Manchester Memoirs, Vol. xli. (1897), No. 12. 13 



APPENDIX. 



Let P n be the declared value of (say) the imports in 

 any year, P n _ x in the preceding year, &c. 



Let A n be the value of a part of them, B n the value of 

 the same at the prices of the preceding year. 



Then we may write 



AJB n =pjp n ^, 



and we may also write A n =p n a n , B n =p n _ x a n , them's being 

 symbols indicating the average price-level, the a's simi- 

 larly denoting movements of quantity. 



As /4 n _! denotes the value of goods of the same kind as 

 those included in A n , B n and ^4 n _i denote the values, 

 on the same price-level, of these goods imported in the 

 two years, and we have B n /A n _ l = a n /a n _ 1 , a result which 

 follows from B n =p Q _ 1 a n , A n . 1 =p n ^ 1 a n ^ 1 . 



Let P n = A n x n , x n denoting the proportion which the 

 value of the total imports bears to those which are 

 included in the above estimates. If also Q n denote the 

 value of the preceding year's imports on the assump- 

 tion that the average change of price in the goods 

 not included in A n is identical with that of the goods 

 which are so included, R n the true value of the whole 

 imports at the prices of the preceding year, we have 



Qn = B n x n , R n = B n y n , 



where y n denotes the proportion of the whole to the part 

 evaluated at the prices of the preceding year. We can 

 accurately determine the x's, but the form of the returns 

 and the nature of some of the goods do not permit us 

 to know the numerical values of the y's. 

 We have 



Pn=pnan*n, Qn=pn-ld n X n , R n =p n -iCl n y n . 



