Understanding the impact of the minimum wage on the Gini index

Using an example from Holland

Thomas Cool
October 2000

Start

Note: This uses the Economics Pack, see Cool (1999).
JEL 99-0820, Journal of Economic Literature, volume 37, no. 3, September 1999.
See the review of the earlier edition of the Pack in the Economic Journal, July 1 1998, v 108 n 449, page 1229-1234.

Needs["Economics`Pack`"]
ResetAll
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[Graphics:Images/index_gr_2.gif]
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Economics[Economic`TaxRegimes]
FindRateNTV     NoTaxVoid       TaxMinimand     WithTaxVoid
NetIncomeLorenz NoTaxVoidRate1  TaxRegime
[Graphics:Images/index_gr_4.gif]
AddLinearRates    FindMinimumWage   PieceWiseCurved
AdjustRates       GrossToNetRatio   PieceWiseLinear
AfterPremium      Indexed           PiecewiseWhich
AverageLevyRatio  Levy              PremiumRates
BenefitLine       LevyPlot          ReplacementRate
Bentham           MarginalRate      SetTaxAndPremium
CleanTaxRates     MinimumIncome     SolveMinimumWage
CurvedTax         MinimumWage       Subsistence
CurvedTaxRates    NetIncome         TaxCredit
DynamicMarginal   NetIncomeFindRoot TaxDeduction
Exemption         NetIncomeInverse  TaxOnWage
FindGrossWage     NetIncomePlot     TaxRates
Economics[Economic`Inequality]
FromWageDensity  NLorenz          ResetWageDensity Total
Gini             Options$Theil    Theil            WageDensity
Lorenz           PensParade       ToPensParade     WageDistribution
LorenzPlot       PensParadePlot
SetOptions[Plot, TextStyle -> {FontSize -> 12}];
SetOptions[NIntegrate, PrecisionGoal -> Automatic];

Note: NIntegrate sometimes gives messages about failed convergence at some points. Setting PrecisionGoal at a lower value like 3 eliminates these messages. However, leaving the value at Automatic still causes better outcomes.

Introduction

Analyses on inequality - e.g. on the Gini coefficient - commonly neglect the impact of the Tax Void.

Inequality analysis distinguishes the "Primary Income Gini" before incomes policies are enacted, and the "After Gini" after incomes policies are enacted - here called the "Gross Gini" and "Net Gini" henceforth. It is a key insight that it is rather impossible to determine the proper Gross Gini. Observations are on actual economic developments that are conditional to the existing incomes policies. If those income policies would not be present, the Gross Gini would be quite different. For the minimum wage, this means that both the Gross Gini and the Net Gini include its presence. The common analysis thus acknowledges that people below the minimum wage become unemployed. The Gross Gini counts them as having no income, and the Net Gini counts them as having some benefit income.

This current paper proposes to depart from this common approach by accounting for the Tax Void that is caused by the minimum wage. The Tax Void is the tax wedge at the minimum wage, i.e. the difference between gross and net wages. Since workers below the minimum wage are unemployed anyway, there are no tax revenues for the Tax Void section of the earnings distribution. Rather strangely, taxes are levied that are not cashed. Rather strangely, such taxes only cause higher gross wages that make people unemployed. Such taxes also could be abolished by a stroke of the pen, since there are no revenues involved. Since unemployment in the Tax Void thus could be avoided, such a minimum wage is a distorted implementation of the social security system and not a proper implementation of the welfare state. If we want to capture the redistributive properties of the welfare state, then we should not include wrong ways of implementation that distort the situation. Inequality measurement, e.g. with the Gini, should account for this distortion.

It would be wrong to regard current Gini's as the proper representations of the welfare state per se. It is so easy to say "So many people need assistence", but one then forgets that such a need is actually absent. It is so easy to say "Income redistribution is so large", but one then forgets that the redistribution is in the reverse direction since unemployment is a 'real tax' on a person. This paper thus intends to issue a warning that current Gini's cannot be used for conclusions on the welfare state per se.

It is useful to note that the minimum wage is an essential ingredient in the Gini analysis. One might argue that the minimum wage would be different from the concept of 'taxes and transfers'. However, as explained, it is the tax system that drives up the gross minimum wage. A discussion about what would be the 'prime cause' would be something like the discussion whether it was Adam or Eve who actually caused the dismissal from paradise. Indeed, conventional analysis uses taxes and transfers and neglects the impact of the Tax Void, but a convention is not an argument in itself, espectially when it appears to be lopsided and to lead to wrong conclusions. Scientific research should not copy the partial rationality of the governmental bureaucracies. This position is reflected in this paper too by using the word 'threshold' for the tax and the word 'exemption' for the system as a whole.

