843611 1.846293 1.846295 1.864156 1.866943 1.866948 1.866946 0.673392 0.673085 x2 1.424358 1.307310 1.309193 1.309196 1.266912 1.268784 1.268787 1.268783 1.098336 1.098023 x3 1.045136 0.992853 0.994266 0.994269 1.016091 1.017582 1.017585 1.017579 1.553407 1.553179 x4 0.972189 0.933001 0.934324 0.934327 1.016091 1.017582 1.017585 1.017576 1.671145 1.670951 y1 0.125371 0.112179 0.109651 0.109642 0.113429 0.110793 0.110793 0.110794 0.040974 0.039919 y2 0.086669 0.079546 0.077753 0.077746 0.077088 0.075296 0.075296 0.075296 0.066831 0.065121 y3 0.063594 0.060413 0.059049 0.059044 0.061827 0.060388 0.060388 0.060384 0.094521 0.092115 y4 0.059155 0.056771 0.055489 0.055485 0.061827 0.060388 0.060388 0.060387 0.101685 0.099100 w1 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 w2 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 w3 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 w4 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 L1 0.350661 0.345234 0.345243 0.345242 0.410874 0.410885 0.410885 0.410887 0.787782 0.787601 L2 0.424090 0.404316 0.404333 0.404332 0.403815 0.403832 0.403833 0.403833 0.555910 0.555457 L3 0.488191 0.452074 0.452099 0.452098 0.411909 0.411933 0.411935 0.411930 0.239287 0.238405 L4 0.503315 0.462827 0.462854 0.462853 0.448325 0.448351 0.448353 0.448347 0.146222 0.145206 38 Table 9. Optimal allocations and supporting taxes when tastes are identical: φ = 0.0258, ρ = 0.99, ω = 0.2689 (monetary Þgures in 100,000 French francs) ULST LITACT LITPDT LITODT GITACT GITPDT GITLDT GITNDT FBADT FB φ/µ 0.093310 0.094242 0.094238 0.094238 0.096635 0.096631 0.096632 0.096630 0.100005 0.100000 τ 0.101251 0.091369 0.091860 0.091860 0.094470 0.094988 0.095000 0.094988 0.099762 0.100000 q1 − 1 0.000000 0.000000 0.094238 0.094547 0.000000 0.096631 0.096585 0.096168 0.000000 0.100000 q2 − 1 0.000000 0.000000 0.094238 0.094547 0.000000 0.096631 0.096585 0.096537 0.000000 0.100000 q3 − 1 0.000000 0.000000 0.094238 0.094547 0.000000 0.096631 0.096585 0.096689 0.000000 0.100000 q4 − 1 0.000000 0.000000 0.094238 0.094547 0.000000 0.096631 0.096585 0.096780 0.000000 0.100000 t1 -0.000000 0.162655 0.158126 0.158123 0.006344 0.000811 0.000009 0.000811 0.000000 -0.000000 t2 0.000000 0.162655 0.158126 0.158123 0.194714 0.190317 0.190305 0.190316 0.000000 -0.000000 t3 -0.000000 0.162655 0.158126 0.158123 0.206686 0.202304 0.202314 0.202301 0.000000 0.000000 t4 0.000000 0.162655 0.158126 0.158123 0.081227 0.076144 0.076158 0.076136 -0.000000 0.000000 T1 0.374586 0.093761 0.094421 0.094405 1.107509 1.113999 1.116975 1.114062 4.914025 4.908511 T2 0.374586 0.093761 0.094421 0.094405 0.063839 0.064225 0.064255 0.064236 1.288835 1.281450 T3 0.374586 0.093761 0.094421 0.094405 -0.096543 -0.096950 -0.096961 -0.096950 -0.882263 -0.891497 T4 0.374586 0.093761 0.094421 0.094405 0.055296 0.055748 0.055733 0.055749 -1.320068 -1.329769 TP1 0.374586 0.604338 0.604190 0.604206 1.131070 1.130999 1.130997 1.130995 4.914025 4.914041 TP2 0.374586 0.408907 0.408885 0.408887 0.437519 0.437494 0.437502 0.437496 1.288835 1.288839 TP3 0.374586 0.312369 0.312409 0.312405 0.153603 0.153632 0.153629 0.153632 -0.882263 -0.882267 TP4 0.374586 0.295550 0.295601 0.295596 0.153603 0.153632 0.153629 0.153632 -1.320068 -1.320074 EV 1 0.000000 -0.253364 -0.253061 -0.253081 -0.755530 -0.755315 -0.755307 -0.755310 -4.539458 -4.539165 EV 2 0.000000 -0.048320 -0.048162 -0.048166 -0.084368 -0.084211 -0.084216 -0.084211 -0.914259 -0.914075 EV 3 0.000000 0.052921 0.053001 0.053003 0.203384 0.203481 0.203480 0.203480 1.256836 1.256946 EV 4 0.000000 0.070555 0.070620 0.070624 0.219296 0.219379 0.219379 0.219378 1.694652 1.694747 1 x 2.745719 2.389296 2.392817 2.392802 2.434924 2.438593 2.439715 2.438614 0.933981 0.932352 x2 1.613981 1.440927 1.443027 1.443020 1.396619 1.398634 1.398645 1.398637 1.248056 1.245895 x3 1.054847 0.972457 0.973854 0.973852 0.996059 0.997519 0.997513 0.997520 1.558995 1.556308 x4 0.957424 0.890840 0.892114 0.892113 0.996059 0.997519 0.997513 0.997522 1.637511 1.634691 y1 0.167070 0.145383 0.142113 0.142101 0.148159 0.144747 0.144815 0.144764 0.056830 0.055296 y2 0.098207 0.087677 0.085703 0.085697 0.084981 0.083018 0.083020 0.083020 0.075941 0.073891 y3 0.064185 0.059172 0.057839 0.057834 0.060608 0.059209 0.059210 0.059209 0.094861 0.092301 y4 