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Constitutional and postconstitutional taxation
European Journal of Political Economy Vol. 18 (2002) 615 – 630 www.elsevier.com/locate/econbase
Constitutional and postconstitutional taxation Thomas Eichner * Department of Economics, University of Siegen, Hoelderlinstr. 3, D-57068 Siegen, Germany Received 2 August 1999; received in revised form 12 November 2001; accepted 3 January 2002
Abstract This paper investigates voting on linear income taxes behind a veil of ignorance over the postconstitutional situation, and voting on linear income taxes by majority rule when the veil has been lifted in a model of endogenous labour supply. Numerical simulations show that the constitutional tax rate (CTR) is positively correlated with the standard deviation of the pretax income distribution, whereas the main determinant of the postconstitutional tax rate (PCTR) is the difference between the median voter’s wage and the wages of members of other population groups. D 2002 Elsevier Science B.V. All rights reserved. JEL classification: H21; J21 Keywords: Voting; Taxation; Constitutional; Postconstitutional choice
1. Introduction Two different approaches have been used to study the implementation of linear income tax rates. The first uses the concept of constitutional choice based on a welfare economics framework. The second is based on the postconstitutional choice, also known as political choice, and uses a political economics framework. Both approaches have a long tradition and have been applied to various investigations. Here, we argue along the line of a framework in which agents differ with regard to their productivity. In the first step, individuals decide on labour supply and in the second step, on taxation. The government implements a linear income tax that is indirectly progressive through redistribution by equal lump-sum transfers. In case of constitutional choice, decisions on taxation take place under uncertainty. If we consider constitutional choice as a normative concept, then individuals make their *
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decision behind a veil of ignorance, that is, as if they do not know their productivity and the position they will have in society after taxation has been chosen. An alternative interpretation of constitutional choice is that agents determine the income taxation for their children and grandchildren. Then, uncertainty consists in the genetically caused abilities of their (unborn) children that play a decisive role for the childrens’ real living standards. From this point of view, constitutional choice is a positive approach. Harsanyi (1955) and Rawls (1971), who were the first to investigate constitutional choice, predicted that selfinterested agents would unanimously choose constitutional rules without being influenced by their particular future interests. In such approach, taxation has the function of a social insurance. The state, as social insurer, claims a part of uncertain pretax income and offers in return a safe payment. Technically speaking, from the ex ante perspective with regard to risk, taxation reduces the variance and the dispersion of posttax incomes. This has been well illustrated by Sinn (1995, 1996). In their labour supply model, Eaton and Rosen (1980) analysed voting on the tax rate at the constitutional level, and pointed out the effects of a change in tax rate on labour supply. Under the assumption that agents do not take the balanced fiscal budget into account, they found that the tax rate is between the laissez-faire tax and the confiscatorial tax. Labour supply changes, relative to the tax rate, are ambiguous. Unfortunately, these are very weak statements. The second approach is the postconstitutional choice. If uncertainty has been resolved, agents are in a decision situation under certainty, which we call the postconstitutional phase. Individuals differ ex post with respect to their productivity, which ex ante is a random parameter and causes uncertainty. Clearly, agents with different abilities to earn money favour different tax rates, so that an agent’s preferred tax rate depends on his or her productivity. This model under certainty was formulated by Romer (1975), Roberts (1977), and Meltzer and Richards (1981). Since decisions are no longer unanimous, the government or the agents must search for a suitable implementation procedure for taxation. In democracies, a well-accepted procedure is majority rule. Along with other authors, Romer (1975, 1977) shows that by applying majority rule, the society implements the tax rate that is the first-best choice of the voter with the median pretax income. Under the mild requirement of hierarchical adherence (see Roberts, 1977, Lemma 1), agents with higher pretax income favour less redistribution, whereas agents with pretax income below the median pretax income favour more redistribution than the redistribution that results from majority voting. Meltzer and Richards (1981) show that the preferred tax rate of the median income taker is less than the confiscatorial tax, which is also a weak result. Having sketched both approaches, it is obvious how closely related these approaches are. Both provide justifications for taxation, but for different reasons in each case. The purpose of this paper is to contribute to both approaches. With help of the benchmark cases: exogenous labour supply and risk-neutrality, and with help of numerical simulations, we point out the main effects, which drive constitutional and postconstitutional tax rates. Use of a parametric utility function and simple discrete wage rate distributions has the advantage over the more general models of Eaton and Rosen (1980) and Meltzer and Richards (1981) that more clear-cut result can be obtained.1
1
Unfortunately, these results are coupled with a loss of generality.
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Before presenting the model, a remark is in order with respect to the timing of decisions. In case of postconstitutional choice, productivities are known and decisions take place under certainty. First, the tax rate is determined by majority rule, and then individuals choose labour supply. By contrast, constitutional tax rates are chosen behind a veil of ignorance where, strictly speaking, agents face uncertainty about their type. In the next stage, the veil of ignorance lifts, productivities become known, and then agents sign labour supply contracts.2 Under both regimes, tax rates are found by backward induction, and under both regimes at the last stage, we have to determine labour supply decisions in the presence of type uncertainty and for given tax rates. That is the reason why Section 2 outlines the model and the labour supply decisions. In Subsection 3.1 we introduce constitutional choice, and in Subsection 3.2 we investigate postconstitutional choice. Subsection 3.3 presents simulation results and a comparison of the tax rates. The paper ends with a summary and a concluding remark in Section 4.
2. The model and labour supply decisions We consider the society as consisting of three different types of individuals. A type i individual is characterised by (exogenous) productivity or wage rate. For simplicity, wage rates are assumed to be discretely distributed with realisations wi and relative P cumulations pi for i = 1, 2, 3. A standard requirement of relative cumulations is that 3i¼1 pi ¼ 1. One possible interpretation of three realisations is that the society is composed of a lower class, a middle class, and an upper class. Each member of group i determines his or her labour effort by solving max U ðyi ; S i Þ s:t: yi ¼ ð1 tÞwi S i þ p: S
ð1Þ
i
The posttax income yi of a person of group i is completely spent on consumption. It is composed of the pretax income wiS i, that is linearly taxed at rate t and finances a lumpsum transfer p. Agents’ tastes are identical and are represented by the additive separable utility function u ¼ U ðy; S Þ ¼ ay by2 cS
2
ð2Þ
with the partial derivatives Uy>0, Uyy V 0, US V 0, US S V 0, and with a>0, b z 0, c z 0.3 The parameters a, b and c take the same values for the three groups i = 1, 2, 3. Individuals are price takers in the labour market and they consider t and p as exogenously given
2 It should be noted that Eaton and Rosen (1980) use a different timing: Agents choose labour supply before the wage rate is known. Eaton and Rosen’s timing is motivated by the assumption that output is a stochastic rather than a deterministic function of labour input, which is an assumption on production technology. However, Eaton and Rosen’s timing for constitutional choice makes no sense in a paper that aims at comparing constitutional and postconstitutional taxation. 3 The property Uy>0 implies that we have to ensure that a>2by.
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information. Taking Eq. (2) into account, we obtain the following first- and second-order condition of the optimisation problem (1) ð1 tÞða 2pbÞwi 2S i c 2S i ð1 tÞ2 bw2i ¼ 0;
ð3Þ
2c 2ð1 tÞ2 bw2i < 0:
ð4Þ
Rearranging Eq. (3) yields group i’s labour effort 9 8 > = < ð1 tÞða 2pbÞwi for ð1 tÞða 2pbÞwi > 0 > 2c þ 2ð1 tÞ2 bw2i ¼: Li ðt; pÞ: Si¼ > > ; : 0 for ð1 tÞða 2pbÞwi V0
ð5Þ
Eq. (5) shows that individuals are voluntarily unemployed if (1 t)wi(a 2pb) V 0. An increase in p may have the consequence that agents choose to subsist on the transfer, which in this case can be interpreted as welfare payment. Since t, p, a, b and c are the same for all agents, the group with the lowest wage rate (defined as w1) has the greatest incentive to exploit the other groups. For the time being, the labour effort Li(t, p) depends on both the exogenous tax rate and the transfer. Procedure (1) implies that individuals do not take the fiscal budget constraint into account. The state offers taxes and transfers with the assumption that the fiscal budget constraint is satisfied. Then, a government’s payments have to equal its receipts, that is, in formal terms p¼t
3 X
pi wi Li ðt; pÞ:
ð6Þ
i¼1
Combining Eqs. (5) and (6) and solving with respect to p, we obtain for the case where all citizens work, tð1 tÞa
3 X
pi w2i
2c þ 2ð1 tÞ2 bw2i 3 X pi w 2
i¼1
p¼
1 þ 2tð1 tÞb
ˆ ¼: PðtÞ
ð7Þ
i
i¼1
2c þ 2ð1 tÞ2 bw2i
which in turn inserted into Eq. (5), yields ˆ Li ðt; PðtÞÞ ¼
"
ð1 tÞawi
½2c þ 2ð1 tÞ2 bw2i 1 þ 2tð1 tÞb
3 X
pi w2i
i¼1
2c þ 2ð1 tÞ2 bw2i
# ¼: Lˆ i ðtÞ:
ð8Þ The function Lˆi(t) is called a labour supply function and specifies the group i’s labour effort, presupposing that the fiscal budget is balanced.