The following Table shows how the proposed analysis differs from the common analysis. We will compare a situation with a Tax Void or the current situation in Holland (first row), with a situation where the Tax Void has been abolished (second row). The first row gives the current redistribution with the distortion included. The second row would give the Pareto improvement with the distortion removed, which would be the proper base for statements on the welfare state. Hence, the redistributive properties of the welfare state should be measured by comparing the Pareto Gross Gini with the Pareto Net Gini (both bottom row). Note in this Table for the Pareto Net Gini that the current unemployment still work, so that benefit payments are saved, so that taxes can be lower in general. The Pareto Net Gini thus would show the lowest Gini overall.


[Graphics:Images/index_gr_5.gif]

Calculations have been done below with rough estimates on Dutch data for 1997. These data are on earnings, and thus exclude other sources of income. They also exclude parttime work since this work in Holland is excluded from the minimum wage - which explains why Holland has so much parttime work. The earnings estimate is documented by Cool (1999) and Cool (2000). The Gini calculations result into the following findings:


[Graphics:Images/index_gr_6.gif] [Graphics:Images/index_gr_7.gif] [Graphics:Images/index_gr_8.gif] [Graphics:Images/index_gr_9.gif]
[Graphics:Images/index_gr_10.gif] [Graphics:Images/index_gr_11.gif] [Graphics:Images/index_gr_12.gif] [Graphics:Images/index_gr_13.gif]
[Graphics:Images/index_gr_14.gif] [Graphics:Images/index_gr_15.gif] [Graphics:Images/index_gr_16.gif] [Graphics:Images/index_gr_17.gif]

Thus we find that the common method would report a difference of 0.2170 or 21.7% points for the redistribution effect, while, actually, the proper redistribution effect of the welfare state (per se) would be only 0.1002 or 10% points (or 10% of the Lorenz area). The distortion by the minimum wage or Tax Void would be measured by the column sums. The difference between the differences (i.e. the 21.7% - 10% = 11.7%) would in fact be a good candidate for the overall measure of distortion.

Caminada and Goudswaard (2000a), in a standard analysis and with full data, report a Gross Gini (of 'primary income') of 0.453 and a Net Gini of 0.338, so that the difference is 0.115. The higher net value likely derives from the fact that their average income would be lower by the inclusion of all other income sources such as pensioners. However, their Gini values still seem to allow room for the Pareto improvement that has been indicated and that will be discussed more below.

As a reminder, it should not be forgotten that the Net Gini is not a welfare index. A scheme that reduces the Net incomes to only $1 for all (e.g. with a huge minimum wage) cannot be judged to be an improvement. So our current focus on the Gini has only limited value.

The major routine used below is:

?TaxRegime
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The Gini analysis

This chapter only introduces the techniques, routines and data that we use. It thus only intends to clarify the concepts involved, and thus we do not attach any meaning to the calculated Gini coefficients.

The Gini from a wage density before tax

We regard the Wage Density only. On the cost side this excludes other labour costs (that also contribute to unemployment), while on the income side it excludes e.g. parttime income and transfers like pensions.

We have a Wage Density, and can directly determine the Distribution, Pen's Parade (as the inverse of the Distribution) and the Lorenz curve (as the share of total income up to a point in Pen's parade). The Gini coefficient then uses the integral of the Lorenz curve.

?Gini
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The routines have been defined for discrete and continuous data, but the present discussion uses only continuous data.

The wage density has been defined in million personyears. Income is defined in DFL 1000. The wage sum thus is in billions or DFL 1000 million. For 1997 $1 is about 1.7 DFL, so if you divide these numbers by 2 then you get approximate dollars.

ResetWageDensity[];
?WageDensity
[Graphics:Images/index_gr_20.gif]
Results[Default, Gross, WageDensity] = wdp = Plot[WageDensity[x], {x, 0, 130}, PlotRange -> All, AxesLabel -> {"Earnings\n1000 DFL", "Million\nPersonyears"}];

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?Total
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LabourSupply = Total[Number] 
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Results[Default, Gross, WageSum] = Total["Wage"]
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?FromWageDensity
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FromWageDensity[x, p]
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Plot[WageDistribution[x], {x, 0, 150}];

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Pen's parade is the inverse of the Distribution. The population proportion thus is cumulated, but the value at that point represents the value for only the subgroup at that point. Think of the population passing by in a parade of one hour, all ordered by the height of their incomes. For a common distribution you have to wait 40 minutes before you encouter people of medium income.