0.058257 0.054205 0.052984 0.052980 0.060608 0.059209 0.059210 0.059207 0.099638 0.096950 w1 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 w2 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 w3 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 w4 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 L1 0.453170 0.432718 0.432733 0.432731 0.512002 0.512027 0.512191 0.512032 0.813991 0.813557 L2 0.473508 0.439639 0.439662 0.439660 0.435465 0.435472 0.435476 0.435473 0.592875 0.591920 L3 0.497154 0.447352 0.447387 0.447383 0.402841 0.402871 0.402868 0.402871 0.256826 0.255078 L4 0.503663 0.449441 0.449478 0.449474 0.438454 0.438487 0.438484 0.438487 0.151100 0.149102 39 Table 10. Optimal allocations and supporting taxes when tastes are identical: φ = 0.0260, ρ = 0.7927, ω = 0.10 (monetary Þgures in 100,000 French francs) ULST LITACT LITPDT LITODT GITACT GITPDT GITLDT GITNDT FBADT FB φ/µ 0.093309 0.094250 0.094249 0.094249 0.096813 0.096812 0.096812 0.096812 0.100002 0.100000 τ 0.101453 0.090524 0.090986 0.090985 0.093850 0.094342 0.094342 0.094342 0.099778 0.100000 q 1 − 1 0.000000 0.000000 0.094249 0.093898 0.000000 0.096812 0.096761 0.096761 0.000000 0.100000 q 2 − 1 0.000000 0.000000 0.094249 0.093898 0.000000 0.096812 0.096761 0.096761 0.000000 0.100000 q 3 − 1 0.000000 0.000000 0.094249 0.093898 0.000000 0.096812 0.096761 0.096761 0.000000 0.100000 q 4 − 1 0.000000 0.000000 0.094249 0.093898 0.000000 0.096812 0.096761 0.096761 0.000000 0.100000 t1 0.000000 0.175423 0.171203 0.171219 0.005240 0.000001 0.000007 0.000007 -0.000000 -0.000000 t2 0.000000 0.175423 0.171203 0.171219 0.213911 0.209780 0.209783 0.209783 0.000000 -0.000000 t3 0.000000 0.175423 0.171203 0.171219 0.227962 0.223911 0.223908 0.223908 0.000000 0.000000 t4 0.000000 0.175423 0.171203 0.171219 0.090585 0.085810 0.085807 0.085807 0.000000 0.000000 T1 0.367690 0.073467 0.073889 0.073887 1.049008 1.054618 1.054605 1.054605 4.949896 4.945227 T2 0.367690 0.073467 0.073889 0.073887 0.036621 0.036864 0.036862 0.036862 1.286892 1.280133 T3 0.367690 0.073467 0.073889 0.073887 -0.118910 -0.119497 -0.119490 -0.119490 -0.901575 -0.910455 T4 0.367690 0.073467 0.073889 0.073887 0.049135 0.049436 0.049444 0.049444 -1.340363 -1.349783 TP1 0.367690 0.577334 0.577296 0.577296 1.066852 1.066819 1.066818 1.066818 4.949896 4.949917 TP2 0.367690 0.402151 0.402144 0.402144 0.434332 0.434332 0.434330 0.434330 1.286892 1.286898 TP3 0.367690 0.310383 0.310393 0.310393 0.159941 0.159950 0.159951 0.159951 -0.901575 -0.901581 TP4 0.367690 0.293885 0.293899 0.293899 0.159941 0.159950 0.159951 0.159951 -1.340363 -1.340372 EV 1 0.000000 -0.231398 -0.231305 -0.231305 -0.698396 -0.698310 -0.698308 -0.698308 -4.582552 -4.582448 EV 2 0.000000 -0.047688 -0.047637 -0.047637 -0.087730 -0.087681 -0.087679 -0.087679 -0.919457 -0.919389 EV 3 0.000000 0.048370 0.048399 0.048399 0.190026 0.190058 0.190058 0.190058 1.269062 1.269107 EV 4 0.000000 0.065622 0.065645 0.065645 0.205923 0.205951 0.205950 0.205950 1.707859 1.707900 1 x 2.467946 2.170258 2.171379 2.171375 2.211655 2.212834 2.212831 2.212831 0.825798 0.824037 x2 1.541744 1.391552 1.392264 1.392262 1.347468 1.348174 1.348174 1.348174 1.190903 1.188496 x3 1.054454 0.983638 0.984135 0.984133 1.005513 1.006032 1.006034 1.006034 1.562061 1.559036 x4 0.966589 0.910305 0.910763 0.910762 1.005513 1.006032 1.006034 1.006034 1.656621 1.653445 y1 0.141816 0.124710 0.123655 0.123659 0.127089 0.125987 0.125987 0.125987 0.047453 0.046903 y2 0.088593 0.079963 0.079286 0.079289 0.077430 0.076758 0.076758 0.076758 0.068433 0.067647 y3 0.060592 0.056523 0.056044 0.056046 0.057780 0.057278 0.057278 0.057278 0.089761 0.088737 y4 0.055543 0.052309 0.051866 0.051868 0.057780 0.057278 0.057278 0.057278 0.095195 0.094111 w1 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 w2 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 w3 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 w4 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 L1 0.410446 0.395951 0.395955 0.395955 0.469467 0.469473 0.469472 0.469472 0.802730 0.802414 L2 0.453371 0.425152 0.425158 0.425158 0.421876 0.421884 0.421883 0.421883 0.577762 0.577039 L3 0.493532 0.449531 0.449541 0.449541 0.407156 0.407164 0.407166 0.407166 0.249721 0.248372 L4 0.503502 0.455202 0.455213 0.455213 0.443151 