100 Irving Fisher — 31athematical investigatioiis 



capacities for pleasure were great would consume the most in order 

 to make the aggregate gain in the whole market a maximum). Or 

 we may destroy all the levers and re-arrange the rear thicknesses 

 until the front and back ordinates are made equal. 



In like manner the minimum disutility would be attained if all 

 marginal disutilities were equal. The maximum gain would then 

 result. This is the maximum gain obtainable -^fifAe?? the amounts of 

 each commodity consumed and produced are fixed and given. If 

 we are permitted to rearrange these amounts also, we shall secure 

 the maximum gain Avhen the marginal utilities equal the marginal 

 disutilities ; i. e. 



Under such a socialistic regime more " necessaries " and less 

 "luxuries" would be consumed and produced than previously. 

 The "rich" or powerful would produce more and consume less than 

 previously ; the poor or weak would consume more and produce 

 less. Yet for each the marginal utilities and disutilities would be 

 equal. 



It is needless to say that these considerations are no plea for 

 socialism, but they serve to clear up a subject sometimes discussed 

 by mathematical economists and reconcile Launhardt's contention* 

 that utility is not a maximum with Aus23itz und Lieben's that it is. 

 The former unconsciously has reference to equation (4) which is not 

 true, the latter to equation (3) which is.f 



IV. ELIMINATION OF VARIABLES. 



The four sets of equations, Part I, Ch, IV, § 10, can be reduced. 



7 T^ 



We may substitute for — — its value F(A J and thus eliminate all mar- 



ginal utilities. Moreover we can get an expression for jt?^, p^, etc., 

 in terms of commodities. First, if m = n the second set of equa- 

 tions are easily solved bj^ determinantsj giving :§ 



* Volkswirthschaftslelire under " Widerliolte Tausch." 



f Auspitz und Lieben appear to overlook this difference of standpoint. 

 Preface, p. xxv. 



\ Burnside and Panton. Theory of Equations, p. 251. 



§ This equation does not mean that any arbitrary values can be assigned to 

 Ai, Bi, etc., and the resulting price of A be so simply exj^ressed ; only when Ai, 

 Bi, etc. satisfy all the conditions of Ch. IV, §10 will the jDrice be expressible aa 

 the quotient of the two determinants. 