PensParadePlot[PlotRange -> {0, 150}];

[Graphics:Images/index_gr_29.gif]

Results[Default, Gross, LorenzPlot] = LorenzPlot[];

[Graphics:Images/index_gr_30.gif]

The Gini from the gross market situation follows from this.

Results[Default, Gross, Gini] = Gini[]
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The Gini after tax

We will study taxes in more detail in a later section. Temporarily we assume them given.

Results[Default, LevyRevenue] = Total[Levy]
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The Lorenz and Gini

We can determine the net income distribution, and then perform the same analysis for the net distribution aswe did for the gross distribution.

Before we can calculate the Lorenz curve for the net incomes, we have to find, as a technical detail, the inverse to net incomes.

NetIncomeInverse[Set, {MinimumWage}]

We now can find the Lorenz for the net incomes, which appears to be a piecewise function derived from the tax brackets.

NetIncomeLorenz[p_] = NetIncomeLorenz[Set, Default, p]
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The following table summarises the net income Lorenz results. The income classes are in gross wages for the tax brackets. If there is a minimum wage - which in this Default case however is not, then it would be included as a point of reference. You can check that the table fits above Lorenz. Take for example the second class. If the population share is 4.8% ≤ p ≤ 65.5% then the base of the net income share is 2% and the remainder depends upon the integral of Lorenz curve between 4.8% and p. Note that PensParade is used to translate these population shares into the relevant ranges for the density.

NetIncomeLorenz[Table]
[Graphics:Images/index_gr_35.gif] [Graphics:Images/index_gr_36.gif] [Graphics:Images/index_gr_37.gif] [Graphics:Images/index_gr_38.gif] [Graphics:Images/index_gr_39.gif] [Graphics:Images/index_gr_40.gif] [Graphics:Images/index_gr_41.gif]
[Graphics:Images/index_gr_42.gif] 0.001` 0.01` 0.009` 0.` 0.` 0.`
[Graphics:Images/index_gr_43.gif] 0.379` 6.399` 4.625` 4.8` 1.5` 2.`
[Graphics:Images/index_gr_44.gif] 4.858` 189.367` 111.142` 65.5` 46.` 49.1`
[Graphics:Images/index_gr_45.gif] 0.154` 8.924` 4.875` 67.4` 48.1` 51.1`
[Graphics:Images/index_gr_46.gif] 0.177` 10.601` 5.752` 69.6` 50.5` 53.6`
[Graphics:Images/index_gr_47.gif] 2.085` 162.967` 86.188` 95.7` 88.8` 90.10000000000001`
[Graphics:Images/index_gr_48.gif] 0.345` 47.733` 23.424` 100.` 100.` 100.`
[Graphics:Images/index_gr_49.gif] 8.` 426.001` 236.015` - - -

This gives us the net Lorenz plot.

Results[Default, Net, LorenzPlot] = LorenzPlot[NetIncomeLorenz];

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Show[Results[Default, Gross, LorenzPlot], 
     Results[Default, Net, LorenzPlot]];

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Results[Default, Net, Gini] = Gini[NetIncomeLorenz]
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Results[Default, Gross, Gini] - Results[Default, Net, Gini] 
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Looking at the density

It is instructive to plot the net income density by using the inverse of NetIncome[wage]. This adds no information to our problem at discussion, but it helps to understand the general issues, while it also shows how you could use the present routines.

ResetWageDensity[];
temp[x_] = WageDensity[x]
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Clear[WageDensity]
WageDensity[w_] := CF * temp[NetIncomeInverse[w]]

When we approximate the net income density in this manner, then we require a correction factor CF, so that the population size remains the same.

CF = 8 / (CF = 1; Total[Number])
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Results[Default, Net, WageDensity] = 
Plot[WageDensity[x], {x, 0, 130}, PlotRange -> All, AxesLabel -> {"Earnings\n1000 DFL", "Million\nPersonyears"}];

[Graphics:Images/index_gr_57.gif]

Show[Results[Default, Gross, WageDensity],
Results[Default, Net, WageDensity] ];

[Graphics:Images/index_gr_58.gif]

Total net income follows from the wage sum minus tax revenue.

Results[Default, Net, WageSum] = 
    Results[Default, Gross, WageSum] -
    Results[Default, LevyRevenue]
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If we apply the direct calculation on net incomes, then we find an error due to the approximation method and numerical approximation itself.

Total[Identity]
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Taxes and premiums, the minimum wage and the Tax Void

Taxes and premiums

We consider levies consisting of taxes and employee premiums. This still excludes VAT and excises, employer premiums en labour market subsidies and other wage costs that are relevant for net incomes and the minimum wage.