0.443160 0.443162 0.443162 0.149060 0.147514 40 Table 11. Optimal allocations and supporting taxes when tastes are identical: φ = 0.0256, ρ = 0.7927, ω = 0.50 (monetary Þgures in 100,000 French francs) ULST LITACT LITPDT LITODT GITACT GITPDT GITLDT GITNDT FBADT FB φ/µ 0.093314 0.094256 0.094249 0.094249 0.096819 0.096811 0.096811 0.096811 0.100009 0.100000 τ 0.101429 0.090469 0.090988 0.090988 0.093791 0.094343 0.094343 0.094343 0.099755 0.100000 q 1 − 1 0.000000 0.000000 0.094249 0.094310 0.000000 0.096811 0.096796 0.096796 0.000000 0.100000 q 2 − 1 0.000000 0.000000 0.094249 0.094310 0.000000 0.096811 0.096796 0.096796 0.000000 0.100000 q 3 − 1 0.000000 0.000000 0.094249 0.094310 0.000000 0.096811 0.096796 0.096796 0.000000 0.100000 q 4 − 1 0.000000 0.000000 0.094249 0.094310 0.000000 0.096811 0.096796 0.096796 0.000000 0.100000 t1 0.000000 0.175922 0.171146 0.171143 0.005937 0.000017 0.000010 0.000010 0.000000 -0.000000 t2 0.000000 0.175922 0.171146 0.171143 0.214416 0.209737 0.209742 0.209742 -0.000000 0.000000 t3 0.000000 0.175922 0.171146 0.171143 0.228425 0.223838 0.223842 0.223842 0.000000 0.000000 t4 0.000000 0.175922 0.171146 0.171143 0.091197 0.085777 0.085782 0.085782 0.000000 0.000000 T1 0.367561 0.072624 0.073301 0.073302 1.046946 1.053521 1.053549 1.053549 4.949801 4.944711 T2 0.367561 0.072624 0.073301 0.073302 0.035735 0.036230 0.036224 0.036224 1.286772 1.279379 T3 0.367561 0.072624 0.073301 0.073302 -0.119572 -0.120030 -0.120034 -0.120034 -0.901714 -0.911454 T4 0.367561 0.072624 0.073301 0.073302 0.048200 0.048783 0.048778 0.048778 -1.340507 -1.350844 TP1 0.367561 0.577774 0.577560 0.577560 1.067159 1.066957 1.066961 1.066961 4.949801 4.949851 TP2 0.367561 0.402114 0.402079 0.402079 0.434213 0.434204 0.434204 0.434204 1.286772 1.286792 TP3 0.367561 0.310098 0.310156 0.310156 0.159695 0.159751 0.159750 0.159750 -0.901714 -0.901729 TP4 0.367561 0.293556 0.293631 0.293631 0.159695 0.159751 0.159750 0.159750 -1.340507 -1.340531 EV 1 0.000000 -0.231929 -0.231399 -0.231399 -0.698713 -0.698206 -0.698210 -0.698210 -4.582638 -4.582099 EV 2 0.000000 -0.047737 -0.047445 -0.047445 -0.087763 -0.087471 -0.087472 -0.087472 -0.919506 -0.919176 EV 3 0.000000 0.048570 0.048732 0.048732 0.190131 0.190309 0.190310 0.190310 1.269040 1.269244 EV 4 0.000000 0.065866 0.066003 0.066003 0.206077 0.206229 0.206230 0.206230 1.707844 1.708023 1 x 2.448258 2.152064 2.158277 2.158281 2.193103 2.199590 2.199596 2.199596 0.819229 0.819207 x2 1.529476 1.380015 1.383960 1.383963 1.336290 1.340208 1.340205 1.340205 1.181434 1.181556 x3 1.046086 0.975590 0.978346 0.978348 0.997261 1.000154 1.000153 1.000153 1.549644 1.549958 x4 0.958925 0.902886 0.905427 0.905429 0.997261 1.000154 1.000153 1.000153 1.643454 1.643823 y1 0.161100 0.141610 0.135765 0.135762 0.144311 0.138202 0.138204 0.138204 0.053907 0.051397 y2 0.100643 0.090808 0.087057 0.087055 0.087931 0.084207 0.084207 0.084207 0.077741 0.074131 y3 0.068835 0.064196 0.061542 0.061541 0.065622 0.062841 0.062841 0.062841 0.101970 0.097244 y4 0.063099 0.059412 0.056955 0.056954 0.065622 0.062841 0.062841 0.062841 0.108143 0.103133 w1 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 w2 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 w3 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 w4 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 L1 0.410373 0.395834 0.395855 0.395855 0.469325 0.469350 0.469352 0.469352 0.802701 0.802359 L2 0.453291 0.424986 0.425023 0.425023 0.421695 0.421737 0.421737 0.421737 0.577698 0.576911 L3 0.493447 0.449312 0.449365 0.449365 0.406938 0.406993 0.406993 0.406993 0.249606 0.248132 L4 0.503416 0.454968 0.455026 0.455026 0.442913 0.442974 0.442973 0.442973 0.148929 0.147239 41 Table 12. Optimal allocations and supporting taxes when tastes are identical: φ = 0.0249, ρ = 0.7927, ω = 0.99 (monetary Þgures in 100,000 French francs) ULST LITACT LITPDT LITODT GITACT GITPDT GITLDT GITNDT FBADT FB φ/µ 0.093321 0.094261 0.094244 0.094248 0.096827 0.096810 0.096810 0.096810 0.100020 0.100000 τ 0.101398 0.090386 0.090982 0.090990 0.093708 0.094344 0.094344 0.094344 0.099723 0.100000 q 1 − 1 0.000000 0.000000 0.094244 0.094167 0.000000 0.096810 0.096806 0.096811 0.000000 0.100000 q 2 − 1 0.000000 0.000000 0.094244 0.094167 0.000000 0.096810 0.096806 0.096808 0.000000 