5^4 



NATURE 



\_Sept. 19, 1889 



the fir-vt assumption a1)ove made: Xrx + p^x,-^ + &c. +/'„jr,.„ 

 = o. In order to determine the maximum oi^r subject to this 

 condition, we obtain (;8) by the calculus of variations {n - \) 

 equations of the form — 



(dK 



\dXr 



f-}\dXr.^ 





(with certain conditions as to the second term of variation). To 

 which is to be added the equation (a). We have thus n equa- 

 tions relating to the rth individual The same being true of 

 each of the m individuals, we have in all mn equations of the 

 forms (a) and (.8). We have also (7), from che condition that 

 everything which is bought is sold, and conversely, n equations 

 of the following form : x,, + x^, + &c. + Xm = o. 



But of the {m + n) equations of the forms (o) and (7) only 

 {in + It - i) are independent. Foriadding the m equations of 

 the form (a) we have : — 



+ pn{Xin + -^2" + 



< T" Xjn) 

 • + x„,, 



"r Xmn ' 



Now, if any (« ~ l) of the equations of the form 7, say all 

 but the t rst, are given, then in the last written equation the 

 coefficients of /a • • • ./n vanish. Therefore the first equation 

 of the form (7), viz. x^ + x.^-^ + &c. + Xm^, is also given. We 

 have thus mn + (« - l) equations to determine mn + (« - i) 

 quantities, viz. the x variables, which are vin in number, and 

 the {n - l)/'s. 



The great lesson to be learnt is this. The equations are 

 simultaneous, and their solutions determinate. That the factors 

 of economic equilibrium are simultaneously determined is a con- 

 ception which few of the literary school have received. The 

 reader is referred to Prof. Walras's " Le9on " 12 (" Econ, 

 Pol.," second ed.) for a lengthier exiosition, 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 distin- 

 guished from " commercial " competition (comp. Sidgwick's 

 " Pol. Econ.," book ii. ch. i.), let us suppose that the ^s of the 

 last note, which primarily denoted commodities ready for im- 

 mediate consumption, include also agencies of producti n : 

 (the use of) land, labour, and capital. We may conceive entre- 

 preneurs 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 sifnultaneously (in the mathematical sense). 



In a primary view of complex exchange it is proper, with 

 Jevons, to regard each portion of commodity sold, each negative 

 variable, say - Xrs, as a deduction from an initial store, say 

 |„. But when we consider production, we regard | as a func- 

 tion of the outlay of the entrepreneur. Sujiposing that the 

 entrepreneur confines himself to the production of a single 

 article, let the gross produce, in money, after replacing 

 capital, be fr{cr, Ir), where fr 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 Cyi - Irp, 

 where t is the rate of interest and p is the rent per acre. Thus 

 the advantage which the entrepreneur seeks to maximize is of 

 the form — 



^^Xr^, Xr.2 . . . [/-(^r, /,•) - Crl - /,-p] - ~r,.rP,- 



.). 



whence ^'^ = t and ^''=p. The first of these equations ex- 



dCr dlr 



presses a well-known proposition 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{cr, h) - /{c,-,^) =fr{Cr, /r) - ^^r x t ; 

 where f(e,:, 0) 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 quantity of land above the 



margin is small as compared with the number of applicants, and 

 that /{cr, 0) is identical with Cr x i, 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. ch. vii. § 2). The second ap- 

 proximations made by him may be usefully expressed in the 

 symbols which have been proposed, or rather in those which the 

 student may construct for himself. I do not put forward those 

 which occur to me as the best — if, indeed, there is any abso- 

 lutely best in the matter of expression. For some purposes it 

 would have been proper to take account of the various qualities 

 of land (as I have elsewhere done — " Brit. Assoc. Rep.," 1886). 

 For other purposes it would be well to put labour hired by the 

 entrepreneur as an independent variable. When this or any other 

 variable is omitted, we are to understand that there is implied 

 the best possible arrangements with respect to the variables 

 which are not expressed. The nature of this implication is 

 shown in the following note. 



{g) So far we have been taking for granted that the entre- 

 preneur does his best, without reference to the motives acting 

 upon him, the pleasures procurable by the sale of his product. 

 Formally it would be proper to take account that the utility- 

 function 4'r involves the ejfort, say e,-, explicitly, as fatigue 

 diminishes advantage, and implicitly, as exertion increases pro- 

 duction. Corresponding to the new variable we have a new 

 equation, thie complete differential of if/,- with reference to er, say 



f^) + {'^:^j\ ^fr = o. It is a nice question how far effort 

 \ derl \djr' der 



should be regarded as an independent variable ; how far the 

 essential principle of piece-work prevails in modern industry. 



{h) Industrial Competition, — The condition that net 

 advantages should be equal in industries between which there is 

 mobility may thus be contemplated. Let us put the advantage 

 of an individual, say No. r, engaged in the occupation j as a 

 function of his net income, the price of the articles in which his 

 expenditure is made, and the disutility of effort. Say <^,v,(fr,,(7r,, 

 TFg . . . . ers\ Pi, p2 • • • — ^rs); where <^,! is a utility-function, 

 not necessarily the same for the same individual in different 

 occupations, since his indulgences may vary with the nature of 

 his employment ; f„— a symbol not identical with the / of the 

 last but one note — is the individual's net earnings in the business 

 s, involving prices Tr,, tt.,, &c. , of all manner of agents of pro- 

 duction, involving also, as stated in note {g), the effort e.-s ;pi,pn, 

 &c., are prices of articles of consumption as a function of which 

 the individual's advantage may be obtained by means of the 

 equations (o) and (j8) in note (^)— eliminating the quantities con- 

 sumed. The last variable in the function <p„, the explicit ers, 

 has a negative sign prefixed, to indicate that the direct effect of 

 increased fatigue is diminished advantage. 



The equation of net advantages imports that the advantage, 

 tprr, of the occupation which the individual chooses is not less 

 than (prs, the advantage of any other occupation open to him. It is 

 important to observe that for all occupations the complete differ- 

 ential with regard to e is zero ; in symbols Ifrj-r + ( / ) ~ °- 

 But this equation conveys no presumption that the final dis- 

 utility in different occupations is the same that ( ^'-- ) = ( -^ 



The equation of final disutility holds only where efforts and 

 sacrifices are capable of being applied in " doses " to any num- 

 ber of occupations. The latter is the only case, I think, con- 

 templated by Jevons in his analysis of cost of production 

 ("Theory," ch. v.). The inquiry, what is meant in general by 

 saying that the cost of production of two articles is equal, must 

 start from right conceptions about final and total utility. But 

 this is not the place to follow up the difficult investigation. I do 

 not attempt here to discuss any matter fully, but only to illus- 

 trate the suitability of the subtle language of mathematics to 

 economical discussions. 



(f) Prof. Walker's Theory of Business Profits. — Prof. 

 Walker's theory as stated in the Quarterly Journal of 

 Economies for April 1887, involves the proposition that the 

 remuneration of the lowest, the least gifted employers, is on 

 a level with that of the labouring class. Concerned as we are 

 here with methods rather than results, it is allowable to posit 