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Proposition. The labour supply function has the following properties: (a) Lˆi(t)>0 for 0 V t < 1, and Lˆi(1) = 0. (b) Suppose b = 0, c>0, then Lˆi(t)=(1 t) awi/(2c). (c) Suppose b>0, c = 0, then Lˆi(t) = a/(2bwi). (d) Suppose b>0, c>0, then 8 9 8 9 >> ><> > > > > > > > !> > > > > 3 = = < > < X pi w 2 i 2 2 i 0: ¼ lim Lˆ t ðtÞ ¼ 0Zc bwi þ bð2c þ 2bwi Þ t!0 > > > 2c þ 2bw2i > > > > > i¼1 > > > > > > ; ; : > : > > <
and lim Lˆ it ðtÞ < 0: t!1
Unfortunately, the slope of the labour supply, function in general, is not determinate in sign. Lˆi(t) decreases in the tax rate if agents have risk neutral preferences (b = 0, see proposition (b)), and it is independent of t if agents do not have disutility from labour (c = 0, compare proposition (c)). In addition, proposition (d) establishes conditions that are necessary and sufficient for the labour supply function to decrease or to increase at t ! 0, and shows that taxation induces agents to reduce labour supply if the tax rate converges to the confiscatorial rate. Figs. 1 –3 illustrate labour supply and posttax income of the different groups for the numerical specification a = 10, b = 1, c = 2, p1 = p2 = p3 = 1/3, w1 = 2, w2 = 2.5, w3 = 4. First of all, at t = 0 groups with less productivity supply less labour than groups with higher
Fig. 1. Labour supply.
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Fig. 2. Labour supply of group 3.
productivity (Lˆ1(0)>Lˆ2(0)>Lˆ3(0)), as Fig. 1 shows. While the labour supply functions of groups 1 and 2 are strictly decreasing in the tax rate, group 3 raises its labour supply if the initial tax rate is not too high, in order to compensate for the income loss caused by redistribution (compare Fig. 2). This results in labour supplies characterised by Lˆ3(t)>Lˆ2(t)>Lˆ1(t) for 0.65 V t < 1. Clearly, it is possible to find other curvatures of labour supply functions, but these cases can be interpreted in a similar way. However, focusing on posttax incomes, Fig. 3 shows decreasing posttax income curves caused by the reduction of the labour supply of groups 1 and 2. Increasing taxation redistributes the additional pretax income of group 3 that emerged from rising labour effort in favour of the lower and middle class. In every case, group 1 benefits from redistribution, as does group 2 in some cases.
Fig. 3. Posttax income.
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3. Constitutional and postconstitutional taxation 3.1. Constitutional taxation We now consider the choice of fiscal policy parameters at the constitutional level. Decisions are taken behind a veil of ignorance. In the constitutional phase, individuals choose the tax rate without knowing their future incomes, that is, without knowing their productivity. An individual determines his or her preferred tax rate by solving max t
3 X
ˆ pi U ðyi ; Lˆ i ðtÞÞ s:t: yi ¼ ð1 tÞwi Lˆ i ðtÞ þ PðtÞ:
ð9Þ
i¼1
The first-order condition is 3 X
pi Uyi ðyi ; Lˆ i ðtÞÞ½wi Lˆ i ðtÞ þ Pˆ t ðtÞ ¼ 0:
ð10Þ
i¼1
For the numerical example provided in Section 2, the constitutional tax rate (CTR) is 0.204072. The total effect of taxation can be divided into four partial effects. Suppose t < 0.4. Then, taxation (i) reduces the expected labour supply (see Fig. 4), (ii) reduces the standard deviation of labour supply (see Fig. 5), (iii) reduces the expected posttax income (see Fig. 6), and (iv) increases the standard deviation of posttax income4 (see Fig. 7). The partial effects5 (i) and (ii) are welfare-enhancing, whereas the partial effects (iii) and (iv) are welfarediminishing. From Fig. 8, we conclude that effects (i) and (ii) overcompensate the effects (iii) and (iv) for tax rates smaller than the constitutionally chosen rate (t = 0.204072).6 Neither we nor the computer programme ‘Mathematica’ are able to provide the expected utility maximising tax rate as a function of the exogenous parameters a, b, c, pi, wi, but, as shown in the examples of Subsection 3.3, we are able to calculate the expected utility maximising tax rates for given values of exogenous parameters. Before presenting the simulation results, we focus on two benchmark cases, namely exogenous labour supply and risk-neutrality.7 4 Effect (iv) is not inconsistent with effect (ii), since in calculating the standard deviation of posttax income, it has to be taken into account that labour supplies are multiplied by wage rates. It can be shown that the standard deviation of wages (wi Lˆi(t)) is increasing in t if t < 0.4. 5 Observe that partial effects are isolated effects. For example, in the partial effect of labour supply, we ignore the impact of labour supply changes on posttax incomes. Then, the reduction of expected labour supply is welfare-enhancing due to US < 0. 6 Eaton and Rosen (1980) demonstrate that the tax on wage serves partially to insure the individual against random wage movements, with the consequence that lump-sum taxation is not efficient. Interestingly, in our numerical example, initial standard deviations of wages and posttax incomes are increasing in the tax rate. If t>0.5, standard deviations of wages and posttax incomes are decreasing in t so that we are able to apply Eaton and Rosen’s (1980) argument with respect to lump-sum taxation, although their timing of decisions is different from our timing. 7 The reader may find the case where labour causes no disutility interesting. However, it can be shown that in this case the groups’ pretax incomes are identical, which implies that individuals are indifferent between all tax rates. This, in my view, seems to be a very atypical case that is neglected in what follows.
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Fig. 4. Expected labour supply.
3.1.1. Exogenous labour supply Taking _ labour supply as exogenously given information_ where labour is assumed to be fixed (S = const.), individuals’ Ppretax income is xi(: = wiS ). For notational convenience, we introduce the mean l :¼ 3i¼1 pi wi and the standard deviation vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u 3 3 uX X pi w2i pi w i r :¼ t i¼1
i¼1
of the wage rate distribution. Then, the posttax income can be written as yi ¼ wi S þ tS ðl wi Þ:
ð11Þ
The decisions with respect to labour supply are dropped and agents directly choose the tax rate that maximises their expected utility. Taking into account Lˆi(t) = S ¯, Pˆt(t) = lS ¯and Eq. (11), the left-hand side of condition (10) becomes 3 X
pi ½a 2b½wi S þ tS ðl wi Þðwi S þ lS Þ ¼: AðtÞ:
ð12Þ
i¼1
P P Using 3i¼1 pi aðwi S þ lS Þ ¼ aS ðl 3i¼1 pi wi Þ ¼ 0 and the definition of r, we obtain after some rearrangements 2
AðtÞ ¼ 2bS r2 ð1 þ tÞ:
ð13Þ
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Fig. 5. Standard deviation of labour supply.
Since A(t) is increasing in t, the preferred tax rate is t = 1. The reason for this result is that redistributive taxation provides social insurance and increasing the tax rate reduces the _ standard deviation of posttax incomes (1 t)rS , which is utility-enhancing for risk-averse agents and induces them to choose the confiscatorial tax rate. In view of the distribution, the constitutional choice with exogenous labour supply results in an egalitarian income distribution.
Fig. 6. Expected posttax income.
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Fig. 7. Standard deviation of posttax income.