0.100000 q 3 − 1 0.000000 0.000000 0.094244 0.094167 0.000000 0.096810 0.096806 0.096814 0.000000 0.100000 q 4 − 1 0.000000 0.000000 0.094244 0.094167 0.000000 0.096810 0.096806 0.096814 0.000000 0.100000 t1 0.000000 0.176746 0.171202 0.171073 0.006952 0.000021 0.000020 0.000022 -0.000000 -0.000000 t2 0.000000 0.176746 0.171202 0.171073 0.215114 0.209680 0.209683 0.209681 -0.000000 0.000000 t3 0.000000 0.176746 0.171202 0.171073 0.229070 0.223748 0.223750 0.223748 0.000000 0.000000 t4 0.000000 0.176746 0.171202 0.171073 0.092049 0.085743 0.085745 0.085743 0.000000 0.000000 T1 0.367388 0.071257 0.072303 0.072507 1.043939 1.052106 1.052114 1.052105 4.949665 4.944014 T2 0.367388 0.071257 0.072303 0.072507 0.034514 0.035375 0.035371 0.035375 1.286609 1.278363 T3 0.367388 0.071257 0.072303 0.072507 -0.120493 -0.120761 -0.120763 -0.120761 -0.901898 -0.912797 T4 0.367388 0.071257 0.072303 0.072507 0.046903 0.047889 0.047887 0.047889 -1.340695 -1.352271 TP1 0.367388 0.578553 0.578082 0.577921 1.067599 1.067153 1.067155 1.067153 4.949665 4.949759 TP2 0.367388 0.402097 0.402020 0.401994 0.434057 0.434035 0.434035 0.434035 1.286609 1.286649 TP3 0.367388 0.309665 0.309793 0.309837 0.159357 0.159481 0.159481 0.159481 -0.901898 -0.901926 TP4 0.367388 0.293048 0.293214 0.293270 0.159357 0.159481 0.159481 0.159481 -1.340695 -1.340742 EV 1 0.000000 -0.232837 -0.231613 -0.231416 -0.699141 -0.697963 -0.697965 -0.697963 -4.582750 -4.581506 EV 2 0.000000 -0.047833 -0.047149 -0.047100 -0.087797 -0.087109 -0.087109 -0.087109 -0.919576 -0.918814 EV 3 0.000000 0.048893 0.049282 0.049253 0.190292 0.190717 0.190718 0.190717 1.269003 1.269476 EV 4 0.000000 0.066263 0.066597 0.066555 0.206304 0.206672 0.206672 0.206672 1.707815 1.708232 1 x 2.420759 2.126455 2.140373 2.140593 2.167167 2.181757 2.181758 2.181757 0.810054 0.812706 x2 1.512337 1.363796 1.372637 1.372750 1.320692 1.329470 1.329468 1.329470 1.168207 1.172211 x3 1.034395 0.964296 0.970475 0.970530 0.985741 0.992227 0.992226 0.992227 1.532298 1.537734 x4 0.948216 0.892478 0.898177 0.898222 0.985741 0.992227 0.992226 0.992227 1.625058 1.630867 y1 0.188060 0.165197 0.152093 0.152119 0.168360 0.154674 0.154675 0.154674 0.062930 0.057451 y2 0.117488 0.105948 0.097538 0.097553 0.102600 0.094252 0.094252 0.094252 0.090754 0.082864 y3 0.080359 0.074913 0.068961 0.068970 0.076579 0.070343 0.070343 0.070343 0.119039 0.108704 y4 0.073664 0.069333 0.063823 0.063831 0.076579 0.070343 0.070343 0.070343 0.126245 0.115287 w1 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 w2 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 w3 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 w4 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 L1 0.410275 0.395662 0.395709 0.395721 0.469126 0.469189 0.469190 0.469189 0.802661 0.802284 L2 0.453186 0.424738 0.424818 0.424841 0.421449 0.421542 0.421541 0.421542 0.577613 0.576740 L3 0.493334 0.448975 0.449094 0.449130 0.406637 0.406762 0.406762 0.406762 0.249452 0.247812 L4 0.503301 0.454608 0.454737 0.454776 0.442587 0.442722 0.442722 0.442722 0.148754 0.146872 42 Table 13. Optimal allocations and supporting taxes when tastes are heterogeneous: φ = 0.0263, ρ = 0.50, ω = 0.2689 (monetary Þgures in 100,000 French francs) ULST LITACT LITPDT LITODT GITACT GITPDT GITLDT GITNDT FBADT FB φ/µ 0.093554 0.094823 0.094818 0.094826 0.098299 0.098294 0.098304 0.098309 0.100005 0.100000 τ 0.100123 0.084869 0.085323 0.084647 0.090218 0.090717 0.089895 0.090135 0.099772 0.100000 q1 − 1 0.000000 0.000000 0.094818 -0.044807 0.000000 0.098294 -0.061639 0.098317 0.000000 0.099995 q2 − 1 0.000000 0.000000 0.094818 -0.044807 0.000000 0.098294 -0.061639 -0.159162 0.000000 0.099995 q3 − 1 0.000000 0.000000 0.094818 -0.044807 0.000000 0.098294 -0.061639 0.116090 0.000000 0.099995 q4 − 1 0.000000 0.000000 0.094818 -0.044807 0.000000 0.098294 -0.061639 -0.045803 0.000000 0.099995 t1 -0.000000 0.242478 0.238417 0.244468 0.004478 0.000004 0.007401 -0.000001 -0.000000 0.000012 t2 0.000000 0.242478 0.238417 0.244468 0.281311 0.277403 0.283855 0.287693 -0.000000 0.000007 t3 -0.000000 0.242478 0.238417 0.244468 0.213482 0.208228 0.216897 0.210225 -0.000000 0.000005 t4 -0.000000 0.242478 0.238417 0.244468 0.151012 0.146113 0.154201 0.151934 -0.000000 0.000004 T1 0.350214 -0.041079 -0.041184 -0.041036 1.467170 