Index numbers are essential tools in accountancy and business analytics, especially in Kenya's dynamic economic environment. They allow professionals to measure changes in variables such as prices, quantities, or values over time, providing insight into inflation trends, cost of living adjustments, and economic performance. Mastery of constructing index numbers enables accountants to support decision-making processes, budgeting, and financial analysis with quantitative evidence.

8.1 Construction of index numbers

This section addresses how to construct index numbers, focusing on methods relevant to accounting and economic data in Kenya. It covers the fundamental formulas and techniques used to compute price, quantity, and value indices, crucial for analyzing financial and economic trends.

8.1.1 Simple Index Numbers by Price

Simple price index numbers compare the price of a single item or a group of items at different time periods. They measure the relative change in price from a base period to a current period. The formula for a simple price index is:

$$ \text{Price Index (PI)} = \frac{P_n}{P_0} \times 100 $$

Where:

• \(P_n\) = Price in the current period
• \(P_0\) = Price in the base period

Worked Examples

Example 1: A hospital paid Ksh 150 for a medical supply in 2023, while in 2020 the price was Ksh 120. Find the price index for 2023 using 2020 as the base year.

Given: \(P_n = 150\), \(P_0 = 120\)

$$ PI = \frac{150}{120} \times 100 $$

$$ = 1.25 \times 100 $$

$$ = 125 $$

Answer: The price index is 125, indicating a 25% price increase since 2020.

Example 2: A school textbook cost Ksh 400 in 2024 and Ksh 350 in 2021. Calculate the price index for 2024 with 2021 as the base.

Given: \(P_n = 400\), \(P_0 = 350\)

$$ PI = \frac{400}{350} \times 100 $$

$$ = 1.1429 \times 100 $$

$$ = 114.29 $$

Answer: The price index is 114.29, representing a 14.29% increase in price.

Example 3: A retail business recorded the price of a commodity as Ksh 180 in 2022 and Ksh 150 in 2019. Find the price index for 2022 using 2019 as the base.

Given: \(P_n = 180\), \(P_0 = 150\)

$$ PI = \frac{180}{150} \times 100 $$

$$ = 1.2 \times 100 $$

$$ = 120 $$

Answer: The price index is 120, showing a 20% price increase.

Example 4: A cooperative sold a product at Ksh 250 in 2025; the base year price in 2020 was Ksh 200. Calculate the price index for 2025.

Given: \(P_n = 250\), \(P_0 = 200\)

$$ PI = \frac{250}{200} \times 100 $$

$$ = 1.25 \times 100 $$

$$ = 125 $$

Answer: The price index is 125, indicating a 25% price rise.

Example 5: A SACCO noted a price of Ksh 90 for a service in 2023, while in 2018 the price was Ksh 75. Find the price index for 2023.

Given: \(P_n = 90\), \(P_0 = 75\)

$$ PI = \frac{90}{75} \times 100 $$

$$ = 1.2 \times 100 $$

$$ = 120 $$

Answer: The price index is 120, reflecting a 20% increase in price.

8.1.2 Simple Index Numbers by Quantity

Simple quantity index numbers measure the relative change in quantity of a single item or group of items between two periods. The formula is:

$$ \text{Quantity Index (QI)} = \frac{Q_n}{Q_0} \times 100 $$

Where:

• \(Q_n\) = Quantity in the current period
• \(Q_0\) = Quantity in the base period

Worked Examples

Example 1: A bank issued 1,200 loan accounts in 2023 compared to 1,000 in 2020. Find the quantity index for 2023.

Given: \(Q_n = 1200\), \(Q_0 = 1000\)

$$ QI = \frac{1200}{1000} \times 100 $$

$$ = 1.2 \times 100 $$

$$ = 120 $$

Answer: The quantity index is 120, showing a 20% increase in loan accounts issued.

Example 2: A county government supplied 600 office chairs in 2024; in 2021, it supplied 500. Calculate the quantity index for 2024.

Given: \(Q_n = 600\), \(Q_0 = 500\)

$$ QI = \frac{600}{500} \times 100 $$

$$ = 1.2 \times 100 $$

$$ = 120 $$

Answer: The quantity index is 120, indicating a 20% rise in chairs supplied.

Example 3: A retail business sold 2,500 units of a product in 2022 compared to 3,000 units in 2019. Find the quantity index for 2022.

Given: \(Q_n = 2500\), \(Q_0 = 3000\)

$$ QI = \frac{2500}{3000} \times 100 $$

$$ = 0.8333 \times 100 $$

$$ = 83.33 $$

Answer: The quantity index is 83.33, showing a 16.67% decrease in quantity sold.

Example 4: A university library purchased 1,000 books in 2025 compared to 800 in 2020. Calculate the quantity index for 2025.

Given: \(Q_n = 1000\), \(Q_0 = 800\)

$$ QI = \frac{1000}{800} \times 100 $$

$$ = 1.25 \times 100 $$

$$ = 125 $$

Answer: The quantity index is 125, representing a 25% increase in books purchased.

Example 5: A hotel served 1,800 meals in 2023 while in 2018 it served 2,000 meals. Find the quantity index for 2023.

Given: \(Q_n = 1800\), \(Q_0 = 2000\)

$$ QI = \frac{1800}{2000} \times 100 $$

$$ = 0.9 \times 100 $$

$$ = 90 $$

Answer: The quantity index is 90, indicating a 10% decrease in meals served.

8.1.3 Weighted Index Numbers: Laspeyres and Paasche Indices

Weighted index numbers account for the relative importance of items by assigning weights, often based on quantities or values. Two common weighted indices are the Laspeyres and Paasche indices.

