Q.Cost of living at two different cities can be compared with the help of:

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): $\frac{(50 \times 20) + (300 \times 1)}{(40 \times 20) + (200 \times 1)} \times 100 = \frac{1000 + 300}{800 + 200} \times 100 = 130$ …