3.1.2. Risk-neutrality Now, we turn to the case in which the marginal returns from income are constant, or in formal terms b = 0, which means that agents are no longer risk-averse but risk-neutral. As can be seen in proposition (b), labour supply simplifies to Lˆi(t)=(1 t)awi/(2c), and the transfer to Pˆ(t) = t(1 t)a(l2 + r2)/(2c). Lˆi(t) and Pˆt(t) inserted into the left-hand side of first-order condition (10) for b = 0, we obtain
3 X i¼1
pi
a2 ½ð1 2tÞðl2 þ r2 Þ ð1 tÞw2i 2c
which can be manipulated to a2 ðl2 þ r2 Þt: 2c
ð14Þ
ð15Þ
(15) shows that if agents are risk-neutral, then the preferred tax rate is zero. It can be shown that taxation reduces expected labour supply, expected posttax income, the standard deviation of labour supply, and the standard deviation of posttax income. Due to riskneutrality, agents do not benefit from the insurance and redistribution function of taxation. They judge taxation to be welfare diminishing, since it is a disincentive to labour supply and thus, reduces posttax incomes. 3.2. Postconstitutional taxation In contrast to constitutional choice, postconstitutional choice is not made unanimously. The veil of ignorance concerning the wage rate has lifted, and each individual knows that he or she belongs to one of the three groups of productivity. On the postconstitutional level, individuals establish tax rates by means of majority voting. The application of
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majority voting implies that the tax rate chosen is the one preferred by the median voter. Formally, the postconstitutional tax rate (PCTR) is determined by ˆ ð16Þ max U ðym ; Lˆ m ðtÞÞ s:t: ym ¼ ð1 tÞwm Lˆ m ðtÞ þ PðtÞ t
with the first-order condition Uym ðym ; Lˆ m ðtÞÞ½wm Lˆ m ðtÞ þ Pˆ t ðtÞ ¼ 0
ð17Þ
where wm denotes the median wage rate and Lˆm(t) the labour supply of the voter with the median income. For the numerical example outlined in Section 2, members of group 2, who are the median voters, favour low tax rates. Their best choice is t = 0.094144, which is the tax rate implemented through majority voting. Group 3, the rich group, favours no redistribution at all, whereas the poor group favours redistribution (its preferred tax rate is t = 0.471681). Analogous to Subsection 3.1, we will examine the benchmark cases of exogenous labour supply and risk neutrality. 3.2.1. Exogenous labour supply _ In the case of exogenously given labour8 (S = S ), the left-hand side of Eq. (17) becomes Uym ðym ; S ÞðlS wm S Þ ¼: BðtÞ:
ð18Þ
Due to B(t)>0 if l>wm, B(t) < 0 if l < wm and Bt = 0, we obtain the ‘bang-bang’ solutions t = 0 and t = 1. This result can be traced back to Foley (1967) _ and Roberts (1977). The median voter equilibrium is t = 0 if the mean pretax income (lS ) is smaller than the median _ voter income (wmS ), which implies that the median voter looses from redistribution. Conversely, the median voter prefers t = 1 if his pretax income is smaller than the average pretax income and thus, he profits from redistribution. Whether wm>l or wm < l depends on the curvature of the wage rate distribution. For right-skewed distributions, we have wm>l, and for left-skewed distributions, which are the empirically relevant cases, we have wm < l. 3.2.2. Risk-neutrality If agents are risk-neutral, labour supply is given by Lˆi(t)=(1 t) awi/(2c), the transfer is given by Pˆ(t) = t(1 t) a (l2 + r2)/(2c) and the left-hand side of condition (17) turns into a2 ½ð1 2tÞðl2 þ r2 Þ ð1 tÞw2m ¼: CðtÞ: 2c
ð19Þ
The preferred tax rate depends on the relation of (l2 + r2) to wm2 . If l2 + r2>wm2 , which is the relevant case from an empirical point of view, we have C(0)>0, C(1/2) < 0 and Ct(t) < 0, 8 A remark is in order to the related work of Meltzer and Richards (1981). They use a more general model encompassing our algebraic example, but in elaborating the determinants of tax rates they gear to pretax incomes. They do not consider the labour supply effects of taxation. Their main result is: ‘‘Using the common economic assumption that elasticities are constant, the tax rate rises as mean income rises relative to the income of the decisive voter.’’ This statement is fully compatible with our results, which are presented in the next subsections, especially in Subsection 3.3, but our contribution is to explain how pretax incomes come about. Starting from wage rates, we are interested in the groups’ labour supplies, which determine the groups’ pretax incomes.
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Fig. 8. Expected utility.
which ensure that the median voter equilibrium lies in the interval ta[0,1/2]. If l2 + r2 < wm2 , it holds that C(t) < 0 for all t and the corner solution t = 0 yields the maximal benefit. However, concentrating on l2 + r2>wm2 , the equation C(t) = 0 can be solved for t, which provides t¼
l2 þ r2 w2m : 2ðl2 þ r2 Þ w2m
ð20Þ
Comparative static analysis reveals that tl ¼ tr ¼
2lw2m ½2ðl2 þ r2 Þ
w2m 2
2rw2m ½2ðl2
þ r2 Þ w2m 2
;
t wm ¼
2wm ðl2 þ r2 Þ ½2ðl2 þ r2 Þ w2m 2
;
:
ð21Þ
The foregoing results are intuitively plausible. Keeping in mind that wm < l, a reduction in the difference l wm reduces the median voter’s net gain from redistribution and is responsible for lower tax rates (t( l) < 0 and twm < 0). The positive correlation of the standard deviation with the preferred tax rate shows that the median voters’ profits caused by redistribution from higher income takers overcompensate the median voters’ losses caused by redistribution to lower income takers. 3.3. Simulation results and comparison In this subsection, especially in Table 1, we report on constitutional and postconstitutional tax rates with endogenous labour supply and risk-averse agents. Table 1 is divided into five blocks. The utility function parameters underlying the simulations9 of rows 1– 5 9
The computer programme for calculations is available from the author upon request.
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Table 1 CTRs and PCTRs for w1 = 2, w3 = 4, a = 10, b = 1, c = 2 l˜
Parameter
p1
p2
p3
w2
l
r
1 2 3 4 5
1/3 1/3 1/3 1/3 1/3
1/3 1/3 1/3 1/3 1/3
1/3 1/3 1/3 1/3 1/3
2.0 2.1 2.5 3.0 3.6
2.67 2.70 2.83 3.00 3.20
0.943 0.920 0.850 0.816 0.864
3.70 3.74 3.86 3.96 4.04
1/3 1/3 1/3 1/3
1/3 1/3 1/3 1/3
1/3 1/3 1/3 1/3
2.1 2.1 2.1 2.1
2.70 2.70 2.70 2.70
0.920 0.920 0.920 0.920
1/3 1/3 1/3 1/3 1/3
1/3 1/3 1/3 1/3 1/3
1/3 1/3 1/3 1/3 1/3
2.1 2.5 2.8 5.0 6.0
4.03 4.17 4.27 5.00 5.33
15 16 17 18
0.45 0.45 0.45 0.45
0.40 0.40 0.40 0.40
0.15 0.15 0.15 0.15
2.1 2.5 3.0 3.5
19 20 21 22
0.65 0.65 0.65 0.65
0.30 0.30 0.30 0.30
0.05 0.05 0.05 0.05
2.1 2.5 3.0 3.5
6 7 8 9
b=0 b = 0.1 b = 0.5 b=1
10 11 12 13 14
w3 = 8 w3 = 8 w3 = 8 w3 = 8 w3 = 8
r˜
CTR
PCTR
0.524 0.501 0.456 0.463 0.499
0.230335 0.218584 0.204072 0.232262 0.291899
0.426558 0.366583 0.094144 0 0
20.34 13.20 6.08 3.74
13.907 6.389 1.360 0.501
0 0.107736 0.200326 0.218584
0.314133 0.350297 0.381615 0.366583
2.805 2.718 2.660 2.449 2.494
3.87 4.00 4.06 4.27 4.31
0.690 0.635 0.621 0.669 0.690
0.725846 0.686710 0.657267 0.670457 0.700212
0.799246 0.764711 0.186675 0 0
2.34 2.50 2.70 2.90
0.699 0.671 0.714 0.831
3.54 3.68 3.80 3.89
0.382 0.383 0.441 0.502
0.002070 0.008479 0.056175 0.122216
0.109638 0 0 0
2.13 2.25 2.40 2.55
0.431 0.461 0.583 0.757
3.42 3.53 3.62 3.68
0.240 0.295 0.392 0.471
0 0 0 0
0.082341 0.183754 0.284727 0.370259
are a = 10, b = 1, c = 2 and the wage rate distribution is assumed to be uniformly distributed with w1 = 2, w3 = 4, with the median voter’s wage rate w2 varied. In columns 6 and 7, we have calculated the mean wageP rate l and the standard deviation of wage rates r. Columns 8 and 9 show the mean l˜ :¼ 3i¼1 pi wi Lˆ i ð0Þ and standard deviation vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u 3 3 uX X pi w2i Lˆi ð0Þ2 pi wi Lˆi ð0Þ r˜ :¼ t i¼1
i¼1
of pretax incomes, and columns 10 and 11 show the associated tax rates. Comparing CTRs with PCTRs demonstrates that the conjecture that PCTRs are generally lower than CTRs is false. At first glance, it is conspicuous that PCTRs are very sensitive to changes of the wage rate of the group with median income. In contrast, CTRs seem to be more robust to changes of w2. Upon closer examination, rows 1 –5 reveal that, with the exception of one case, higher standard deviations emerge with higher CTRs, whereas increasing means l and l˜ does not necessarily result in higher CTRs. Comparing r and CTRs show that the direction of changes of r is not always in accordance with the