The following uses the format {b, r} = {bracket level in 1000 DFL, marginal rate starting there}, while the first bracket level is the threshold (with zero taxes below).

SetTaxAndPremium[]
TaxRates
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PremiumRates
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Levy[50]
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Exemption is defined not just as tax exemption, but as the "system exemption", where a minimum wage can protect subsistence as well. Here we take exemption as subsistence * replacement rate, where the latter rate is taken as 10%.

Exemption
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The (gross) minimum wage can thus be derived from taxes, premiums, net subsistence and the replacement rate.

Results["MinimumWage"] = FindMinimumWage[]
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For the levy revenue we have to start from the minimum wage (while we didn't in the Default regime).

ResetWageDensity[];
Results[Default, LevyRevenue] 
[Graphics:Images/index_gr_66.gif]
Results[WithTaxVoid, LevyRevenue] = 
    Total[Levy, MinimumWage]
[Graphics:Images/index_gr_67.gif]
[Graphics:Images/index_gr_68.gif]

The situation can be clarified by the "Levy Plot". This includes the following elements:

  1.  the levy line

  2.  the 45-degrees line, so that the difference between the levy and the 45-degree line gives net income

  3.  the benefit line parallel to the 45-degrees line, i.e. that part of net income that is required for subsistence and the replacement rate

  4.  the minimum wage given by the intersection of levy and benefit line.

lp = LevyPlot[0, 130, Premium, Tax, 
    AxesOrigin -> {0,0}, PlotStyle -> Thickness[.007],
    FilledPlot -> True,
    Fills -> { Hue[0.55, 1, .7], Hue[0.11, .7, 1]},
    PlotRange -> All, TextStyle -> {FontSize -> 12}];

[Graphics:Images/index_gr_69.gif]

Show[lp, Graphics[{Text["M", {30, 80}], 
    Text["Levy", {70, 10}], Text["Net", {90, 55}],
    Text["B", {110, 80}]},
   TextStyle -> {FontFamily -> "Helvetica",
   FontSize -> 14}]];

[Graphics:Images/index_gr_70.gif]

It is also instructive to regard the Net income plot.

NetIncomePlot[];

[Graphics:Images/index_gr_71.gif]

A note on the minimum wage and unemployment

That people below the minimum wage are unemployed, is purely a matter of definition. If a minimum wage is enacted, then the employer has the option for a potential employee to pay the minimum or not to hire the person. If the person is hired, then apparently his or her total contribution to the company is adequate. He or she then belongs to the Above Group. If the person is not hired, then he or she belongs to the Below Group. This is purely a matter of definition.

This is not quite the same issue as discussed in the literature. The discussion in the literature e.g. in Brittain is whether the introduction of the minimum wage causes too much unemployment, or e.g. in the US whether raising it causes more unemployment. The literature thus accepts the definition that people below the minimum wage are unemployed, and only discuss the Below and Above group sizes and the properties that cause the movement across the boundary.

For an unemployed person, the reasoning is: I am unemployed, thus apparently I cannot earn even the minimum wage. If I am an unemployed dentist, then for a while I can aspire at the minimum wage that exists for dentists. But after this, my resources and society's patience with me run out, and I will have to apply for jobs that offer the general minimum wage.

Tax Void

Though tax wedges are important in general, it is useful to give a special name to the tax wedge at the minimum wage. My choice has been "Tax Void": here taxes are levied that will not be cashed, since people are not be allowed to work below the minimum wage - and thus those workers don't earn anything that can be taxed. Those taxes also could be easily be abolished since there is no revenue.

Tax Void Diagram

The "Tax Void Diagram" shows the area between the net and gross minimum, both for taxes and for employment.

mni = NetIncome[MinimumWage];
dfid =  DisplayFunction -> Identity;
wdfill = FilledPlot[WageDensity[x], {x, mni, MinimumWage}, Fills -> Hue[.9], Axes -> False, dfid];
wdna = Show[wdp, Axes -> False, dfid];
wd = Show[wdna, wdfill /. (AxesFront -> True) :> ToSequence[], dfid];
lp2 = Show[lp, Axes -> False, dfid];
Show[GraphicsArray[{wd, lp2, wdna}, GraphicsSpacing -> -1], DisplayFunction -> $DisplayFunction];

[Graphics:Images/index_gr_72.gif]

Tax Void consequences

The OECD has computed Dutch unemployment as approximately 25%. Of course this is not all minimum wage unemployment, but in our discussion we will presume this as a 'worst case'. For example, it appears that people have been given disability pensions since administrators thought that "they would not get jobs anyhow". With a better management of the minimum wage, those people however might have found jobs. For our 'worst case' discussion, we collect all such welfare state problems under the heading of the minimum wage.

