TKANS ACTIONS OF SECTION F. 687 



(with certain conditions as to the second term of variation). To wliich is to be added 

 the equation («). We have thus n equations relatino^ to the t"> individual. The same 

 being true of each of the m. individuals, we have in all m n equations of the forms (a) 

 and (;8). We have also (7), from the condition that everjtliing which is bought is 

 sold, and conversely, n equations of the following form : w^, + r„, + ^-c. + Xj^ =0. 



But of the (/« + «) equations of the forms (//) and (7) only (m + n-l) are 

 independent. For adding the m equations of the form («) we have : 



(j-„ + a-2,+ . . . +.r„„ \ 

 +A>(«12+a'22+ • • • +3-m-. \ ^o. 



+A.(a',n + »2n+ • • • +■'•„,„) 



Now, if any (» - 1) of the equations of the form 7, say all but the first, are given, 

 then in the last written equation the coefficients of p., . . . . y^n vanish. Therefore 

 the first equation of the form (7), viz. a?,, H-a-.,, +4-t'. + a!,„„ is also given. We have 

 thus w« + (/i — 1) equations to determine ?«« + (;»— 1) quantities, viz. the a- variables, 

 which are mn in number, and the (» — l)^^'s. 



The great lesson to be learnt is this. The equations arc gimiiltaneons, and their 

 solution determinate. That the factors of economic equilibrium are simultaneously 

 determined is a conception which few of the literary school have received. The 

 reader is referred to Prof. Walras' ' Legon ' 12 (' Econ. Pol.' 2nd ed.) for a lengthier 

 exposition, and for a more accurate one to Messrs. Auspitz and Lieben's Appendix IV. 



(/) Commercial Competition. — Abstracting that change of occupations which 

 Cairnes ascribed to ' Industrial ' as distinguished from ' Commercial,' competition 

 (comp. Sidgwick's 'Pol. Econ.' book ii. ch. 1), let us suppose that the x's of the last 

 note, which primarily denoted commodities ready for immediate consumption, include 

 also agencies of production, (the use of) land, labour, and capital. We may conceive 

 entrepreneurs buying these agencies from landlords, labourers, and capitalists, and 

 selling finished products to the public. We have thus the appropriate idea of rent, 

 wages, interest, and (normal) prices determined mnultaneondy (in the mathema- 

 tical sense). 



In a primary view of complex exchange it is proper with Jevons to regard each 

 portion of commodity sold, each negative variable, say — avj, as a deduction from an 

 initial store, say \rs- But when we consider production, we regard | as a function of 

 the outlay of the entrepreneur. Supposing that the entrepreneur confines himself to 

 the production of a single article, let the gross produce, in money, after replacing 

 cai)ital, be fricr, I,), w^here /,. is a function depending on the individual's skill, 

 energy, opportunities, &c., Cr is the amount of capital borrowed by him, and Ir the 

 number of acres of a certain quality which he rents. The net produce is obtained 

 by deducting from this quantity the payments ^;.n- Z,.p, where 1 is the rate of interest 

 and p is the rent per acre. Thus the advantage which the entrepreneur seeks to 

 maximise is of the form 



Wa^n. »r2 • • -C/rCc,., lr)-Crl-lrp]-XrrPrr, • • ■) ! 



whence 4^ = « and -,v" = p. The first of these equations expresses a well-known pro- 

 dc,. dlr 



position regarding the final utility of capital. The second equation expresses a less 

 familiar condition with respect to the number of acres which will be rented on an 

 ideal supposition of the homogeneity and divisibility of land above the margin of 

 cultivation. 



What then, and where, is the Ricardian theory of rent ? Its symbolic statement 

 is Irp =f(er, Ir) -f(Cr< <>) =/r(Pr. ^r) - Cr X ' ; where /(cv, o) is the gross produce of 

 Cr capital laid out by the individual numbered r, on land below the margin obtainable 

 for nothing in as large quantities as desired. It will be found that these equations 

 postulate that the quantitj' of land above the margin is small as compared with the 

 number of applicants, and that /(cr, o) is identical with Cr x 1, which are the common 

 Ricardian assumptions. The validity of these assumptions as a first approximation, 

 the need of correction where greater accuracy is required (truths which some 

 minds seem incapable of holding together), have been admirably pointed out by 

 Mr. Sidgwick ('Pol. Econ.,' Book ii. chap. 7, § 2). The second approximations 

 made by him may be usefully ex])ressed in the symbols which have been proposed, 

 or rather in those which the student may construct for himself. I do not put 



