SOME DOCTRINES OF POLITICAL ECONOMY. 



133 



18. For the class of commodities D, (m negative,) if the price rises, the quantity effectually 

 demanded diminishes in a higher ratio. 



In the case supposed before, in which a fall of price to one-half produced a fourfold 

 amount of sale, we might at first, perhaps, suppose that the rule applies in the inverse order 

 of change, and that if the price be again doubled, the quantity demanded will again fall to 

 one-fourth, and the money demand to one-half. But this result will not be given by the 

 formula. For in this case m was = - 2 ; and if when p'q = pq{\ - 2a?) we suppose q to be 



-q, we have 1 + x = 4(1 - Zoo), which would give a? = - , denoting an increase of price of 



4 ^ 



one-third, and a money demand diminished to one-third. 



19. This apparent inconsistency arises from the formula p'q =pq (l + rnx) being made to 

 rest on a given medium standard price and quantity, p and q, and to express the changes by 

 an increment or decrement x. Hence large changes are not proportionally the same above 

 and below the standard point. 



1 / x\ 

 When m = , we have the money demand = pq I 1 I , and hence, 



since 



1 - 



(X\ 2 

 1 I ■> P = P + ">)y we nave 9 ~ 1 • 



5 

 When the quantity increases by one and one-half times its value, we have this = - , whence 



x = — , and the price is reduced to one-half, as in the case above stated. When the quantity 



m 

 l - - 



2 1. .1 



decreases by one-half, we have — ^ = - , which would give x = - , and the price is in- 



1 T X >£ ~ 



creased by one-half. 



x 



1 



2 2 2 



When the quantity decreases by one-third, we have — — = - ; whence x = - , the price 



.9 .5 



is increased in the ratio - : the money demand is diminished in the ratio - . 



20. It appears from what has been said, that we have four classes of commodities, which 

 differ according to different values of m, the susceptibility of change in the price by change 



money demand. It would be easy to devise a formula which 

 should more nearly represent Mr. Tooke's progression; but 

 even if his numbers were derived from facts, they would, of 

 themselves, be insecure grounds for generalization. And the 

 nature of the progression makes it probable that the progression 

 is hypothetical merely : the third difference being constant : as 

 appears thus: 



3 8 16 28 45 



3 5 8 12 17 



2 3 4 5 



1 1 1 



The general term of the series 3, 8, 16, 28, 45 is 

 n(w+l)(w+2) |2n _ 

 6 