OnBenefit = Total[Number, 0, MinimumWage]
[Graphics:Images/index_gr_73.gif]
OnBenefit / LabourSupply
[Graphics:Images/index_gr_74.gif]

A different tax structure would increase employment by those employable on benefit.

EmployableOnBenefit = 
    Total[Number, Exemption, MinimumWage]
[Graphics:Images/index_gr_75.gif]
EmployableOnBenefit / LabourSupply
[Graphics:Images/index_gr_76.gif]

Abolishing taxes below the minimum wage would not cost anything, since people are unemployed, and thus don't pay taxes anyway. The scope for a tax reduction comes from the current unemployment benefits.

TaxVoidBurden = EmployableOnBenefit * Subsistence
[Graphics:Images/index_gr_77.gif]

The levy revenue thus can be reduced by this percentage.

TaxVoidBurden / Results[WithTaxVoid, LevyRevenue] *
    100 Procent
[Graphics:Images/index_gr_78.gif]

Note that it would be wrong for the calculation of the burden to include the new taxes that could be paid by the newly employed. This figure is only fictional.

Total[Levy, Exemption, MinimumWage]
[Graphics:Images/index_gr_79.gif]

However, the taxes for the newly employed enter into another calculation. In a situation without a Tax Void, the Tax Void Burden is avoided, and thus allows a reduction of taxes. Since the newly employed will partake in the tax base, the actual reduction to the current working population is even larger.

The conventional analysis

Some authors may be tempted to regard benefit payments as real income - for example as these have been paid for by 'insurance'. The unemployment that we are considering here however is a system unemployment that cannot be insured. Thus the benefit incomes of the unemployed below the minimum wage are pure transfers. Properly they have no market income. In actual observations it may happen that the institutions who do the transfers are called 'insurance organisations', but qua economic function these are pure transfers.

We will use the tax system above, and thus do not need to plot the tax and net income functions again.

Gross incomes with a binding minimum wage

In this regime, the minimum wage is binding, and part of the taxes and premiums are required to pay for those on benefit.

If we disregard all below the minimum wage, then we get an estimate of the proper market Gini.

ResetWageDensity[];
FindMinimumWage[]
[Graphics:Images/index_gr_80.gif]
Results[WithTaxVoid, Gross, Employment] = 
    Total[Number, MinimumWage]
[Graphics:Images/index_gr_81.gif]
Results[WithTaxVoid, Gross, WageSum] = 
    Total["Wage", MinimumWage]
[Graphics:Images/index_gr_82.gif]
Results[WithTaxVoid, Gross, FromWageDensity] = 
    FromWageDensity[x, p, WageDensity, MinimumWage]
[Graphics:Images/index_gr_83.gif]
[Graphics:Images/index_gr_84.gif]

Since apparently 27.3 % of the labour force is below the minimum, the Distribution starts there.

Plot[WageDistribution[x], {x, 0, 150}, AxesOrigin -> 0];

[Graphics:Images/index_gr_85.gif]

Note how we have defined the situation. All people below the minimum have a gross income of 0, and thus they populate an area at 0 with infinite height and still a surface of 27.3%. An alternative approach would be to eliminate them from the discussion altogether, but this is against the basic premisse that we should take account of what is happening. By consequence:

PensParadePlot[PlotRange -> {0, 150}];

[Graphics:Images/index_gr_86.gif]

Results[WithTaxVoid, Gross, LorenzPlot] = LorenzPlot[];

[Graphics:Images/index_gr_87.gif]

The Lorenz plot shows why the welfare state is such a major topic of discussion.

The Gini from the gross market situation follows from this.

Results[WithTaxVoid, Gross, Gini] = Gini[] 
[Graphics:Images/index_gr_88.gif]

Net incomes with a binding minimum wage

In this case, the people below the mininimum wage receive a net benefit. It is useful to adjust the levy scheme so that it reflects the case that these benefits are not taxed. The rates in the range between subsistence and minimum wage must be adjusted such that payments for taxes and premiums remain the same.

adj = AdjustRates[WithTaxVoid]
[Graphics:Images/index_gr_89.gif]

We can find that this gives the same minimum wage.

SetTaxAndPremium[Premium, Tax]/. adj
FindMinimumWage[]
[Graphics:Images/index_gr_90.gif]

The implied levy and net income plots are as follows.

LevyPlot[];

[Graphics:Images/index_gr_91.gif]

NetIncomePlot[];

[Graphics:Images/index_gr_92.gif]

Steps are required to reconstruct the Lorenz function.

