Weighted Price Index – From Intuition to Precision

Imagine you are tracking how the cost of living changes in your city. You pick a basket of goods: bread, milk, petrol, and movie tickets. A simple price index would just average the price changes of these four items. But that would be misleading — if petrol doubles in price but bread barely changes, the simple average treats both changes equally. In reality, petrol might take up a much larger share of a typical household's monthly spending. A weighted price index fixes this by giving each item a weight proportional to its importance in the basket.

The core idea is simple: not all price changes matter equally. A 10% rise in the price of rice hurts a family far more than a 10% rise in the price of gold, because rice is bought daily and gold rarely. The weight captures this relative importance.

The Precise Statement

A weighted price index is a number that measures the average change in prices of a fixed set of goods and services, where each price change is multiplied by a weight that reflects the item's relative importance in the total expenditure of the base period (or current period, depending on the formula).

The two most common forms are:

Laspeyres Price Index (uses base-period quantities as weights):

Paasche Price Index (uses current-period quantities as weights):

Where:

• $p_0$ = price in the base year
• $p_n$ = price in the current year
• $q_0$ = quantity consumed in the base year
• $q_n$ = quantity consumed in the current year

The weight for each item is its expenditure share: $w_i = \frac{p_0 q_0}{\sum p_0 q_0}$ (for Laspeyres) or $w_i = \frac{p_0 q_n}{\sum p_0 q_n}$ (for Paasche).

Why the Weight Matters

Consider a simple example. A student's monthly expenses are:

Simple (unweighted) average of price relatives:

• Rice relative: $\frac{50}{40} = 1.25$ (25% increase)
• Movie relative: $\frac{250}{200} = 1.25$ (25% increase)
• Simple index: $\frac{1.25 + 1.25}{2} \times 100 = 125$

Weighted (Laspeyres) index:

• Rice weight: $40 \times 10 = 400$
• Movie weight: $200 \times 2 = 400$
• Weighted index: $\frac{(50 \times 10) + (250 \times 2)}{(40 \times 10) + (200 \times 2)} \times 100 = \frac{500 + 500}{400 + 400} \times 100 = 125$

Here both give the same result because the weights are equal. But if the student buys 20 kg of rice and only 1 movie ticket:

• Rice weight: $40 \times 20 = 800$
• Movie weight: $200 \times 1 = 200$
• Weighted index: $\frac{(50 \times 20) + (250 \times 1)}{(40 \times 20) + (200 \times 1)} \times 100 = \frac{1000 + 250}{800 + 200} \times 100 = 125$

Still 125? That's because both items rose by exactly 25%. The real power of weighting shows when price changes differ. Suppose rice rises 25% but movie tickets rise 50% (to ₹300):

• Simple index: $\frac{1.25 + 1.50}{2} \times 100 = 137.5$
• Weighted index (rice-heavy basket): $\frac{(50 \times 20) + (300 \times 1)}{(40 \times 20) + (200 \times 1)} \times 100 = \frac{1000 + 300}{800 + 200} \times 100 = 130$

The weighted index (130) is lower than the simple index (137.5) because the heavily weighted rice rose less than the lightly weighted movie ticket. The weighted index correctly reflects that the student's overall cost increase is moderated by the fact that the most important item (rice) had a smaller price rise.

Which Weight to Use? Laspeyres vs Paasche

Laspeyres uses base-year quantities. It answers: "How much more would the base-year basket cost today?" It tends to overstate inflation because it ignores that consumers switch to cheaper substitutes when prices rise.

Paasche uses current-year quantities. It answers: "How much more would the current basket have cost in the base year?" It tends to understate inflation because it reflects actual substitution behaviour.

In practice, most official indices (like India's CPI and WPI) use the Laspeyres formula because base-period data is easier to collect and update only periodically.

The Intuition in One Sentence

A weighted price index is like a customised inflation measure for a specific household or economy: it tells you how much the cost of a typical basket has changed, where "typical" is defined by how much people actually spend on each item.