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direction of CTR changes. More specifically, if r is diminished from 0.943 to 0.850, the CTR decreases from 0.230335 to 0.204072, but further diminishing r to 0.816 implies an increase in CTR to 0.232262. It is interesting to observe that, although r is reduced from 0.850 to 0.816, labour is supplied such that r˜ increases from 0.456 to 0.463. Comparing r˜ and CTRs demonstrates that the direction of changes of the pretax income standard deviation tallies with the direction of CTR changes. Focusing on PCTRs, we found that the greater the difference between the median voter’s wage rate and the poor group’s wage rate, the lower the PCTR, and the greater the difference between the median voter’s wage rate and the rich group’s wage rate, the higher the PCTR. On the other hand, group 2 benefits from transfers of group 3, and on the other hand, group 2 is placed in disadvantage by payments to group 1. The difference between the wage rate of group 2 and the wage rates of the other groups indicates whether this group is a net-tax payer or, rather, a netsubsidy receiver. A net-tax payer prefers a low PCTR while a net-subsidy receiver prefers a high PCTR. In the next block, rows 6– 9, we investigate the influence of the risk-aversion parameter b, which is now successively raised from 0 to 1. All other parameters remain unchanged. Obviously, reducing the risk-aversion diminishes the CTR.10 This observation is in accordance with the fact that risk-neutral agents prefer zero tax rates. In addition, from l˜ and r˜ , we infer that reducing risk-aversion induces individuals to supply more labour; both l˜ and r˜ increase, while l and r remain constant. In summary, we have two effects: on the one hand, reducing b increases labour supply, which increases r˜ . According to our argument above, this would imply higher CTRs. On the other hand, reducing b implies that agents benefit less from the insurance function of taxation (from reducing r˜ ). The fact that CTRs are increasing in b led us to conclude that the second effect overcompensates for the first effect. Now, consider PCTRs. In all of the examples, the median voter’s wage rate is 2.1. Keeping in mind that l2 + r2>wm2 , we can apply the formula in Eq. (20) and obtain for a risk-neutral society (b = 0), the tax rate t = 0.314133. As can be seen in rows 6– 9, for small b’s the PCTR increases in b, and for large b’s, it decreases in b. The changes of labour supply are responsible for the development of PCTRs. Increasing b has the effect that agents supply less labour. Especially if b < 0.5, the labour supply of group 2 approximates to the labour supply of group 1, whereas group 3 offers significantly more labour than group 2. If b>0.5, increasing b has the effect that the labour supply of group 3 falls under the labour level of group 2, with the consequence that the net-payments to group 2 diminish and group 2 prefers less redistribution. In rows 10 – 14, we study the effects of increasing w3. The simulations show that CTRs at w3 = 8 are higher than those at w3 = 4, which can be explained by reduced standard deviations r and r˜ . Comparing the values of r, r˜ , and CTR, again r˜ instead of r is the corresponding measure for CTRs. Both in rows 10 and 14, r˜ takes the value 0.690. Since at w2 = 2.1, CTR is higher than the CTR at w2 = 6.0, one could presume that for given r˜ , 10 The effect of risk-aversion on CTRs has also been studied by Varian (1980) using an algebraic example involving quadratic utility. His model, however, is different from mine. It is a two-period model with exogenous and partly uncertain incomes where agents decide on the allocation of saving and consumption in the two periods. However, in both models the impact of risk-aversion on the CTR is the same.
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CTR is decreasing in l or l˜ , in general. This presumption is shown to be false by rows 2 and 4, where r˜ is approximately 0.5, since the CTR at l˜ = 3.74 is lower than the CTR at l˜ = 4.04. With regard to PCTRs, we infer from rows 10 – 14 that the greater the wage rate difference to the rich group, the higher the PCTR, since the greater the wage rate difference, the greater the payments of group 3 to the median voters. Rows 15 – 22 show the consequences of abandoning uniform distributions. All these simulations show that CTRs are lower than those of rows 1 – 5 and 10– 14, which goes back to lower pretax income standard deviations. Comparing rows 15 –18 to 19 –22 shows that CTRs in rows 19 – 22 are 0 and thus, lower than those of rows 15– 18. This can be explained by the fact that in the examples of rows 19 –22, the expected labour supply significantly decreases upon an increase in the tax rate such that taxation reduces expected posttax income, with the consequence that agents reject taxation at all. Concerning PCTRs, the examples of rows 15 –22 show that the groups’ shares of the whole population have a major impact on PCTRs. In the examples of rows 15 –18, PCTRs are clearly lower than the corresponding PCTRs of rows 2 –5. Responsible for this development is the greater share of group 1 and the smaller share of group 3, with the consequence that the median voter, group 2, receives less from group 3 and pays more to group 1. In the examples of rows 19– 22, the lower class rather than the middle class is the median voter. An increase in the wage rate of group 2 raises the payments of group 2 to group 1 and thus, the preferred PCTR of group 1 is increasing in the wage rate of group 2. The PCTR of the example in row 19 is relatively low in comparison to that of row 2. We have expected that shifting the median voter into group 1 would lead to higher tax rates, but note that an increase in the PCTR implies a reduction of group 2’s labour supply such that the transfers available for distribution may be diminished even if group 3 raises its labour supply.
4. Summary and concluding remark This paper has considered influences on the choice of constitutional tax rates (CTRs) and postconstitutional tax rates (PCTRs), and has offered a comparison of these tax rates. The main findings are: (a) On the constitutional level, agents choose the confiscatorial tax rate if labour supply is exogenous. They favour no taxation at all if they are risk-neutral. (b) On the postconstitutional level, the preferred PCTR is either zero or the confiscatorial tax if labour supply is exogenous. If agents are risk-neutral, the PCTR lies in the interval [0,1/2]. (c) CTRs are not positively correlated with mean wage rates, mean pretax incomes or standard deviation of the wage rate distribution, but CTRs are positively correlated with the standard deviation of the pretax income distribution. (d) PCTRs are the result of a redistribution conflict between net-tax payer and net-subsidy receiver. The main determinant of PCTR is the difference between the median voter’s wage rate and the wage rates of members of the other population groups. (e) The conjecture that CTRs are generally lower than PCTRs is false. (f) Reducing risk aversion diminishes CTRs, whereas PCTR changes are ambiguous.
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To keep the analysis tractable, quite restrictive assumptions have been required, in particular about the functional form of individual preferences and the distribution of wage rates. Unfortunately, there seem to be no satisfactory alternatives, which would allow one to avoid these difficulties.
Acknowledgements I am grateful to two anonymous referees for suggestions that have led to significant improvements of the paper. I also wish to thank Ru¨diger Pethig for the discussions, comments, and encouragement.
References Eaton, J., Rosen, H.S., 1980. Labor supply, uncertainty, and efficient taxation. Journal of Public Economics 14, 365 – 374. Foley, D., 1967. Resource allocation and the public sector. Yale Economic Essays 7, 45 – 98. Harsanyi, J.C., 1955. Cardinal welfare, individualistic ethics, and interpersonal comparison of utility. Journal of Political Economy 63, 309 – 321. Meltzer, A., Richards, S., 1981. A rational theory of the size of the government. Journal of Political Economy 89, 914 – 927. Rawls, J., 1971. A Theory of Justice. Harvard Univ. Press, Cambridge, MA. Roberts, K., 1977. Voting over income tax schedules. Journal of Public Economics 8, 329 – 340. Romer, T., 1975. Individual welfare, majority voting and the properties of a linear income tax. Journal of Public Economics 4, 163 – 185. Romer, T., 1977. Majority voting on tax parameters: some further results. Journal of Public Economics 7, 127 – 134. Sinn, H.-W., 1995. A theory of the welfare state. Scandinavian Journal of Economics 97, 495 – 526. Sinn, H.-W., 1996. Social insurance, incentives, and risk taking. International Tax and Public Finance 3, 258 – 280. Varian, H.R., 1980. Redistributive taxation. Journal of Public Economics 14, 49 – 68.