1.470213 1.465175 1.491132 4.593594 4.589182 T2 0.350214 -0.041079 -0.041184 -0.041036 -0.091130 -0.091799 -0.090690 -0.090133 1.227373 1.220676 T3 0.350214 -0.041079 -0.041184 -0.041036 -0.200540 -0.200343 -0.200658 -0.205053 -0.794798 -0.802853 T4 0.350214 -0.041079 -0.041184 -0.041036 -0.131835 -0.131987 -0.131731 -0.134509 -1.262938 -1.273755 TP1 0.350214 0.663720 0.661114 0.665009 1.483417 1.479645 1.485895 1.500512 4.593594 4.593661 TP2 0.350214 0.392544 0.392563 0.392536 0.408107 0.407953 0.408211 0.407459 1.227373 1.227398 TP3 0.350214 0.257432 0.257433 0.257429 0.034249 0.034579 0.034032 0.032560 -0.794798 -0.794769 TP4 0.350214 0.249128 0.250125 0.248635 0.034249 0.035640 0.033335 0.028884 -1.262938 -1.262993 EV 1 0.000000 -0.342245 -0.339300 -0.343797 -1.132784 -1.128873 -1.135486 -1.150068 -4.244349 -4.244164 EV 2 0.000000 -0.059785 -0.059691 -0.059906 -0.083126 -0.082865 -0.083414 -0.082776 -0.877969 -0.877822 EV 3 0.000000 0.080862 0.080973 0.080759 0.306129 0.306009 0.306122 0.307469 1.144334 1.144417 EV 4 0.000000 0.089976 0.088996 0.090403 0.311831 0.310490 0.312615 0.317223 1.612513 1.612633 1 x 2.451883 2.140377 2.144409 2.138362 2.046549 2.050323 2.044027 2.042917 0.944884 0.944645 x2 1.452982 1.316226 1.318088 1.315286 1.288711 1.290605 1.287436 1.285768 1.138822 1.138517 x3 0.968752 0.918708 0.920014 0.918050 1.005428 1.007158 1.004272 1.006967 1.386680 1.386341 x4 0.923278 0.884353 0.885217 0.883910 0.994324 0.995302 0.993653 0.996054 1.541256 1.540963 y1 0.117478 0.102553 0.100273 0.103727 0.098057 0.095792 0.099627 0.095446 0.045273 0.044116 y2 0.087788 0.079526 0.077722 0.080455 0.077863 0.076036 0.079129 0.081393 0.068807 0.067048 y3 0.057937 0.054944 0.053698 0.055586 0.060130 0.058734 0.061098 0.058470 0.082931 0.080813 y4 0.066144 0.063356 0.061892 0.064110 0.071234 0.069529 0.072415 0.072264 0.110417 0.107602 w1 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 w2 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 w3 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 w4 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 L1 0.402468 0.400686 0.400569 0.400748 0.500129 0.499817 0.500339 0.501624 0.769729 0.769546 L2 0.429081 0.405780 0.405798 0.405776 0.402691 0.402671 0.402713 0.402677 0.552524 0.552061 L3 0.458305 0.409769 0.409789 0.409763 0.366073 0.366294 0.365938 0.365471 0.224613 0.223805 L4 0.485321 0.433588 0.433732 0.433522 0.398436 0.398676 0.398289 0.397492 0.140830 0.139684 43 Table 14. Optimal allocations and supporting taxes when tastes are heterogeneous: φ = 0.0258, ρ = 0.99, ω = 0.2689 (monetary Þgures in 100,000 French francs) ULST LITACT LITPDT LITODT GITACT GITPDT GITLDT GITNDT FBADT FB φ/µ 0.093302 0.094218 0.094213 0.094221 0.096606 0.096602 0.096610 0.096614 0.100005 0.100000 τ 0.101178 0.091474 0.091953 0.091210 0.094527 0.095038 0.094286 0.094491 0.099754 0.100000 q1 − 1 0.000000 0.000000 0.094213 -0.050328 0.000000 0.096602 -0.044238 0.096651 0.000000 0.100002 q2 − 1 0.000000 0.000000 0.094213 -0.050328 0.000000 0.096602 -0.044238 -0.133291 0.000000 0.100002 q3 − 1 0.000000 0.000000 0.094213 -0.050328 0.000000 0.096602 -0.044238 0.114746 0.000000 0.100002 q4 − 1 0.000000 0.000000 0.094213 -0.050328 0.000000 0.096602 -0.044238 -0.035758 0.000000 0.100002 t1 0.000000 0.160187 0.155798 0.162615 0.004406 0.000009 0.006485 -0.000002 0.000000 0.000008 t2 0.000000 0.160187 0.155798 0.162615 0.198430 0.194156 0.200457 0.204232 -0.000000 0.000004 t3 0.000000 0.160187 0.155798 0.162615 0.161787 0.156786 0.164154 0.156989 0.000000 0.000002 t4 -0.000000 0.160187 0.155798 0.162615 0.096566 0.091144 0.099132 0.097831 -0.000000 0.000002 T1 0.376306 0.098873 0.099424 0.098570 1.128300 1.130659 1.127204 1.146162 4.887622 4.883166 T2 0.376306 0.098873 0.099424 0.098570 0.040115 0.040132 0.040105 0.040393 1.307431 1.300261 T3 0.376306 0.098873 0.099424 0.098570 -0.036071 -0.035568 -0.036304 -0.037486 -0.868513 -0.877381 T4 0.376306 0.098873 0.099424 0.098570 0.044384 0.045438 0.043891 0.041278 -1.324003 -1.335661 TP1 0.376306 0.603868 0.601324 0.605282 1.144751 1.141867 1.146126 1.157304 4.887622 4.887689 TP2 0.376306 0.405339 0.405409 0.405300 0.412036 0.411877 0.412110 0.411753 1.307431 1.307458 TP3 0.376306 0.311438 0.311446 0.311434 0.163506 0.163618 0.163454 0.162783 -0.868513 -0.868480 