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Compiled by: Jane Kamau
An index number is a statistical measure that shows the change in a variable, or a group of related variables, from one period (the base period) to another (the current period), expressed as a percentage of the base. The base period is given the value 100, so an index of 112 means a 12% rise and an index of 95 a 5% fall. Procurement officers use index numbers to follow changes in the prices they pay, the quantities they buy and the total value of purchases, and to adjust budgets and contract prices. This topic sets out the formulae; their full computation and comparison follow in 8.2.
A simple index, also called a relative, measures the change in one item.
Price relative = (p1 ÷ p0) × 100
Quantity relative = (q1 ÷ q0) × 100
Value relative = (p1 × q1) ÷ (p0 × q0) × 100
Example: a ream of printing paper cost Ksh 500 in the base year and Ksh 560 in the current year.
Price relative = (560 ÷ 500) × 100 = 112
The price of paper rose by 12%.
These combine several items but give each the same importance.
Simple aggregate price index = (Σp1 ÷ Σp0) × 100
Simple average of price relatives = Σ[(p1 ÷ p0) × 100] ÷ n
Example: a stationery basket in the base and current years.
| Item | p0 (Ksh) | q0 | p1 (Ksh) | q1 | Price relative |
|---|---|---|---|---|---|
| Paper (ream) | 500 | 200 | 560 | 220 | 112 |
| Toner (cartridge) | 4,000 | 10 | 4,200 | 8 | 105 |
| Files (pack) | 300 | 50 | 360 | 60 | 120 |
Simple aggregate price index = (560 + 4,200 + 360) ÷ (500 + 4,000 + 300) × 100
= 5,120 ÷ 4,800 × 100
= 106.67
Simple average of price relatives = (112 + 105 + 120) ÷ 3
= 337 ÷ 3
= 112.33
The two answers differ because the aggregate index is dominated by the expensive item (toner), whose price rose least, while the average of relatives treats every item equally. Neither reflects how much of each item is actually bought, which is why weighted indices are preferred.
Weighted indices give each item an importance (weight) according to how much of it is bought. The four formulae used in this unit are:
| Index | Price index formula | Weights used |
|---|---|---|
| Laspeyres' | (Σp1q0 ÷ Σp0q0) × 100 | Base period quantities |
| Paasche's | (Σp1q1 ÷ Σp0q1) × 100 | Current period quantities |
| Fisher's ideal | √(Laspeyres' × Paasche's) | Both, by geometric mean |
| Marshall (Marshall-Edgeworth) | [Σp1(q0 + q1) ÷ Σp0(q0 + q1)] × 100 | Sum of base and current quantities |
The matching quantity indices are obtained by interchanging p and q, so that prices become the weights:
| Index | Quantity index formula |
|---|---|
| Laspeyres' | (Σq1p0 ÷ Σq0p0) × 100 |
| Paasche's | (Σq1p1 ÷ Σq0p1) × 100 |
| Fisher's ideal | √(Laspeyres' quantity × Paasche's quantity) |
| Marshall | [Σq1(p0 + p1) ÷ Σq0(p0 + p1)] × 100 |
A value index measures the change in total spending, which combines changes in both prices and quantities.
Value index = (Σp1q1 ÷ Σp0q0) × 100
Using the stationery basket:
Σp1q1 = 560 × 220 + 4,200 × 8 + 360 × 60 = 123,200 + 33,600 + 21,600 = 178,400
Σp0q0 = 500 × 200 + 4,000 × 10 + 300 × 50 = 100,000 + 40,000 + 15,000 = 155,000
Value index = 178,400 ÷ 155,000 × 100 = 115.10
Total spending on the basket rose by about 15.1%, partly because prices rose and partly because more paper and files were bought.
Where only price relatives and the value of each item's spending are known, a weighted average of relatives is used:
Weighted average of price relatives = Σ(R × w) ÷ Σw
where R is each item's price relative and w its weight, usually the base period value p0q0. With the stationery basket, w = 100,000, 40,000 and 15,000:
Σ(R × w) = 112 × 100,000 + 105 × 40,000 + 120 × 15,000 = 11,200,000 + 4,200,000 + 1,800,000 = 17,200,000
Index = 17,200,000 ÷ 155,000 = 110.97
This equals the Laspeyres' price index, since weighting relatives by base values is algebraically the same as Laspeyres' formula.