Constitutional and postconstitutional taxation Thomas Eichner * Department of Economics, University of Siegen, Hoelderlinstr. 3, D-57068 Siegen, Germany Received 2 August 1999; received in revised form 12 November 2001; accepted 3 January 2002
Abstract This paper investigates voting on linear income taxes behind a veil of ignorance over the postconstitutional situation, and voting on linear income taxes by majority rule when the veil has been lifted in a model of endogenous labour supply. Numerical simulations show that the constitutional tax rate (CTR) is positively correlated with the standard deviation of the pretax income distribution, whereas the main determinant of the postconstitutional tax rate (PCTR) is the difference between the median voter’s wage and the wages of members of other population groups. D 2002 Elsevier Science B.V. All rights reserved. JEL classification: H21; J21 Keywords: Voting; Taxation; Constitutional; Postconstitutional choice
1. Introduction Two different approaches have been used to study the implementation of linear income tax rates. The first uses the concept of constitutional choice based on a welfare economics framework. The second is based on the postconstitutional choice, also known as political choice, and uses a political economics framework. Both approaches have a long tradition and have been applied to various investigations. Here, we argue along the line of a framework in which agents differ with regard to their productivity. In the first step, individuals decide on labour supply and in the second step, on taxation. The government implements a linear income tax that is indirectly progressive through redistribution by equal lump-sum transfers. In case of constitutional choice, decisions on taxation take place under uncertainty. If we consider constitutional choice as a normative concept, then individuals make their *
Tel.: +49-271-740-3164; fax: +49-271-740-2732. E-mail address: [email protected]. (T. Eichner).
0176-2680/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 7 6 - 2 6 8 0 ( 0 2 ) 0 0 111 - 8
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decision behind a veil of ignorance, that is, as if they do not know their productivity and the position they will have in society after taxation has been chosen. An alternative interpretation of constitutional choice is that agents determine the income taxation for their children and grandchildren. Then, uncertainty consists in the genetically caused abilities of their (unborn) children that play a decisive role for the childrens’ real living standards. From this point of view, constitutional choice is a positive approach. Harsanyi (1955) and Rawls (1971), who were the first to investigate constitutional choice, predicted that selfinterested agents would unanimously choose constitutional rules without being influenced by their particular future interests. In such approach, taxation has the function of a social insurance. The state, as social insurer, claims a part of uncertain pretax income and offers in return a safe payment. Technically speaking, from the ex ante perspective with regard to risk, taxation reduces the variance and the dispersion of posttax incomes. This has been well illustrated by Sinn (1995, 1996). In their labour supply model, Eaton and Rosen (1980) analysed voting on the tax rate at the constitutional level, and pointed out the effects of a change in tax rate on labour supply. Under the assumption that agents do not take the balanced fiscal budget into account, they found that the tax rate is between the laissez-faire tax and the confiscatorial tax. Labour supply changes, relative to the tax rate, are ambiguous. Unfortunately, these are very weak statements. The second approach is the postconstitutional choice. If uncertainty has been resolved, agents are in a decision situation under certainty, which we call the postconstitutional phase. Individuals differ ex post with respect to their productivity, which ex ante is a random parameter and causes uncertainty. Clearly, agents with different abilities to earn money favour different tax rates, so that an agent’s preferred tax rate depends on his or her productivity. This model under certainty was formulated by Romer (1975), Roberts (1977), and Meltzer and Richards (1981). Since decisions are no longer unanimous, the government or the agents must search for a suitable implementation procedure for taxation. In democracies, a well-accepted procedure is majority rule. Along with other authors, Romer (1975, 1977) shows that by applying majority rule, the society implements the tax rate that is the first-best choice of the voter with the median pretax income. Under the mild requirement of hierarchical adherence (see Roberts, 1977, Lemma 1), agents with higher pretax income favour less redistribution, whereas agents with pretax income below the median pretax income favour more redistribution than the redistribution that results from majority voting. Meltzer and Richards (1981) show that the preferred tax rate of the median income taker is less than the confiscatorial tax, which is also a weak result. Having sketched both approaches, it is obvious how closely related these approaches are. Both provide justifications for taxation, but for different reasons in each case. The purpose of this paper is to contribute to both approaches. With help of the benchmark cases: exogenous labour supply and risk-neutrality, and with help of numerical simulations, we point out the main effects, which drive constitutional and postconstitutional tax rates. Use of a parametric utility function and simple discrete wage rate distributions has the advantage over the more general models of Eaton and Rosen (1980) and Meltzer and Richards (1981) that more clear-cut result can be obtained.1
1
Unfortunately, these results are coupled with a loss of generality.
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Before presenting the model, a remark is in order with respect to the timing of decisions. In case of postconstitutional choice, productivities are known and decisions take place under certainty. First, the tax rate is determined by majority rule, and then individuals choose labour supply. By contrast, constitutional tax rates are chosen behind a veil of ignorance where, strictly speaking, agents face uncertainty about their type. In the next stage, the veil of ignorance lifts, productivities become known, and then agents sign labour supply contracts.2 Under both regimes, tax rates are found by backward induction, and under both regimes at the last stage, we have to determine labour supply decisions in the presence of type uncertainty and for given tax rates. That is the reason why Section 2 outlines the model and the labour supply decisions. In Subsection 3.1 we introduce constitutional choice, and in Subsection 3.2 we investigate postconstitutional choice. Subsection 3.3 presents simulation results and a comparison of the tax rates. The paper ends with a summary and a concluding remark in Section 4.
2. The model and labour supply decisions We consider the society as consisting of three different types of individuals. A type i individual is characterised by (exogenous) productivity or wage rate. For simplicity, wage rates are assumed to be discretely distributed with realisations wi and relative P cumulations pi for i = 1, 2, 3. A standard requirement of relative cumulations is that 3i¼1 pi ¼ 1. One possible interpretation of three realisations is that the society is composed of a lower class, a middle class, and an upper class. Each member of group i determines his or her labour effort by solving max U ðyi ; S i Þ s:t: yi ¼ ð1 tÞwi S i þ p: S
ð1Þ
i
The posttax income yi of a person of group i is completely spent on consumption. It is composed of the pretax income wiS i, that is linearly taxed at rate t and finances a lumpsum transfer p. Agents’ tastes are identical and are represented by the additive separable utility function u ¼ U ðy; S Þ ¼ ay by2 cS
2
ð2Þ
with the partial derivatives Uy>0, Uyy V 0, US V 0, US S V 0, and with a>0, b z 0, c z 0.3 The parameters a, b and c take the same values for the three groups i = 1, 2, 3. Individuals are price takers in the labour market and they consider t and p as exogenously given
2 It should be noted that Eaton and Rosen (1980) use a different timing: Agents choose labour supply before the wage rate is known. Eaton and Rosen’s timing is motivated by the assumption that output is a stochastic rather than a deterministic function of labour input, which is an assumption on production technology. However, Eaton and Rosen’s timing for constitutional choice makes no sense in a paper that aims at comparing constitutional and postconstitutional taxation. 3 The property Uy>0 implies that we have to ensure that a>2by.
