[45] Relation (13.1b) gives the Bentham tax function, that has exemption x and marginal rate r. We will write Bentham[y] for the Bentham tax, Tax[y] for a special nonlinear function, and T[y] for a general function.
[46] Welfare states commonly distinguish the minimum earning wage and the minimum on benefit. In Holland the latter is 70% of the former, thus some S = 0.7 B. But then there are subsidies that apply to people on benefit - and the poverty trap discussion starts. Here it suffices to take S = B. Chapter 39 deals with the argument in reduced form fashion.
[47] In chapter 28 we will develop the formula for the influence of indexation on minimum wage M.
[48] This graph gives the theoretical values for the Dutch minimum wage, if indexation since 1951 had been rigorously applied with inflation for exemption and net average income for subsistence. The actual minimum wage however was different, but within range. OECD (2000:40) Chart 2.1 graphs the observed real minimum with 1975 = 100 with for example {1970, 77}, {1978, 108}, {1996, 85}.
[49] Common themes in tax theory are merit versus demerit goods and that one would tax the less mobile factor labour rather than capital. These themes have less priority than the tax void. The main reason that remains for a VAT (or a profit tax) is that the government wants to monitor the economic process.
[50] These are virtual subsidies only: while handed out, they are immediately cashed in under the tax rule.
[51] Note the bureaucratic mind-set: There is a tax system and thus people are supposed to pay taxes. Benefits are established at a net level but are recalculated to a gross level so that the Ministry of Finance is happy again that it can levy taxes. By consequence it also seems as if much more is paid on benefits.
[52] In general http://econwpa.wustl.edu. More specifically Colignatus (1996a, c) at http://econwpa.wustl.edu/months/get/9604.html, and Colignatus (1998a) at http://wueconb.wustl.edu/eprints/get/papers/9808/9808002.abs
[53] The OECD (1998) “Employment Outlook” Table 2.3 gives an international comparison of the level of the minimum wage in relation to the median wage. The situation in the Netherlands may be a yardstick to interprete these data. The table shows that the minimum wage in the Netherlands (in 1997) was 55.9% of full-time median earnings (excluding overtime and bonusses). Applying that rate to the 2002 values in Table 3 gives an estimate of the 2002 median of € 27,975. However, the proper subsistence wage should rather be € 12,516, i.e. net of taxes and premiums. The ratio thus is rather 44.7% than 55.9%. The rate could even be lower when we consider VAT and other taxes and the possibility of some employment subsidy, so that 30% could well be attainable. With this yardstick, the OECD levels of the minimum wage are strikingly high.
[54] The analysis in chapter 13 holds in theory for full time workers. In reality, only part of this Tax Void Unemployment will be on benefit, since a part will substitute for part-time work (at a wage lower than full-time subsistence). A practical question is also whether the tax statute really must, and if so can, distinguish properly between full-timers and part-timers. These questions need to be answered, and definitely so when a practical run is done with a general equilibrium model.