TP4 0.376306 0.302740 0.303677 0.302218 0.163506 0.164665 0.162954 0.159187 -1.324003 -1.324062 EV 1 0.000000 -0.250496 -0.247536 -0.252269 -0.767641 -0.764587 -0.769197 -0.780374 -4.512441 -4.512165 EV 2 0.000000 -0.042549 -0.042505 -0.042666 -0.058253 -0.057979 -0.058462 -0.058118 -0.931951 -0.931767 EV 3 0.000000 0.055897 0.055999 0.055777 0.202777 0.202851 0.202685 0.203297 1.244159 1.244250 EV 4 0.000000 0.065292 0.064353 0.065745 0.210033 0.208947 0.210488 0.214370 1.699676 1.699809 1 x 2.788346 2.432142 2.437242 2.429238 2.470586 2.474669 2.468589 2.468434 0.959422 0.958068 x2 1.589485 1.421932 1.423830 1.420825 1.378967 1.380911 1.377998 1.376544 1.222305 1.220148 x3 1.037762 0.958239 0.959549 0.957476 1.009691 1.011446 1.008826 1.011234 1.529319 1.526650 x4 0.971339 0.905102 0.905703 0.904729 0.998539 0.999552 0.998023 1.000133 1.664114 1.660910 y1 0.133600 0.116533 0.113983 0.118021 0.118374 0.115666 0.119727 0.115373 0.045969 0.044743 y2 0.096036 0.085912 0.083969 0.087046 0.083316 0.081390 0.084277 0.086432 0.073851 0.071855 y3 0.062064 0.057308 0.056014 0.058063 0.060385 0.059008 0.061072 0.058736 0.091462 0.088992 y4 0.069588 0.064842 0.063333 0.065722 0.071536 0.069855 0.072375 0.072356 0.119219 0.115978 w1 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 w2 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 w3 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 w4 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 L1 0.454669 0.434583 0.434584 0.434583 0.514698 0.514490 0.514799 0.515718 0.812361 0.812014 L2 0.467847 0.434119 0.434124 0.434116 0.425300 0.425268 0.425315 0.425393 0.590777 0.589841 L3 0.491334 0.441690 0.441698 0.441686 0.410600 0.410763 0.410523 0.410324 0.250394 0.248694 L4 0.513433 0.461066 0.461076 0.461060 0.446900 0.447077 0.446816 0.446209 0.166405 0.164049 44 Table 15. Optimal allocations and supporting taxes when tastes are heterogeneous: φ = 0.0260, ρ = 0.7927, ω = 0.10 (monetary Þgures in 100,000 French francs) ULST LITACT LITPDT LITODT GITACT GITPDT GITLDT GITNDT FBADT FB φ/µ 0.093369 0.094385 0.094383 0.094390 0.097097 0.097094 0.097109 0.097111 0.100002 0.100000 τ 0.100935 0.089627 0.090077 0.088527 0.093199 0.093683 0.092038 0.092915 0.099771 0.100000 q1 − 1 0.000000 0.000000 0.094383 -0.222203 0.000000 0.097094 -0.228754 0.097121 0.000000 0.100000 q2 − 1 0.000000 0.000000 0.094383 -0.222203 0.000000 0.097094 -0.228754 -0.240173 0.000000 0.100000 q3 − 1 0.000000 0.000000 0.094383 -0.222203 0.000000 0.097094 -0.228754 0.148378 0.000000 0.100000 q4 − 1 0.000000 0.000000 0.094383 -0.222203 0.000000 0.097094 -0.228754 -0.123229 0.000000 0.100000 t1 0.000000 0.183260 0.179181 0.193329 0.005005 0.000845 0.014394 0.000005 -0.000000 -0.000000 t2 0.000000 0.183260 0.179181 0.193329 0.226155 0.222182 0.235771 0.235720 0.000000 -0.000000 t3 -0.000000 0.183260 0.179181 0.193329 0.178274 0.173470 0.189986 0.173945 -0.000000 0.000000 t4 0.000000 0.183260 0.179181 0.193329 0.113473 0.108469 0.125683 0.118319 0.000000 -0.000000 T1 0.365903 0.057487 0.057751 0.056615 1.210085 1.212223 1.207572 1.240693 4.818766 4.814604 T2 0.365903 0.057487 0.057751 0.056615 -0.006087 -0.006313 -0.005414 -0.004642 1.285934 1.279244 T3 0.365903 0.057487 0.057751 0.056615 -0.085076 -0.084764 -0.085847 -0.089095 -0.850836 -0.859090 T4 0.365903 0.057487 0.057751 0.056615 -0.008103 -0.007522 -0.009539 -0.013202 -1.318983 -1.329899 TP1 0.365903 0.614731 0.612362 0.620784 1.228368 1.225414 1.235634 1.250784 4.818766 4.818809 TP2 0.365903 0.398386 0.398444 0.398274 0.406924 0.406787 0.407299 0.407150 1.285934 1.285948 TP3 0.365903 0.293965 0.293947 0.293951 0.126685 0.126833 0.126319 0.125456 -0.850836 -0.850825 TP4 0.365903 0.285508 0.286398 0.283242 0.126685 0.127840 0.123823 0.118485 -1.318983 -1.319014 EV 1 0.000000 -0.273416 -0.270770 -0.280677 -0.861819 -0.858808 -0.869690 -0.884369 -4.453878 -4.453792 EV 2 0.000000 -0.047127 -0.047154 -0.047476 -0.065209 -0.065041 -0.066023 -0.065615 -0.920823 -0.920757 EV 3 0.000000 0.062121 0.062175 0.061783 0.229141 0.229106 0.228950 0.229976 1.216098 1.216136 EV 4 0.000000 0.071300 0.070356 0.073413 0.235927 0.234777 0.238467 0.244152 1.684280 1.684339 1 x 2.657915 2.320967 2.323802 2.313423 2.319610 2.321696 2.315007 2.312844 0.939731 0.938200 x2 1.539002 1.382893 1.383509 