in the theory of VI alue and prices. 15 



We may call the perpendicular direction III S the "maximum 

 direction." It has the important property that its components III a 

 and III /? are proportional to the marginal utilities of A and B. 

 This follows from a theorem^ of vector calculus or thus : III a and 

 III /? are inversely proportional to OAg and OBg, that is directly 

 proportional to the prices of A and B and therefore proportional to 

 their marginal utilities. f 



If (fig. 25) the separate curve systems of all individuals I, II, etc. 

 are drav^^n, and the lines AB drawn in each case, they will be paral- 

 lel. For the prices are uniform among all individuals and OA and 

 OB in each case are inversely as the prices. 



Since the normals to these lines will also be parallel, this theorem 

 may be stated: The '' maximii7n directions " of all are alike. 



§16. 



These methods apply to the comparison of any tioo commodities 

 and afford a means of graphically representing statistical relations 

 connecting the demands for two articles so far as the variations in 

 the quantities of other articles can be eliminated. 



The same principles apply to the production of two articles. Hides 

 and tallow are completing articles from a jjroducer's standpoint. 

 Likewise coke and coal gas, mutton and wool, and in general any 

 article and its " secondary product." 



On the other hand most articles are competing or substitutes from 

 a producer's point of view. The difficulty of producing cloth is 

 greatly increased if the same individual produces books. This is 

 the root of the principle of division of labor and leads to that im- 

 portant contrast between production and consumption once before 

 alluded to. This and other contrasts will be mentioned in Appen- 

 dix II, § 8. Marshall and others are fond of using the expression 

 " fundamental symmetry of supply and demand." This notion must 

 be supplemented by that of a " fundamental asymmetry." As social 

 organization progresses each man (and each community or nation) 

 tends to become producer oi fewer things but consumer of ouore. 



* Gibbs, Vector Analysis, §§ 50-53. 



f For by similar tnanp;les: :r^r:r-^ = z~--r- = -^ ~ = — 

 ' -^ ^ III /? Ill ji OAs pi. 