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information. Taking Eq. (2) into account, we obtain the following first- and second-order condition of the optimisation problem (1) ð1 tÞða 2pbÞwi 2S i c 2S i ð1 tÞ2 bw2i ¼ 0;
ð3Þ
2c 2ð1 tÞ2 bw2i < 0:
ð4Þ
Rearranging Eq. (3) yields group i’s labour effort 9 8 > = < ð1 tÞða 2pbÞwi for ð1 tÞða 2pbÞwi > 0 > 2c þ 2ð1 tÞ2 bw2i ¼: Li ðt; pÞ: Si¼ > > ; : 0 for ð1 tÞða 2pbÞwi V0
ð5Þ
Eq. (5) shows that individuals are voluntarily unemployed if (1 t)wi(a 2pb) V 0. An increase in p may have the consequence that agents choose to subsist on the transfer, which in this case can be interpreted as welfare payment. Since t, p, a, b and c are the same for all agents, the group with the lowest wage rate (defined as w1) has the greatest incentive to exploit the other groups. For the time being, the labour effort Li(t, p) depends on both the exogenous tax rate and the transfer. Procedure (1) implies that individuals do not take the fiscal budget constraint into account. The state offers taxes and transfers with the assumption that the fiscal budget constraint is satisfied. Then, a government’s payments have to equal its receipts, that is, in formal terms p¼t
3 X
pi wi Li ðt; pÞ:
ð6Þ
i¼1
Combining Eqs. (5) and (6) and solving with respect to p, we obtain for the case where all citizens work, tð1 tÞa
3 X
pi w2i
2c þ 2ð1 tÞ2 bw2i 3 X pi w 2
i¼1
p¼
1 þ 2tð1 tÞb
ˆ ¼: PðtÞ
ð7Þ
i
i¼1
2c þ 2ð1 tÞ2 bw2i
which in turn inserted into Eq. (5), yields ˆ Li ðt; PðtÞÞ ¼
"
ð1 tÞawi
½2c þ 2ð1 tÞ2 bw2i 1 þ 2tð1 tÞb
3 X
pi w2i
i¼1
2c þ 2ð1 tÞ2 bw2i
# ¼: Lˆ i ðtÞ:
ð8Þ The function Lˆi(t) is called a labour supply function and specifies the group i’s labour effort, presupposing that the fiscal budget is balanced.
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Proposition. The labour supply function has the following properties: (a) Lˆi(t)>0 for 0 V t < 1, and Lˆi(1) = 0. (b) Suppose b = 0, c>0, then Lˆi(t)=(1 t) awi/(2c). (c) Suppose b>0, c = 0, then Lˆi(t) = a/(2bwi). (d) Suppose b>0, c>0, then 8 9 8 9 >> ><> > > > > > > > !> > > > > 3 = = < > < X pi w 2 i 2 2 i 0: ¼ lim Lˆ t ðtÞ ¼ 0Zc bwi þ bð2c þ 2bwi Þ t!0 > > > 2c þ 2bw2i > > > > > i¼1 > > > > > > ; ; : > : > > <
and lim Lˆ it ðtÞ < 0: t!1
Unfortunately, the slope of the labour supply, function in general, is not determinate in sign. Lˆi(t) decreases in the tax rate if agents have risk neutral preferences (b = 0, see proposition (b)), and it is independent of t if agents do not have disutility from labour (c = 0, compare proposition (c)). In addition, proposition (d) establishes conditions that are necessary and sufficient for the labour supply function to decrease or to increase at t ! 0, and shows that taxation induces agents to reduce labour supply if the tax rate converges to the confiscatorial rate. Figs. 1 –3 illustrate labour supply and posttax income of the different groups for the numerical specification a = 10, b = 1, c = 2, p1 = p2 = p3 = 1/3, w1 = 2, w2 = 2.5, w3 = 4. First of all, at t = 0 groups with less productivity supply less labour than groups with higher
Fig. 1. Labour supply.
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Fig. 2. Labour supply of group 3.
productivity (Lˆ1(0)>Lˆ2(0)>Lˆ3(0)), as Fig. 1 shows. While the labour supply functions of groups 1 and 2 are strictly decreasing in the tax rate, group 3 raises its labour supply if the initial tax rate is not too high, in order to compensate for the income loss caused by redistribution (compare Fig. 2). This results in labour supplies characterised by Lˆ3(t)>Lˆ2(t)>Lˆ1(t) for 0.65 V t < 1. Clearly, it is possible to find other curvatures of labour supply functions, but these cases can be interpreted in a similar way. However, focusing on posttax incomes, Fig. 3 shows decreasing posttax income curves caused by the reduction of the labour supply of groups 1 and 2. Increasing taxation redistributes the additional pretax income of group 3 that emerged from rising labour effort in favour of the lower and middle class. In every case, group 1 benefits from redistribution, as does group 2 in some cases.
Fig. 3. Posttax income.
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3. Constitutional and postconstitutional taxation 3.1. Constitutional taxation We now consider the choice of fiscal policy parameters at the constitutional level. Decisions are taken behind a veil of ignorance. In the constitutional phase, individuals choose the tax rate without knowing their future incomes, that is, without knowing their productivity. An individual determines his or her preferred tax rate by solving max t
3 X
ˆ pi U ðyi ; Lˆ i ðtÞÞ s:t: yi ¼ ð1 tÞwi Lˆ i ðtÞ þ PðtÞ:
ð9Þ
i¼1
The first-order condition is 3 X
pi Uyi ðyi ; Lˆ i ðtÞÞ½wi Lˆ i ðtÞ þ Pˆ t ðtÞ ¼ 0:
ð10Þ
i¼1
For the numerical example provided in Section 2, the constitutional tax rate (CTR) is 0.204072. The total effect of taxation can be divided into four partial effects. Suppose t < 0.4. Then, taxation (i) reduces the expected labour supply (see Fig. 4), (ii) reduces the standard deviation of labour supply (see Fig. 5), (iii) reduces the expected posttax income (see Fig. 6), and (iv) increases the standard deviation of posttax income4 (see Fig. 7). The partial effects5 (i) and (ii) are welfare-enhancing, whereas the partial effects (iii) and (iv) are welfarediminishing. From Fig. 8, we conclude that effects (i) and (ii) overcompensate the effects (iii) and (iv) for tax rates smaller than the constitutionally chosen rate (t = 0.204072).6 Neither we nor the computer programme ‘Mathematica’ are able to provide the expected utility maximising tax rate as a function of the exogenous parameters a, b, c, pi, wi, but, as shown in the examples of Subsection 3.3, we are able to calculate the expected utility maximising tax rates for given values of exogenous parameters. Before presenting the simulation results, we focus on two benchmark cases, namely exogenous labour supply and risk-neutrality.7 4 Effect (iv) is not inconsistent with effect (ii), since in calculating the standard deviation of posttax income, it has to be taken into account that labour supplies are multiplied by wage rates. It can be shown that the standard deviation of wages (wi Lˆi(t)) is increasing in t if t < 0.4. 5 Observe that partial effects are isolated effects. For example, in the partial effect of labour supply, we ignore the impact of labour supply changes on posttax incomes. Then, the reduction of expected labour supply is welfare-enhancing due to US < 0. 6 Eaton and Rosen (1980) demonstrate that the tax on wage serves partially to insure the individual against random wage movements, with the consequence that lump-sum taxation is not efficient. Interestingly, in our numerical example, initial standard deviations of wages and posttax incomes are increasing in the tax rate. If t>0.5, standard deviations of wages and posttax incomes are decreasing in t so that we are able to apply Eaton and Rosen’s (1980) argument with respect to lump-sum taxation, although their timing of decisions is different from our timing. 7 The reader may find the case where labour causes no disutility interesting. However, it can be shown that in this case the groups’ pretax incomes are identical, which implies that individuals are indifferent between all tax rates. This, in my view, seems to be a very atypical case that is neglected in what follows.
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Fig. 4. Expected labour supply.
3.1.1. Exogenous labour supply Taking _ labour supply as exogenously given information_ where labour is assumed to be fixed (S = const.), individuals’ Ppretax income is xi(: = wiS ). For notational convenience, we introduce the mean l :¼ 3i¼1 pi wi and the standard deviation vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u 3 3 uX X pi w2i pi w i r :¼ t i¼1
i¼1
of the wage rate distribution. Then, the posttax income can be written as yi ¼ wi S þ tS ðl wi Þ:
ð11Þ
The decisions with respect to labour supply are dropped and agents directly choose the tax rate that maximises their expected utility. Taking into account Lˆi(t) = S ¯, Pˆt(t) = lS ¯and Eq. (11), the left-hand side of condition (10) becomes 3 X
pi ½a 2b½wi S þ tS ðl wi Þðwi S þ lS Þ ¼: AðtÞ:
ð12Þ
i¼1
P P Using 3i¼1 pi aðwi S þ lS Þ ¼ aS ðl 3i¼1 pi wi Þ ¼ 0 and the definition of r, we obtain after some rearrangements 2
AðtÞ ¼ 2bS r2 ð1 þ tÞ:
ð13Þ
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Fig. 5. Standard deviation of labour supply.
