Q.Using value weights $w=p_0q_0$ for Commodities A, B, C, D (as in the earlier worked examples), calculate the Price Index by the Weighted Average of Price Relatives method, and show that it agrees with Laspeyres' Index calculated earlier.

Weighting each commodity's price relative by its base-year value $w=p_0q_0$ and averaging should reproduce Laspeyres' Index exactly, since both formulas reduce to the same ratio algebraically.

Step 1 — Tabulate the weight $w=p_0q_0$ and the price relative for each commodity.

Step 2 — Apply the weighted-average formula.

Dual-check / proof of equivalence: since $w\times\text{Relative}=p_0q_0\times\frac{p_1}{p_0}\times100=p_1q_0\times100$, we have $\Sigma(w\times\text{Relative})=100\times\Sigma p_1q_0=100\times347=34{,}700$ — exactly matching the table above — and $\Sigma w=\Sigma p_0q_0=300$, so this formula is algebraically identical to $\dfrac{\Sigma p_1q_0}{\Sigma p_0q_0}\times100$, i.e. Laspeyres' Index (115.67, from the earlier worked example).

The weight must be $w=p_0q_0$ (the base-year VALUE, price times quantity) — using $q_0$ alone, or $p_1q_0$, as the weight is what breaks the equivalence with Laspeyres and gives a wrong answer.