Indirect and Chain Indices

Having dealt with binary comparisons in Part I, we now move on to consider multiple comparisons, i.e. comparisons involving more than two years. The main problem here is how to ensure that all possible binary comparisons between a set of N years are consistent with each other. The value, quantity and price relatives for single commodities all meet this requirement of being consistent with each other, because they all multiply according to the transitive law. Thus for commodity i and any three years r, s and t, we find that (5.1) V s i V r i × V t i V t i = V t i V r i ( r , s , t = 1 , … , N ) $$\frac{{V_s^i}}{{V_r^i}} \times \frac{{V_t^i}}{{V_t^i}} = \frac{{V_t^i}}{{V_r^i}}\quad \left( {r,s,t = 1, \ldots ,N} \right)$$ (5.2) Q s i Q r i × Q t i Q s i = Q t i Q r i ( r , s , t = 1 , … , N ) $$\frac{{Q_s^i}}{{Q_r^i}} \times \frac{{Q_t^i}}{{Q_s^i}} = \frac{{Q_t^i}}{{Q_r^i}}\quad \left( {r,s,t = 1, \ldots ,N} \right)$$ and (5.3) P s i P r i × P t i P s i = P t i P r i ( r , s , t = 1 , … , N ) $$\frac{{P_s^i}}{{P_r^i}} \times \frac{{P_t^i}}{{P_s^i}} = \frac{{P_t^i}}{{P_r^i}}\quad \left( {r,s,t = 1, \ldots ,N} \right)$$