1.381031 1.342692 1.343370 1.340851 1.341290 1.188624 1.186228 x3 1.013062 0.943158 0.943616 0.941822 1.004424 1.005221 1.002288 1.004319 1.480245 1.477312 x4 0.955561 0.897983 0.897768 0.898300 0.993909 0.993845 0.993871 0.997332 1.627965 1.624326 y1 0.120267 0.105021 0.104205 0.107343 0.104959 0.104085 0.107507 0.103687 0.042521 0.042050 y2 0.087814 0.078906 0.078233 0.080805 0.076613 0.075944 0.078521 0.078664 0.067822 0.067043 y3 0.057217 0.053269 0.052816 0.054547 0.056729 0.056250 0.058098 0.055944 0.083603 0.082646 y4 0.064650 0.060754 0.060194 0.062322 0.067244 0.066620 0.069011 0.068369 0.110142 0.108853 w1 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 w2 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 w3 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 w4 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 L1 0.433417 0.419168 0.419119 0.419282 0.503563 0.503323 0.504281 0.505545 0.799679 0.799409 L2 0.452166 0.422093 0.422093 0.422076 0.414388 0.414359 0.414488 0.414586 0.576889 0.576172 L3 0.478036 0.429510 0.429505 0.429486 0.395374 0.395530 0.394997 0.394669 0.237327 0.236036 L4 0.502159 0.450763 0.450805 0.450625 0.430328 0.430497 0.429917 0.429005 0.151840 0.150043 45 Table 16. Optimal allocations and supporting taxes when tastes are heterogeneous: φ = 0.0256, ρ = 0.7927, ω = 0.50 (monetary Þgures in 100,000 French francs) ULST LITACT LITPDT LITODT GITACT GITPDT GITLDT GITNDT FBADT FB φ/µ 0.093382 0.094394 0.094386 0.094389 0.097101 0.097091 0.097100 0.097105 0.100009 0.100000 τ 0.100852 0.089558 0.090063 0.089611 0.093143 0.093659 0.093176 0.093267 0.099747 0.100000 q1 − 1 0.000000 0.000000 0.094386 0.010971 0.000000 0.097091 0.011178 0.096894 0.000000 0.100000 q2 − 1 0.000000 0.000000 0.094386 0.010971 0.000000 0.097091 0.011178 -0.068749 0.000000 0.100000 q3 − 1 0.000000 0.000000 0.094386 0.010971 0.000000 0.097091 0.011178 0.106592 0.000000 0.100000 q4 − 1 0.000000 0.000000 0.094386 0.010971 0.000000 0.097091 0.011178 0.009877 0.000000 0.100000 t1 -0.000000 0.183195 0.178575 0.182766 0.004844 0.000778 0.004713 0.000607 0.000000 0.000000 t2 0.000000 0.183195 0.178575 0.182766 0.225437 0.221502 0.225524 0.228790 0.000000 0.000000 t3 0.000000 0.183195 0.178575 0.182766 0.178236 0.172966 0.177989 0.173469 0.000000 -0.000000 t4 0.000000 0.183195 0.178575 0.182766 0.113399 0.107912 0.113188 0.112339 0.000000 -0.000000 T1 0.365742 0.057487 0.057987 0.057350 1.209750 1.210671 1.209223 1.222817 4.806153 4.801648 T2 0.365742 0.057487 0.057987 0.057350 -0.004831 -0.005894 -0.005902 -0.005141 1.281472 1.274165 T3 0.365742 0.057487 0.057987 0.057350 -0.085045 -0.084745 -0.085553 -0.086645 -0.862309 -0.871338 T4 0.365742 0.057487 0.057987 0.057350 -0.008041 -0.007451 -0.008611 -0.010046 -1.305966 -1.317955 TP1 0.365742 0.614454 0.611675 0.614296 1.227440 1.224610 1.227764 1.236094 4.806153 4.806285 TP2 0.365742 0.398207 0.398247 0.398234 0.406969 0.406674 0.406884 0.406538 1.281472 1.281522 TP3 0.365742 0.293842 0.293878 0.293794 0.126637 0.126758 0.126501 0.125805 -0.862309 -0.862254 TP4 0.365742 0.285385 0.286417 0.285453 0.126637 0.127855 0.126632 0.123973 -1.305966 -1.306076 EV 1 0.000000 -0.273120 -0.269778 -0.272904 -0.860955 -0.857807 -0.861213 -0.869729 -4.441562 -4.441102 EV 2 0.000000 -0.046968 -0.046768 -0.046969 -0.065159 -0.064739 -0.065172 -0.064744 -0.916631 -0.916307 EV 3 0.000000 0.062194 0.062382 0.062267 0.229096 0.229271 0.229235 0.229794 1.227323 1.227496 EV 4 0.000000 0.071372 0.070438 0.071317 0.235880 0.234813 0.235892 0.238643 1.671016 1.671267 1 x 2.641028 2.306336 2.313743 2.306995 2.305350 2.311082 2.305401 2.306867 0.938574 0.938604 x2 1.526847 1.372020 1.375821 1.372369 1.332635 1.336044 1.332575 1.329856 1.180875 1.180979 x3 1.005160 0.935815 0.938405 0.936076 0.996559 0.999662 0.996755 0.999680 1.472980 1.473131 x4 0.946743 0.889704 0.891810 0.889916 0.984721 0.987266 0.984853 0.986292 1.607687 1.608130 y1 0.136845 0.119503 0.114600 0.118887 0.119452 0.114327 0.118792 0.114129 0.048632 0.046370 y2 0.099763 0.089647 0.085931 0.089182 0.087073 0.083344 0.086587 0.090042 0.077157 0.073573 y3 0.065009 0.060524 0.058015 0.060212 0.064453 0.061726 0.064108 0.061462 0.095265 0.090841 y4 0.073348 0.068929 0.066046 0.068571 0.076291 0.073025 0.075878 0.076038 0.124555 0.118791 w1 