NetIncomeInverse[Set, {MinimumWage}];
NetIncomeLorenz[y_] = 
    NetIncomeLorenz[Set, WithTaxVoid, y];
Results[WithTaxVoid, Net, LorenzPlot] = 
    LorenzPlot[NetIncomeLorenz];

[Graphics:Images/index_gr_93.gif]

Results[WithTaxVoid, Net, Gini] =
    Gini[NetIncomeLorenz]
[Graphics:Images/index_gr_94.gif]

And we see the so-called "huge effect of the welfare state".

Results[WithTaxVoid, Gross, Gini] - Results[WithTaxVoid, Net, Gini]
[Graphics:Images/index_gr_95.gif]
NetIncomeLorenz[Table]
[Graphics:Images/index_gr_96.gif] [Graphics:Images/index_gr_97.gif] [Graphics:Images/index_gr_98.gif] [Graphics:Images/index_gr_99.gif] [Graphics:Images/index_gr_100.gif] [Graphics:Images/index_gr_101.gif] [Graphics:Images/index_gr_102.gif]
[Graphics:Images/index_gr_103.gif] 2.184` 0.` 41.5` 27.3` 0.` 17.2`
[Graphics:Images/index_gr_104.gif] 1.674` 67.393` 39.297` 48.199999999999996` 18.3` 33.6`
[Graphics:Images/index_gr_105.gif] 1.381` 70.931` 39.585` 65.5` 37.5` 50.`
[Graphics:Images/index_gr_106.gif] 0.154` 8.924` 4.875` 67.4` 40.` 52.1`
[Graphics:Images/index_gr_107.gif] 0.177` 10.601` 5.752` 69.6` 42.8` 54.400000000000006`
[Graphics:Images/index_gr_108.gif] 2.085` 162.967` 86.188` 95.7` 87.` 90.3`
[Graphics:Images/index_gr_109.gif] 0.345` 47.733` 23.424` 100.` 100.` 100.`
[Graphics:Images/index_gr_110.gif] 8.` 368.548` 240.621` - - -

The Pareto improved situation: But with rate 1

We now set the thresholds at exemption (= subsistence * replacement rate), which allows the move of the minimum wage to subsistence.

Note: When the minimum is adjusted, then this does not mean that those workers with a productivity at the original gross wage would earn less. The gross wage distribution does not change. Of course, there is a discusison in the literature that it would change - somewhat.

Determine the new tax schedule

    adj = AdjustRates[NoTaxVoid]
[Graphics:Images/index_gr_111.gif]

Clearly, the minimum wage now equals subsistence * replacement rate.

    SetTaxAndPremium[Premium, Tax] /. adj; 
    FindMinimumWage[]
[Graphics:Images/index_gr_112.gif]

But we also find a tax rate of 100%.

Results[NoTaxVoidRate1, LevyPlot] = LevyPlot[];

[Graphics:Images/index_gr_113.gif]

Results[NoTaxVoidRate1, NetIncomePlot] =
    NetIncomePlot[];

[Graphics:Images/index_gr_114.gif]

A tax rate of 100% is not so bad as sometimes suggested. Nevertheless, even when one dislikes it, it still is useful to regard its consequences.

The gross wage results

ResetWageDensity[];
FromWageDensity[x, Share, WageDensity, MinimumWage]
[Graphics:Images/index_gr_115.gif]
[Graphics:Images/index_gr_116.gif]
Plot[WageDistribution[x], {x, 0, 150}];

[Graphics:Images/index_gr_117.gif]

PensParadePlot[PlotRange -> {0, 150}];

[Graphics:Images/index_gr_118.gif]

Results[NoTaxVoidRate1, Gross, LorenzPlot] = LorenzPlot[];

[Graphics:Images/index_gr_119.gif]

The Gini from the gross market situation follows from this.

Results[NoTaxVoidRate1, Gross, Gini] = Gini[]
[Graphics:Images/index_gr_120.gif]

Net income at the new situation

NetIncomeInverse[Set, {MinimumWage}];
NetIncomeLorenz[y_] = NetIncomeLorenz[Set, NoTaxVoidRate1, y];
Results[NoTaxVoidRate1, Net, LorenzPlot] = LorenzPlot[NetIncomeLorenz];

[Graphics:Images/index_gr_121.gif]

We find that the Gini is much more equal.