Since A(t) is increasing in t, the preferred tax rate is t = 1. The reason for this result is that redistributive taxation provides social insurance and increasing the tax rate reduces the _ standard deviation of posttax incomes (1 t)rS , which is utility-enhancing for risk-averse agents and induces them to choose the confiscatorial tax rate. In view of the distribution, the constitutional choice with exogenous labour supply results in an egalitarian income distribution.
Fig. 6. Expected posttax income.
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Fig. 7. Standard deviation of posttax income.
3.1.2. Risk-neutrality Now, we turn to the case in which the marginal returns from income are constant, or in formal terms b = 0, which means that agents are no longer risk-averse but risk-neutral. As can be seen in proposition (b), labour supply simplifies to Lˆi(t)=(1 t)awi/(2c), and the transfer to Pˆ(t) = t(1 t)a(l2 + r2)/(2c). Lˆi(t) and Pˆt(t) inserted into the left-hand side of first-order condition (10) for b = 0, we obtain
3 X i¼1
pi
a2 ½ð1 2tÞðl2 þ r2 Þ ð1 tÞw2i 2c
which can be manipulated to a2 ðl2 þ r2 Þt: 2c
ð14Þ
ð15Þ
(15) shows that if agents are risk-neutral, then the preferred tax rate is zero. It can be shown that taxation reduces expected labour supply, expected posttax income, the standard deviation of labour supply, and the standard deviation of posttax income. Due to riskneutrality, agents do not benefit from the insurance and redistribution function of taxation. They judge taxation to be welfare diminishing, since it is a disincentive to labour supply and thus, reduces posttax incomes. 3.2. Postconstitutional taxation In contrast to constitutional choice, postconstitutional choice is not made unanimously. The veil of ignorance concerning the wage rate has lifted, and each individual knows that he or she belongs to one of the three groups of productivity. On the postconstitutional level, individuals establish tax rates by means of majority voting. The application of
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majority voting implies that the tax rate chosen is the one preferred by the median voter. Formally, the postconstitutional tax rate (PCTR) is determined by ˆ ð16Þ max U ðym ; Lˆ m ðtÞÞ s:t: ym ¼ ð1 tÞwm Lˆ m ðtÞ þ PðtÞ t
with the first-order condition Uym ðym ; Lˆ m ðtÞÞ½wm Lˆ m ðtÞ þ Pˆ t ðtÞ ¼ 0
ð17Þ
where wm denotes the median wage rate and Lˆm(t) the labour supply of the voter with the median income. For the numerical example outlined in Section 2, members of group 2, who are the median voters, favour low tax rates. Their best choice is t = 0.094144, which is the tax rate implemented through majority voting. Group 3, the rich group, favours no redistribution at all, whereas the poor group favours redistribution (its preferred tax rate is t = 0.471681). Analogous to Subsection 3.1, we will examine the benchmark cases of exogenous labour supply and risk neutrality. 3.2.1. Exogenous labour supply _ In the case of exogenously given labour8 (S = S ), the left-hand side of Eq. (17) becomes Uym ðym ; S ÞðlS wm S Þ ¼: BðtÞ:
ð18Þ
Due to B(t)>0 if l>wm, B(t) < 0 if l < wm and Bt = 0, we obtain the ‘bang-bang’ solutions t = 0 and t = 1. This result can be traced back to Foley (1967) _ and Roberts (1977). The median voter equilibrium is t = 0 if the mean pretax income (lS ) is smaller than the median _ voter income (wmS ), which implies that the median voter looses from redistribution. Conversely, the median voter prefers t = 1 if his pretax income is smaller than the average pretax income and thus, he profits from redistribution. Whether wm>l or wm < l depends on the curvature of the wage rate distribution. For right-skewed distributions, we have wm>l, and for left-skewed distributions, which are the empirically relevant cases, we have wm < l. 3.2.2. Risk-neutrality If agents are risk-neutral, labour supply is given by Lˆi(t)=(1 t) awi/(2c), the transfer is given by Pˆ(t) = t(1 t) a (l2 + r2)/(2c) and the left-hand side of condition (17) turns into a2 ½ð1 2tÞðl2 þ r2 Þ ð1 tÞw2m ¼: CðtÞ: 2c
ð19Þ
The preferred tax rate depends on the relation of (l2 + r2) to wm2 . If l2 + r2>wm2 , which is the relevant case from an empirical point of view, we have C(0)>0, C(1/2) < 0 and Ct(t) < 0, 8 A remark is in order to the related work of Meltzer and Richards (1981). They use a more general model encompassing our algebraic example, but in elaborating the determinants of tax rates they gear to pretax incomes. They do not consider the labour supply effects of taxation. Their main result is: ‘‘Using the common economic assumption that elasticities are constant, the tax rate rises as mean income rises relative to the income of the decisive voter.’’ This statement is fully compatible with our results, which are presented in the next subsections, especially in Subsection 3.3, but our contribution is to explain how pretax incomes come about. Starting from wage rates, we are interested in the groups’ labour supplies, which determine the groups’ pretax incomes.
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Fig. 8. Expected utility.
which ensure that the median voter equilibrium lies in the interval ta[0,1/2]. If l2 + r2 < wm2 , it holds that C(t) < 0 for all t and the corner solution t = 0 yields the maximal benefit. However, concentrating on l2 + r2>wm2 , the equation C(t) = 0 can be solved for t, which provides t¼
l2 þ r2 w2m : 2ðl2 þ r2 Þ w2m
ð20Þ
Comparative static analysis reveals that tl ¼ tr ¼
2lw2m ½2ðl2 þ r2 Þ
w2m 2
2rw2m ½2ðl2
þ r2 Þ w2m 2
;
t wm ¼
2wm ðl2 þ r2 Þ ½2ðl2 þ r2 Þ w2m 2
;
:
ð21Þ
The foregoing results are intuitively plausible. Keeping in mind that wm < l, a reduction in the difference l wm reduces the median voter’s net gain from redistribution and is responsible for lower tax rates (t( l) < 0 and twm < 0). The positive correlation of the standard deviation with the preferred tax rate shows that the median voters’ profits caused by redistribution from higher income takers overcompensate the median voters’ losses caused by redistribution to lower income takers. 3.3. Simulation results and comparison In this subsection, especially in Table 1, we report on constitutional and postconstitutional tax rates with endogenous labour supply and risk-averse agents. Table 1 is divided into five blocks. The utility function parameters underlying the simulations9 of rows 1– 5 9
The computer programme for calculations is available from the author upon request.
