746 REPORT — 1901. 



view, wlilch thinks tlie latoiirev's necessities of efficiency to be comparatively very 

 high, and which leads to a strong progression. 



As type for the usual scales of progressive taxation the following scheme may 

 serve : — 



Incomes from Os. till 500.?. pay per cent. 



500s. „ 2,500.?. „ 1 

 „ 2,.500s. „ S,500s. „ 2 „ 

 „ 8,500.?. „ 20,500s. „ 3 

 ,, above 20,500«. 4 „ 



The scale is, from a technical point of view, very crude, involving discon- 

 tinuities in taxation at every rung of the ladder. We can avoid these if we state 

 that— 



The 500 first .?. of every income pay per cent. 



„ 2.000 next s. „ „ 1 „ 



„ 6,000 „ „ ., 2 



„ 12,000 „ „ „ 3 



All following s. „ „ 4 „ 



But this scale can just as well he obtained by the method of deductions; we 

 have only to state that — 



The first 500*. shall have the right to deduce 100 per cent. 



„ next 2,000s. „ „ „ 75 „ 



„ „ 6,000«. „ „ „ 50 



„ „ 12,000s. „ „ „ 25 



All following s. ,, „ ,, ,, 



and that of the remainders a constant rate of 4 per cent, shall be paid. Generally, 

 if in the different groups the tax is to be paid at a rate of p^, p.^, p^ . . .p^ 

 per cent., the same result can be obtained by levying the tax at a constant rate P, 

 not less than any of the ^j, but granting deductions within the difl'erent groups of 



>p—^ percent., —ff-^~- per cent., and so on. 



However, even this method is primitive, and involves too much arbitrariness 

 in fixing the deductions for the different incomes. It is better to let the deduction 

 y increase with the income .v as a function of the form 



_ux + 



This contains three independent elements, to which comes the constant tax- 

 percentage P, so that the arbitrariness of the progressive scale now is reduced to the 

 choice of four elements. We denote by ethe tax-free mhiimutn of subsistence, by 

 e7n the upper limit of the deductions, i.e., of the necessaries of efhciency, and call 

 it the 7naximum of subsistence; the arithmetical medium between these two is the 



' medium of subsistence,' and is equal to e ~^ . We denote further by u and 



V the income and the deduction counted in e as a unity, so that .r = eu and y = ev. 

 We can then put 



n- 



v = m- (in — 1) p= , 



where n signifies that value of the income ?<, for which the deduction is equal to 

 the medium of subsistence. Thus we have arrived at a set of formulas where 

 each of the four constants n, on, e, and P has a clear and definite sense. 



We can reduce the arbitrariness involved in the construction of a progressive 

 scale still more if we decjde once for all that n = m + 1, i.e., that the medium of 