Results[NoTaxVoidRate1, Net, Gini] = Gini[NetIncomeLorenz]
[Graphics:Images/index_gr_122.gif]
Results[NoTaxVoidRate1, Gross, Gini] - Results[NoTaxVoidRate1, Net, Gini]
[Graphics:Images/index_gr_123.gif]

However, in this model we have forgotten that we don't have to pay benefits to the former unemployed. The total net income that we presume is the same as the Tax Void case, and that is inconsistent. (Though there is some small difference, due to numeric approximations.) We can, and must, use these saved benefits to adjust the tax schedule.

NetIncomeLorenz[Table]
[Graphics:Images/index_gr_124.gif] [Graphics:Images/index_gr_125.gif] [Graphics:Images/index_gr_126.gif] [Graphics:Images/index_gr_127.gif] [Graphics:Images/index_gr_128.gif] [Graphics:Images/index_gr_129.gif] [Graphics:Images/index_gr_130.gif]
[Graphics:Images/index_gr_131.gif] 0.42` 0.` 7.989` 5.3` 0.` 3.3000000000000003`
[Graphics:Images/index_gr_132.gif] 0.087` 1.852` 1.808` 6.3` 0.4` 4.`
[Graphics:Images/index_gr_133.gif] 1.428` 39.91` 29.845` 24.2` 10.` 16.2`
[Graphics:Images/index_gr_134.gif] 1.718` 66.587` 39.223` 45.7` 25.900000000000002` 32.300000000000004`
[Graphics:Images/index_gr_135.gif] 0.215` 9.739` 5.554` 48.4` 28.199999999999996` 34.599999999999994`
[Graphics:Images/index_gr_136.gif] 0.254` 11.908` 6.747` 51.5` 31.` 37.4`
[Graphics:Images/index_gr_137.gif] 3.311` 217.962` 117.424` 92.9` 83.1` 85.5`
[Graphics:Images/index_gr_138.gif] 0.567` 70.81` 35.382` 100.` 100.` 100.`
[Graphics:Images/index_gr_139.gif] 8.` 418.767` 243.972` - - -

The Pareto improved situation

Welfare analysis suggest that we must use the saved benefits to reduce taxes. If politicians want to raise taxes again for other purposes,  e.g. to pay off the National Debt, then this would need political debate. Ceteris Paribus, however, it is welfare improving to find a better marginal rate for the first bracket after the thresholds. Reducing taxes keeps premiums as they were: with the same revenue at the original gross minimum wage levels. This is in line with the idea that the 'premiums' below the minimum wage are not premiums in the first place (they are statutory, but are not cashed).

Determine the new tax schedule

The new levy sum can be derived from the current revenue minus the saved benefits.

LevySum = Results[WithTaxVoid, LevyRevenue] - EmployableOnBenefit * Subsistence
[Graphics:Images/index_gr_140.gif]

We use Rate as the unknown parameter.

Clear[Rate]; adj = AdjustRates[Rate]
[Graphics:Images/index_gr_141.gif]
SetTaxAndPremium[Premium, Tax] /. adj;
Rate = FindRateNTV[LevySum]
[Graphics:Images/index_gr_142.gif]
[Graphics:Images/index_gr_143.gif]
[Graphics:Images/index_gr_144.gif]
[Graphics:Images/index_gr_145.gif]
[Graphics:Images/index_gr_146.gif]

And thus we find:

TaxRates
[Graphics:Images/index_gr_147.gif]
LevyPlot[];

[Graphics:Images/index_gr_148.gif]

NetIncomePlot[];

[Graphics:Images/index_gr_149.gif]

The gross wage results

In case there is any doubt:

FindMinimumWage[]
[Graphics:Images/index_gr_150.gif]

Given this minimum, the results for the WageDensity are the same as for the Rate = 1 case. But we reproduce them here anyway.

ResetWageDensity[];
Results[NoTaxVoid, Gross, Employment] = 
    Total[Number, Exemption]
[Graphics:Images/index_gr_151.gif]
Results[NoTaxVoid, Gross, WageSum] = 
    Total["Wage", Exemption]
[Graphics:Images/index_gr_152.gif]
FromWageDensity[x, p, WageDensity, Exemption] 
[Graphics:Images/index_gr_153.gif]
[Graphics:Images/index_gr_154.gif]
Plot[WageDistribution[x], {x, 0, 150}];

[Graphics:Images/index_gr_155.gif]

PensParadePlot[PlotRange -> {0, 150}];

[Graphics:Images/index_gr_156.gif]

Results[NoTaxVoid, Gross, LorenzPlot] = LorenzPlot[];

[Graphics:Images/index_gr_157.gif]

The Gini from the gross market situation follows from this.

