Index Numbers — Practice

Question 1

Calculate the Simple Aggregative Price Index for the following data (Base year 2020, Current year 2024): Item A: P₀ = 10, P₁ = 12. Item B: P₀ = 20, P₁ = 25. Item C: P₀ = 30, P₁ = 36.

Accepted Answer

Simple Aggregative Price Index = (ΣP₁ / ΣP₀) × 100. ΣP₁ = 12 + 25 + 36 = 73. ΣP₀ = 10 + 20 + 30 = 60. P₀₁ = (73 / 60) × 100 = 121.67. Interpretation: Prices in 2024 are 21.67% higher than in 2020 (base year). Limitation: This index gives equal weight to all items regardless of how much consumers spend on them.

Question 2

Calculate Laspeyres and Paasche Price Indexes from the following data: Item A: P₀ = 5, P₁ = 6, Q₀ = 100, Q₁ = 90. Item B: P₀ = 10, P₁ = 12, Q₀ = 50, Q₁ = 60.

Accepted Answer

Laspeyres Index (base year quantities): ΣP₁Q₀ = (6×100) + (12×50) = 600 + 600 = 1200. ΣP₀Q₀ = (5×100) + (10×50) = 500 + 500 = 1000. Laspeyres = (1200/1000) × 100 = 120. Paasche Index (current year quantities): ΣP₁Q₁ = (6×90) + (12×60) = 540 + 720 = 1260. ΣP₀Q₁ = (5×90) + (10×60) = 450 + 600 = 1050. Paasche = (1260/1050) × 100 = 120. Note: Here both are equal at 120. In general, Laspeyres ≥ Paasche because consumers substitute toward cheaper goods, reducing the actual cost of living below what the Laspeyres formula suggests.

Question 3

Using the data: Item A: P₀=4, P₁=5, Q₀=60, Q₁=50. Item B: P₀=8, P₁=10, Q₀=40, Q₁=45. Item C: P₀=12, P₁=15, Q₀=30, Q₁=25. Calculate (a) Laspeyres Index, (b) Paasche Index, (c) Fisher's Ideal Index. State why Fisher's Index is called 'Ideal'.

Accepted Answer

(a) Laspeyres Index: ΣP₁Q₀ = (5×60)+(10×40)+(15×30) = 300+400+450 = 1150. ΣP₀Q₀ = (4×60)+(8×40)+(12×30) = 240+320+360 = 920. Laspeyres = (1150/920)×100 = 125. (b) Paasche Index: ΣP₁Q₁ = (5×50)+(10×45)+(15×25) = 250+450+375 = 1075. ΣP₀Q₁ = (4×50)+(8×45)+(12×25) = 200+360+300 = 860. Paasche = (1075/860)×100 = 125. (c) Fisher's Ideal = √(Laspeyres × Paasche) = √(125 × 125) = √15625 = 125. Why 'Ideal': Fisher's Index is called 'Ideal' because it satisfies two important mathematical tests: (1) Time Reversal Test: P₀₁ × P₁₀ = 1 — if you compute the index forward and backward in time, the results are consistent. (2) Factor Reversal Test: the product of the price index and quantity index equals the value index. Neither Laspeyres nor Paasche individually satisfies both tests, but Fisher's (their geometric mean) does. It is also free from the upward bias of Laspeyres and the downward bias of Paasche.

Practice — Index Numbers

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