Index Numbers

  1. Taking 1965 as the base year, with an index number of $100$, calculate an index number for 1972, based on weighted average of price relatives derived from the following:
    Commodities A B C
    Weights $34$ $26$ $40$
    Price per unit in 1965 $16$ $20$ $32$
    Price per unit in 1972 $24$ $19$ $36$

  2. The weights are now changed so that the weight for A is $40$ and the total weight is $100$. If the value of the index number for 1972 is now $120.5$, calculate the weights applied to B and C.


  1. Let base price per unit $= P_o$

    Let current price per unit $= P_i$

    Commodities $P_o \; \left(1965\right)$ $P_i \; \left(1972\right)$ Price relatives for 1972 $x = \dfrac{P_i}{P_o} \times 100$ Weights $\left(w\right)$ $w \cdot x$
    A $16$ $24$ $150$ $34$ $5100$
    B $20$ $19$ $95$ $26$ $2470$
    C $32$ $36$ $112.5$ $40$ $4500$


    $\Sigma w = 100$, $\;$ $\Sigma \left(w \cdot x\right) = 12070$

    Index number for 1972 $= \dfrac{\Sigma \left(w \cdot x\right)}{\Sigma w} = \dfrac{12070}{100} = 120.70$


  2. New index number for 1972 $= 120.5$

    New weight for A $= 40$

    Total weight $= \Sigma w = 100$

    Let new weight for B $= x$

    New weight for C $= 100 - 40 - x = 60 - x$

    Commodities $P_o \; \left(1965\right)$ $P_i \; \left(1972\right)$ Price relatives for 1972 $x = \dfrac{P_i}{P_o} \times 100$ Weights $\left(w\right)$ $w \cdot x$
    A $16$ $24$ $150$ $40$ $6000$
    B $20$ $19$ $95$ $x$ $95 x$
    C $32$ $36$ $112.5$ $60 - x$ $112.5 \left(60 - x\right)$


    $\begin{aligned} \Sigma \left(w \cdot x\right) & = 6000 + 95 x + 112.5 \left(60 - x\right) \\\\ & = 6000 + 95 x + 6750 - 112.5 x \\\\ & = 12750 - 17.5 x \end{aligned}$

    $\therefore \;$ New index number for 1972 $= \dfrac{\Sigma \left(w \cdot x\right)}{\Sigma w} = \dfrac{12750 - 17.5 x}{100} = 120.5$

    i.e. $\;$ $12750 - 17.5 x = 12050$

    i.e. $\;$ $17.5 x = 700$ $\implies$ $x = 40$

    $\therefore \;$ New weight for B $= 40$

    New weight for C $= 60 - x = 60 - 40 = 20$