116

A TREATISE ON MONEY

BE. II

4. The Chain Method

The “Chain Method” of compiling a series of Index-Numbers, which was first introduced by Marshall, isan attempt to deal with the problem of changes in thecharacter of consumption by assuming that the differ-ences are small between any two consecutive positionsin the series of positions to be compared. It alsoassumes—though this is not generally stated—that thesuccessive small errors involved are non-cumulative.A series of comparisons are made, the first of whichassumes that the composite appropriate to the firstposition is virtually equivalent to the compositeappropriate to the second position, and the next ofwhich assumes that the composite appropriate to thesecond position is virtually equivalent to the com-posite appropriate to the third.

The method is as follows :

Let pi p 2 be the prices in the first and secondpositions of the composite appropriate tothe first position ;

q 2 q 3 the prices in the second and third positionsof the composite appropriate to the secondposition ;

r 3 r 4 the prices in the third and fourth positionsof the composite appropriate to the thirdposition ; and so on.

Let n x n 2 n 2 be the series of Index-Numbers comparingthe price-levels in the successive positions.

Then the Chain Method computes n 2 , n 3 in terms ofn x thus :

n 2 =2? • n ly

Pi

n 3 ^ -n 2 =

■Vi

■ P - • Wi,

Pi

r 4

n 4 =- • n 3 =

n

q ? _ p 3

r 3

n '

<h' Pi'

w,

and so on.