A chain index compares each period with the one immediately before it:
Link relative = (price in this period ÷ price in previous period) × 100
Chain index for a period = (link relative of the period × chain index of the previous period) ÷ 100
Example: the price of a ream was Ksh 500, 530 and 560 in three successive years.
Link relatives: year 2 = 530 ÷ 500 × 100 = 106; year 3 = 560 ÷ 530 × 100 = 105.66
Chain index: year 1 = 100; year 2 = 106; year 3 = 105.66 × 106 ÷ 100 = 112.00, which agrees with the fixed base price relative of 112.
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Create a free accountThis chapter introduced the fundamental formulae used to compute index numbers, providing a mathematical basis for their calculation. It then detailed the computation methods for various index numbers, starting with Laspeyre's index which uses base period quantities as weights. Paasche's index was explained next, highlighting the use of current period quantities for weighting. The chapter also covered Fisher's ideal index, which combines both Laspeyre's and Paasche's indices to provide a more accurate measure. Additionally, the Marshal reports method was discussed as another approach to index number calculation. Finally, the chapter explored how index numbers are applied in decision making, emphasizing their role in analyzing economic and financial data for informed choices. Throughout, the focus was on practical computation and interpretation within real-world contexts.
Example 1: Laspeyres Price Index
A procurement manager tracks the price changes of two items:
| Item | Price in Base Year (Ksh) | Price in Current Year (Ksh) | Quantity in Base Year |
|---|---|---|---|
| A | 100 | 120 | 50 |
| B | 200 | 180 | 30 |
Laspeyres Price Index formula:$$ I_L = \frac{\sum P_1 Q_0}{\sum P_0 Q_0} \times 100 $$
Calculate numerator:$$ \sum P_1 Q_0 = (120 \times 50) + (180 \times 30) = 6000 + 5400 = 11400 $$
Calculate denominator:$$ \sum P_0 Q_0 = (100 \times 50) + (200 \times 30) = 5000 + 6000 = 11000 $$
Calculate index:$$ I_L = \frac{11400}{11000} \times 100 = 103.64 $$
Answer: 103.64 (Prices increased by 3.64%)
Example 2: Paasche Price Index
Suppose current year quantities are:
| Item | Price in Base Year (Ksh) | Price in Current Year (Ksh) | Quantity in Current Year |
|---|---|---|---|
| A | 100 | 120 | 60 |
| B | 200 | 180 | 25 |
Paasche Price Index formula:$$ I_P = \frac{\sum P_1 Q_1}{\sum P_0 Q_1} \times 100 $$
Calculate numerator:$$ \sum P_1 Q_1 = (120 \times 60) + (180 \times 25) = 7200 + 4500 = 11700 $$
Calculate denominator:$$ \sum P_0 Q_1 = (100 \times 60) + (200 \times 25) = 6000 + 5000 = 11000 $$
Calculate index:$$ I_P = \frac{11700}{11000} \times 100 = 106.36 $$
Answer: 106.36 (Prices increased by 6.36%)
Example 3: Fisher's Ideal Index
Given Laspeyres Index \(I_L = 103.64\) and Paasche Index \(I_P = 106.36\):
Fisher's Ideal Index formula:$$ I_F = \sqrt{I_L \times I_P} $$
Calculate:$$ I_F = \sqrt{103.64 \times 106.36} = \sqrt{11027.02} = 105.03 $$
Answer: 105.03 (Average price increase of 5.03%)
Example 4: Marshal-Edgeworth Price Index
Suppose total expenditure in base year is Ksh 50,000 and in current year is Ksh 55,000, and quantities are equal.
Marshal-Edgeworth formula:$$ I_{ME} = \frac{E_1 + E_0}{2 E_0} \times 100 $$
Calculate:$$ I_{ME} = \frac{55,000 + 50,000}{2 \times 50,000} \times 100 = \frac{105,000}{100,000} \times 100 = 105 $$
Answer: 105 (Prices increased by 5%)
A procurement officer recorded the price of a commodity as Ksh 120 in the base year and Ksh 150 in the current year. Calculate the Laspeyre’s Price Index. (2 marks)
The quantity of an item purchased in the base year was 200 units at Ksh 50 per unit, and in the current year, 180 units at Ksh 60 per unit. Calculate the Paasche’s Price Index. (3 marks)
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