Question 14.4: Refer to the problem scenario of Example 14.1. Management Qu......

1   Using the Laspeyres weighted average of price relatives method:
Table 14.5 shows the calculations of the above three steps based on the data in Table 14.2.

Management Interpretation

The ‘Price relatives’ column in Table 14.5 shows that a bar of soap increased in price by 7.7% from 2010 to 2011. Similarly, deodorant increased by 8.9% over the same period, while a tube of toothpaste increased in price by 7.2% over the past year.

In terms of the base period values, deodorant is the highest-valued item in the toiletries basket (i.e. R351.60 out of R511.81 = 68.7%), followed by toothpaste (R88.06 out of R511.81 = 17.2%), with soap being the lowest-valued item (i.e. only R72.15 out of R511.81 = 14.1%).

The weighting of each individual item’s price relative by its respective base period weight is shown in the ‘Weighted price relatives’ column. Finally, to derive the composite price index, the sum of the weighted price relatives (R55 499.83) is divided by the base value of the basket (R511.81).

Thus the Laspeyres (price relatives) composite price index is \frac{55499.83}{511.81} = 108.4 (using Formula 14.9). Note that this index value is exactly the same as for the Laspeyres method of weighted aggregates.

This calculation is equivalent to working out the percentage value of each item in the basket (i.e. for soap: \frac{R72.15}{R511.81} = 14.1%; for deodorant: \frac{R351.6}{R511.81} = 68.7%; and for toothpaste: \frac{R88.06}{R511.81} = 17.2%) and then calculating a weighted average of the price relatives.

Weighted average index = (107.7 × 0.141) + (108.9 × 0.687) + (107.2 × 0.172) = 108.4

2   Using the Paasche weighted average of price relatives method: Table 14.6 summarises the above three steps based on the data in Table 14.2.

From Table 14.6, it can be seen that the price relative for each toiletries item is the same as in Table 14.5, since it is based on the same Formula 14.2.

The calculation of the base period value for each item in the basket, however, changes. It is found by multiplying each base period price, p_0, by its current period quantity consumed, q_1, to give the value (weight) of each item in the basket (i.e. p_0 \times q_1). As seen from these base period values, deodorant is the highest-valued item in the toiletries basket (i.e. R263.7 out of R442.34 = 59.6%), while soap is the least-valued item (i.e. only R78 out of R442.34 = 17.6%).

The last column first multiplies each item’s price relative by its base value to produce a weighted price relative per item, and then divides the sum of these weighted price relatives (47 906.14) by the value of the basket (442.34) to give the composite price index.

Thus the Paasche (price relatives) composite price index is \frac{47906.14}{442.34} = 108.3 (using Formula 14.10). Note that this index value is the same as for the Paasche method of weighted aggregates.