Relative indices

The formulas for the relative price indices are:

Weighted by quantity

(Σ(Rel × Q₀) / ΣQ₀) × 100

Weighted by value

(Σ(Rel × Vo) / ΣVo) × 100

In these formulas:

• Rel is the relative price for each product, calculated as P1/P0.
• Q₀ is the base-year quantity.
• V0 (written as Vo in the formula) is the base-year value, calculated as Q0 × P0.

Let’s work through an example to see how the formulas are applied.

Calculate the base weighted relative price index using weights of:

1. Quantities
2. Value (in our case, revenue)

Here, the base year is 2020 and the current year is 2024.

Solution

First, calculate the components needed for the formulas.

Based on quantities

Use the quantity-weighted formula:

Σ(Rel × Q₀) / ΣQ₀) × 100

Calculations

$$\frac{802,20}{650}\times 100$$

$$=123,41$$

Explanations The table provides the two totals you need:

• Σ(Rel × Q₀) = 802,20
• ΣQ₀ = 650

Substitute these totals into the formula and multiply by 100 to convert the result into an index.

Based on value

Use the value-weighted formula:

(Σ(Rel × Vo) / ΣVo) × 100

Calculations

$$\frac{12160}{9780}\times 100$$

$$=124,34$$

Explanations Again, the table provides the totals used in the formula:

• Σ(Rel × V0) = 12160
• ΣV0 = 9780

Substitute these totals into the formula and multiply by 100.

In an exam, use the weighting method the question asks for (either quantity weights or value weights).

Sometimes you’ll be asked to use current-year weights instead of base-year weights. In that case, you must rebuild the table using current-year quantities or values.

The next example shows the same idea using current weightings (2024).

Based on current quantities

(Σ(Rel × Qᵢ) / ΣQᵢ) × 100

Calculations

$$\frac{1461,95}{1177}\times 100$$

$$=124,21$$

Current values

(Σ(Rel × Vᵢ) / ΣVᵢ) × 100

Calculations

$$\frac{28356,56}{22600}\times 100$$

$$=125,47$$

Quantity indices

Instead of measuring change using prices, you can measure change using the quantities purchased over time.

For quantity indices, the weights are prices. Use the following formula:

( Σ( W × (Qᵢ / Q₀) ) / ΣW ) × 100

Let’s use an example to see how the formula works.

Using 2020 as the base year, calculate the relative quantity indices.

Solution

Calculations

The formula is shown above:

Calculations

$$\frac{291,93}{229}\times 100$$

$$=127,48$$

Choice of weightings, either current or base weightings

• The current weightings are more up to date, which makes them preferable; however, they are not readily available since you will have to wait until the end of the year for them to be fully produced or earlier the following year to be more realistic.
• Base weighted indices are readily available, which should speed up the process. However, we should note that the base weightings are not up to date, and as a result, they may exaggerate inflation during calculations.
• Obtaining current weights is costly than using the base weights because of the reasons mentioned in the first point.
• The current weightings are more difficult to compare year by year since they may change due to new items being added into the basket.

Index numbers can be used to calculate inflation movements as below:

Yamal receives a salary of $100 000 per year, and his salary is index-linked. The inflation index at the beginning of 2024 was 240 (end of 2023) and 300 at the end of 2024. Calculate Yamal’s salary increase based on the inflation index.

Solution

First, rebase the inflation index to the beginning of the year. That rebased starting point becomes 100.

Calculations

$$\$100000\times 1,15$$

$$=\$115384,6$$

$$-\$100000$$

$$=\$15384,62$$