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Table 1 CTRs and PCTRs for w1 = 2, w3 = 4, a = 10, b = 1, c = 2 l˜
Parameter
p1
p2
p3
w2
l
r
1 2 3 4 5
1/3 1/3 1/3 1/3 1/3
1/3 1/3 1/3 1/3 1/3
1/3 1/3 1/3 1/3 1/3
2.0 2.1 2.5 3.0 3.6
2.67 2.70 2.83 3.00 3.20
0.943 0.920 0.850 0.816 0.864
3.70 3.74 3.86 3.96 4.04
1/3 1/3 1/3 1/3
1/3 1/3 1/3 1/3
1/3 1/3 1/3 1/3
2.1 2.1 2.1 2.1
2.70 2.70 2.70 2.70
0.920 0.920 0.920 0.920
1/3 1/3 1/3 1/3 1/3
1/3 1/3 1/3 1/3 1/3
1/3 1/3 1/3 1/3 1/3
2.1 2.5 2.8 5.0 6.0
4.03 4.17 4.27 5.00 5.33
15 16 17 18
0.45 0.45 0.45 0.45
0.40 0.40 0.40 0.40
0.15 0.15 0.15 0.15
2.1 2.5 3.0 3.5
19 20 21 22
0.65 0.65 0.65 0.65
0.30 0.30 0.30 0.30
0.05 0.05 0.05 0.05
2.1 2.5 3.0 3.5
6 7 8 9
b=0 b = 0.1 b = 0.5 b=1
10 11 12 13 14
w3 = 8 w3 = 8 w3 = 8 w3 = 8 w3 = 8
r˜
CTR
PCTR
0.524 0.501 0.456 0.463 0.499
0.230335 0.218584 0.204072 0.232262 0.291899
0.426558 0.366583 0.094144 0 0
20.34 13.20 6.08 3.74
13.907 6.389 1.360 0.501
0 0.107736 0.200326 0.218584
0.314133 0.350297 0.381615 0.366583
2.805 2.718 2.660 2.449 2.494
3.87 4.00 4.06 4.27 4.31
0.690 0.635 0.621 0.669 0.690
0.725846 0.686710 0.657267 0.670457 0.700212
0.799246 0.764711 0.186675 0 0
2.34 2.50 2.70 2.90
0.699 0.671 0.714 0.831
3.54 3.68 3.80 3.89
0.382 0.383 0.441 0.502
0.002070 0.008479 0.056175 0.122216
0.109638 0 0 0
2.13 2.25 2.40 2.55
0.431 0.461 0.583 0.757
3.42 3.53 3.62 3.68
0.240 0.295 0.392 0.471
0 0 0 0
0.082341 0.183754 0.284727 0.370259
are a = 10, b = 1, c = 2 and the wage rate distribution is assumed to be uniformly distributed with w1 = 2, w3 = 4, with the median voter’s wage rate w2 varied. In columns 6 and 7, we have calculated the mean wageP rate l and the standard deviation of wage rates r. Columns 8 and 9 show the mean l˜ :¼ 3i¼1 pi wi Lˆ i ð0Þ and standard deviation vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u 3 3 uX X pi w2i Lˆi ð0Þ2 pi wi Lˆi ð0Þ r˜ :¼ t i¼1
i¼1
of pretax incomes, and columns 10 and 11 show the associated tax rates. Comparing CTRs with PCTRs demonstrates that the conjecture that PCTRs are generally lower than CTRs is false. At first glance, it is conspicuous that PCTRs are very sensitive to changes of the wage rate of the group with median income. In contrast, CTRs seem to be more robust to changes of w2. Upon closer examination, rows 1 –5 reveal that, with the exception of one case, higher standard deviations emerge with higher CTRs, whereas increasing means l and l˜ does not necessarily result in higher CTRs. Comparing r and CTRs show that the direction of changes of r is not always in accordance with the
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direction of CTR changes. More specifically, if r is diminished from 0.943 to 0.850, the CTR decreases from 0.230335 to 0.204072, but further diminishing r to 0.816 implies an increase in CTR to 0.232262. It is interesting to observe that, although r is reduced from 0.850 to 0.816, labour is supplied such that r˜ increases from 0.456 to 0.463. Comparing r˜ and CTRs demonstrates that the direction of changes of the pretax income standard deviation tallies with the direction of CTR changes. Focusing on PCTRs, we found that the greater the difference between the median voter’s wage rate and the poor group’s wage rate, the lower the PCTR, and the greater the difference between the median voter’s wage rate and the rich group’s wage rate, the higher the PCTR. On the other hand, group 2 benefits from transfers of group 3, and on the other hand, group 2 is placed in disadvantage by payments to group 1. The difference between the wage rate of group 2 and the wage rates of the other groups indicates whether this group is a net-tax payer or, rather, a netsubsidy receiver. A net-tax payer prefers a low PCTR while a net-subsidy receiver prefers a high PCTR. In the next block, rows 6– 9, we investigate the influence of the risk-aversion parameter b, which is now successively raised from 0 to 1. All other parameters remain unchanged. Obviously, reducing the risk-aversion diminishes the CTR.10 This observation is in accordance with the fact that risk-neutral agents prefer zero tax rates. In addition, from l˜ and r˜ , we infer that reducing risk-aversion induces individuals to supply more labour; both l˜ and r˜ increase, while l and r remain constant. In summary, we have two effects: on the one hand, reducing b increases labour supply, which increases r˜ . According to our argument above, this would imply higher CTRs. On the other hand, reducing b implies that agents benefit less from the insurance function of taxation (from reducing r˜ ). The fact that CTRs are increasing in b led us to conclude that the second effect overcompensates for the first effect. Now, consider PCTRs. In all of the examples, the median voter’s wage rate is 2.1. Keeping in mind that l2 + r2>wm2 , we can apply the formula in Eq. (20) and obtain for a risk-neutral society (b = 0), the tax rate t = 0.314133. As can be seen in rows 6– 9, for small b’s the PCTR increases in b, and for large b’s, it decreases in b. The changes of labour supply are responsible for the development of PCTRs. Increasing b has the effect that agents supply less labour. Especially if b < 0.5, the labour supply of group 2 approximates to the labour supply of group 1, whereas group 3 offers significantly more labour than group 2. If b>0.5, increasing b has the effect that the labour supply of group 3 falls under the labour level of group 2, with the consequence that the net-payments to group 2 diminish and group 2 prefers less redistribution. In rows 10 – 14, we study the effects of increasing w3. The simulations show that CTRs at w3 = 8 are higher than those at w3 = 4, which can be explained by reduced standard deviations r and r˜ . Comparing the values of r, r˜ , and CTR, again r˜ instead of r is the corresponding measure for CTRs. Both in rows 10 and 14, r˜ takes the value 0.690. Since at w2 = 2.1, CTR is higher than the CTR at w2 = 6.0, one could presume that for given r˜ , 10 The effect of risk-aversion on CTRs has also been studied by Varian (1980) using an algebraic example involving quadratic utility. His model, however, is different from mine. It is a two-period model with exogenous and partly uncertain incomes where agents decide on the allocation of saving and consumption in the two periods. However, in both models the impact of risk-aversion on the CTR is the same.
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CTR is decreasing in l or l˜ , in general. This presumption is shown to be false by rows 2 and 4, where r˜ is approximately 0.5, since the CTR at l˜ = 3.74 is lower than the CTR at l˜ = 4.04. With regard to PCTRs, we infer from rows 10 – 14 that the greater the wage rate difference to the rich group, the higher the PCTR, since the greater the wage rate difference, the greater the payments of group 3 to the median voters. Rows 15 – 22 show the consequences of abandoning uniform distributions. All these simulations show that CTRs are lower than those of rows 1 – 5 and 10– 14, which goes back to lower pretax income standard deviations. Comparing rows 15 –18 to 19 –22 shows that CTRs in rows 19 – 22 are 0 and thus, lower than those of rows 15– 18. This can be explained by the fact that in the examples of rows 19 –22, the expected labour supply significantly decreases upon an increase in the tax rate such that taxation reduces expected posttax income, with the consequence that agents reject taxation at all. Concerning PCTRs, the examples of rows 15 –22 show that the groups’ shares of the whole population have a major impact on PCTRs. In the examples of rows 15 –18, PCTRs are clearly lower than the corresponding PCTRs of rows 2 –5. Responsible for this development is the greater share of group 1 and the smaller share of group 3, with the consequence that the median voter, group 2, receives less from group 3 and pays more to group 1. In the examples of rows 19– 22, the lower class rather than the middle class is the median voter. An increase in the wage rate of group 2 raises the payments of group 2 to group 1 and thus, the preferred PCTR of group 1 is increasing in the wage rate of group 2. The PCTR of the example in row 19 is relatively low in comparison to that of row 2. We have expected that shifting the median voter into group 1 would lead to higher tax rates, but note that an increase in the PCTR implies a reduction of group 2’s labour supply such that the transfers available for distribution may be diminished even if group 3 raises its labour supply.
4. Summary and concluding remark This paper has considered influences on the choice of constitutional tax rates (CTRs) and postconstitutional tax rates (PCTRs), and has offered a comparison of these tax rates. The main findings are: (a) On the constitutional level, agents choose the confiscatorial tax rate if labour supply is exogenous. They favour no taxation at all if they are risk-neutral. (b) On the postconstitutional level, the preferred PCTR is either zero or the confiscatorial tax if labour supply is exogenous. If agents are risk-neutral, the PCTR lies in the interval [0,1/2]. (c) CTRs are not positively correlated with mean wage rates, mean pretax incomes or standard deviation of the wage rate distribution, but CTRs are positively correlated with the standard deviation of the pretax income distribution. (d) PCTRs are the result of a redistribution conflict between net-tax payer and net-subsidy receiver. The main determinant of PCTR is the difference between the median voter’s wage rate and the wage rates of members of the other population groups. (e) The conjecture that CTRs are generally lower than PCTRs is false. (f) Reducing risk aversion diminishes CTRs, whereas PCTR changes are ambiguous.
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To keep the analysis tractable, quite restrictive assumptions have been required, in particular about the functional form of individual preferences and the distribution of wage rates. Unfortunately, there seem to be no satisfactory alternatives, which would allow one to avoid these difficulties.
Acknowledgements I am grateful to two anonymous referees for suggestions that have led to significant improvements of the paper. I also wish to thank Ru¨diger Pethig for the discussions, comments, and encouragement.
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