Results[NoTaxVoid, Gross, Gini] = Gini[]
[Graphics:Images/index_gr_158.gif]

Net income at the new situation

NetIncomeInverse[Set, {MinimumWage}];
NetIncomeLorenz[y_] = 
    NetIncomeLorenz[Set, NoTaxVoid, y];
Results[NoTaxVoid, Net, LorenzPlot] = 
    LorenzPlot[NetIncomeLorenz];

[Graphics:Images/index_gr_159.gif]

Results[NoTaxVoid, Net, Gini] = Gini[NetIncomeLorenz]
[Graphics:Images/index_gr_160.gif]
Results[NoTaxVoid, Gross, Gini] - Results[NoTaxVoid, Net, Gini]
[Graphics:Images/index_gr_161.gif]

Conclusions

Results

dofor = TaxRegime[Cases]
[Graphics:Images/index_gr_162.gif]
(Results[#1, Difference, Gini] = 
Results[#1, Gross, Gini] - Results[#1, Net, Gini])& /@ TaxRegime[Cases];
Bank[TableHeadings] = {dofor, 
                   {Gross, Net, Difference}, {Gini}};
Bank[Data] = Outer[Results, Sequence @@ Bank[TableHeadings]];
Bank[Range] = {};

Bank[Show]
[Graphics:Images/index_gr_163.gif] [Graphics:Images/index_gr_164.gif] [Graphics:Images/index_gr_165.gif] [Graphics:Images/index_gr_166.gif]
[Graphics:Images/index_gr_167.gif]
0.2763263929549882`
0.23269260306973405`
0.043633789885254126`
[Graphics:Images/index_gr_168.gif]
0.4274431466078643`
0.2103991882648213`
0.21704395834304302`
[Graphics:Images/index_gr_169.gif]
0.2973841411428987`
0.19824587760637546`
0.09913826353652322`
[Graphics:Images/index_gr_170.gif]
0.2973841411428987`
0.1971014606082171`
0.10028268053468159`

Which are the results mentioned in the introduction.


[Graphics:Images/index_gr_171.gif] [Graphics:Images/index_gr_172.gif] [Graphics:Images/index_gr_173.gif] [Graphics:Images/index_gr_174.gif]
[Graphics:Images/index_gr_175.gif] [Graphics:Images/index_gr_176.gif] [Graphics:Images/index_gr_177.gif] [Graphics:Images/index_gr_178.gif]
[Graphics:Images/index_gr_179.gif] [Graphics:Images/index_gr_180.gif] [Graphics:Images/index_gr_181.gif] [Graphics:Images/index_gr_182.gif]

As said, we have lumped all unemployment into minimum wage unemployment, and thus this has given only a 'worst case' result.

One final note: There appears to be little difference between the Net Gini values and Net Lorenz plots. This is a bit disappointing to the present author, since I had thought that the Pareto improvement would also show up clearly in the Net situation. But the Gini is not a full welfare analysis. The measure neglects unemployment. For a proper decision, we should look at the social welfare function. However, if unemployment could be entered as a real tax, and similarly for other issues, then the Current Net Gini would be larger, and then the originally expected larger difference would show up. This exercise however has not been performed.

Show[Results[Default, Gross, LorenzPlot], 
    Results[WithTaxVoid, Gross, LorenzPlot],
    Results[NoTaxVoidRate1, Gross, LorenzPlot],
    Results[NoTaxVoid, Gross, LorenzPlot]];

[Graphics:Images/index_gr_183.gif]

Show[Results[Default, Net, LorenzPlot], 
    Results[WithTaxVoid, Net, LorenzPlot],
    Results[NoTaxVoidRate1, Net, LorenzPlot],
    Results[NoTaxVoid, Net, LorenzPlot]];

[Graphics:Images/index_gr_184.gif]

Appendices

Literature

Caminada, Koen, and Kees Goudswaard (2000a), "Redistribution of Income and Social Policy. A budget incidence analysis for the Netherlands in an international perspective", research paper Leiden University

Caminada, Koen, and Kees Goudswaard (2000b), "International Trends in Income Inequality and Social Policy", research paper Leiden University

Cool, Thomas (1999), "The Economics Pack, Applications of Mathematica", Thomas Cool Consultancy & Econometrics, ISBN 90-804774-1-9

Cool, Thomas (2000), "Definition and Reality in the General Theory of Political Economy", Samuel van Houten Genootschap, ISBN 90-802263-2-7

Galbraith, James K. (1998), "Created Unequal", Free Press

Lambert, Peter J. (1985), "Advanced mathematics for economists", Blackwell

Odink, J.G. (1985), "Inkomensherverdeling" (Income redistribution), Wolters-Noordhoff


Converted by Mathematica      October 27, 2000