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 w2 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 w3 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 w4 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 L1 0.433352 0.419109 0.419071 0.419093 0.503467 0.503161 0.503428 0.504135 0.798623 0.798334 L2 0.452083 0.422022 0.422051 0.422002 0.414489 0.414350 0.414346 0.414435 0.576237 0.575458 L3 0.477946 0.429439 0.429478 0.429406 0.395311 0.395477 0.395216 0.395078 0.234972 0.233568 L4 0.502057 0.450681 0.450773 0.450652 0.430259 0.430439 0.430156 0.429772 0.154430 0.152463 46 Table 17. Optimal allocations and supporting taxes when tastes are heterogeneous: φ = 0.0248, ρ = 0.7927, ω = 0.99 (monetary Þgures in 100,000 French francs) ULST LITACT LITPDT LITODT GITACT GITPDT GITLDT GITNDT FBADT FB φ/µ 0.093396 0.094401 0.094383 0.094390 0.097108 0.097089 0.097098 0.097100 0.100021 0.100000 τ 0.100783 0.089466 0.090046 0.089794 0.093031 0.093650 0.093384 0.093398 0.099715 0.100000 q1 − 1 0.000000 0.000000 0.094383 0.051659 0.000000 0.097089 0.052291 0.097108 0.000000 0.100000 q2 − 1 0.000000 0.000000 0.094383 0.051659 0.000000 0.097089 0.052291 -0.001731 0.000000 0.100000 q3 − 1 0.000000 0.000000 0.094383 0.051659 0.000000 0.097089 0.052291 0.101827 0.000000 0.100000 q4 − 1 0.000000 0.000000 0.094383 0.051659 0.000000 0.097089 0.052291 0.048257 0.000000 0.100000 t1 -0.000000 0.183665 0.178292 0.180610 0.006217 0.000737 0.002484 0.000001 -0.000000 -0.000000 t2 0.000000 0.183665 0.178292 0.180610 0.226030 0.220932 0.223223 0.226097 -0.000000 -0.000000 t3 0.000000 0.183665 0.178292 0.180610 0.178782 0.172532 0.175316 0.172907 0.000000 0.000000 t4 -0.000000 0.183665 0.178292 0.180610 0.113997 0.107448 0.110367 0.110305 -0.000000 0.000000 T1 0.365537 0.056639 0.057493 0.057133 1.205169 1.208826 1.209696 1.219063 4.798694 4.793752 T2 0.365537 0.056639 0.057493 0.057133 -0.005919 -0.005904 -0.005988 -0.006089 1.278141 1.270009 T3 0.365537 0.056639 0.057493 0.057133 -0.085898 -0.085058 -0.085472 -0.086488 -0.878301 -0.888377 T4 0.365537 0.056639 0.057493 0.057133 -0.009010 -0.007756 -0.008358 -0.009662 -1.293488 -1.306935 TP1 0.365537 0.614868 0.611507 0.612952 1.227866 1.223959 1.225733 1.231502 4.798694 4.798959 TP2 0.365537 0.398076 0.398089 0.398083 0.406727 0.406508 0.406598 0.406473 1.278141 1.278242 TP3 0.365537 0.293460 0.293581 0.293532 0.126285 0.126572 0.126435 0.125728 -0.878301 -0.878180 TP4 0.365537 0.284981 0.286215 0.285684 0.126285 0.127789 0.127115 0.125315 -1.293488 -1.293714 EV 1 0.000000 -0.273626 -0.269234 -0.270979 -0.861433 -0.856774 -0.858737 -0.864873 -4.434513 -4.433448 EV 2 0.000000 -0.046928 -0.046352 -0.046493 -0.065140 -0.064309 -0.064569 -0.064305 -0.913672 -0.912920 EV 3 0.000000 0.062471 0.062879 0.062772 0.229273 0.229623 0.229556 0.230002 1.242979 1.243382 EV 4 0.000000 0.071673 0.070795 0.071253 0.236085 0.235005 0.235580 0.237421 1.658202 1.658775 1 x 2.617342 2.284754 2.299104 2.292896 2.282951 2.296511 2.291230 2.294440 0.932963 0.935427 x2 1.509862 1.356319 1.364958 1.361217 1.317284 1.325863 1.322036 1.317230 1.168924 1.172876 x3 0.994112 0.925307 0.931134 0.928610 0.985308 0.992044 0.989054 0.992250 1.462701 1.467624 x4 0.934455 0.877965 0.883588 0.881152 0.971667 0.978298 0.975355 0.975897 1.581817 1.588451 y1 0.160111 0.139766 0.128629 0.133440 0.139655 0.128170 0.133264 0.128052 0.057072 0.052070 y2 0.116471 0.104627 0.096298 0.099896 0.101615 0.093312 0.096963 0.101786 0.090171 0.082328 y3 0.075907 0.070653 0.065024 0.067456 0.075234 0.069109 0.071804 0.068829 0.111686 0.101971 y4 0.085472 0.080305 0.073915 0.076675 0.088875 0.081638 0.084822 0.085193 0.144684 0.132207 w1 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 7.254181 w2 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 4.407053 w3 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 3.004338 w4 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 2.760310 L1 0.433266 0.418984 0.418964 0.418971 0.503223 0.502971 0.503189 0.503709 0.797985 0.797672 L2 0.451973 0.421829 0.421902 0.421868 0.414251 0.414264 0.414244 0.414220 0.575722 0.574862 L3 0.477827 0.429186 0.429292 0.429245 0.395038 0.395337 0.395193 0.395031 0.231694 0.230139 L4 0.501923 0.450403 0.450572 0.450497 0.429962 0.430287 0.430130 0.429809 0.156871 0.154672 47