• The Laspeyres price index uses base period quantities as weights:

$$ LPI = \frac{\Sigma (P_n \times Q_0)}{\Sigma (P_0 \times Q_0)} \times 100 $$

• The Paasche price index uses current period quantities as weights:

$$ PPI = \frac{\Sigma (P_n \times Q_n)}{\Sigma (P_0 \times Q_n)} \times 100 $$

Where:

• \(P_n\), \(P_0\) = Prices in current and base periods
• \(Q_n\), \(Q_0\) = Quantities in current and base periods

Worked Examples

Example 1 (Laspeyres): A SACCO sells two products. In 2020 (base year), prices and quantities were: Product A: Ksh 100, 50 units; Product B: Ksh 200, 30 units. In 2023, prices changed to: Product A: Ksh 120, Product B: Ksh 240. Calculate the Laspeyres price index.

Given:

Calculate numerator:

$$ \Sigma (P_n \times Q_0) = (120 \times 50) + (240 \times 30) = 6000 + 7200 = 13200 $$

Calculate denominator:

$$ \Sigma (P_0 \times Q_0) = (100 \times 50) + (200 \times 30) = 5000 + 6000 = 11000 $$

Calculate index:

$$ LPI = \frac{13200}{11000} \times 100 = 1.2 \times 100 = 120 $$

Answer: The Laspeyres price index is 120, indicating a 20% price increase.

Example 2 (Paasche): Using the same data, assume quantities in 2023 are: Product A: 60 units, Product B: 25 units. Calculate the Paasche price index.

Given:

Calculate numerator:

$$ \Sigma (P_n \times Q_n) = (120 \times 60) + (240 \times 25) = 7200 + 6000 = 13200 $$

Calculate denominator:

$$ \Sigma (P_0 \times Q_n) = (100 \times 60) + (200 \times 25) = 6000 + 5000 = 11000 $$

Calculate index:

$$ PPI = \frac{13200}{11000} \times 100 = 1.2 \times 100 = 120 $$

Answer: The Paasche price index is 120, reflecting a 20% price increase.

Example 3 (Laspeyres): A retail business tracks two products. Base year (2021) data: Product X: price Ksh 80, quantity 40; Product Y: price Ksh 150, quantity 60. Current year (2024) prices: Product X: Ksh 100, Product Y: Ksh 180. Calculate the Laspeyres price index.

Given:

Numerator:

$$ (100 \times 40) + (180 \times 60) = 4000 + 10800 = 14800 $$

Denominator:

$$ (80 \times 40) + (150 \times 60) = 3200 + 9000 = 12200 $$

Index:

$$ LPI = \frac{14800}{12200} \times 100 = 1.2131 \times 100 = 121.31 $$

Answer: The Laspeyres price index is 121.31, indicating a 21.31% price increase.

Example 4 (Paasche): For the same products, current year quantities are: Product X: 50, Product Y: 55. Calculate the Paasche price index.

Numerator:

$$ (100 \times 50) + (180 \times 55) = 5000 + 9900 = 14900 $$

Denominator:

$$ (80 \times 50) + (150 \times 55) = 4000 + 8250 = 12250 $$

Index:

$$ PPI = \frac{14900}{12250} \times 100 = 1.2163 \times 100 = 121.63 $$

Answer: The Paasche price index is 121.63, showing a 21.63% price increase.

Example 5 (Laspeyres): A cooperative sells three products. Base year prices and quantities: A: Ksh 50, 100 units; B: Ksh 120, 80 units; C: Ksh 200, 60 units. Current prices: A: Ksh 55, B: Ksh 130, C: Ksh 210. Calculate the Laspeyres price index.

Numerator:

$$ (55 \times 100) + (130 \times 80) + (210 \times 60) = 5500 + 10400 + 12600 = 28500 $$

Denominator:

$$ (50 \times 100) + (120 \times 80) + (200 \times 60) = 5000 + 9600 + 12000 = 26600 $$

Index:

$$ LPI = \frac{28500}{26600} \times 100 = 1.0714 \times 100 = 107.14 $$

Answer: The Laspeyres price index is 107.14, indicating a 7.14% price increase.

8.1.4 Construction of Value Index Numbers

Value index numbers measure changes in total value by combining price and quantity changes. They are particularly useful for accountants analyzing turnover or sales value changes over time. The formula is:

$$ \text{Value Index (VI)} = \frac{\Sigma (P_n \times Q_n)}{\Sigma (P_0 \times Q_0)} \times 100 $$

Where:

• \(P_n\), \(Q_n\) = Price and quantity in current period
• \(P_0\), \(Q_0\) = Price and quantity in base period

Worked Examples

Example 1: A farm sold 100 bags of maize at Ksh 2,500 each in 2023. In 2020, it sold 120 bags at Ksh 2,000 each. Calculate the value index for 2023.

Given:

$$ \Sigma (P_n \times Q_n) = 2,500 \times 100 = 250,000 $$

$$ \Sigma (P_0 \times Q_0) = 2,000 \times 120 = 240,000 $$

Value index:

$$ VI = \frac{250,000}{240,000} \times 100 = 1.0417 \times 100 = 104.17 $$

Answer: The value index is 104.17, showing a 4.17% increase in sales value.

Example 2: A retail store sold 500 units of a product at Ksh 150 in 2024, while in 2021 it sold 450 units at Ksh 140. Calculate the value index.

Given:

$$ \Sigma (P_n \times Q_n) = 150 \times 500 = 75,000 $$

$$ \Sigma (P_0 \times Q_0) = 140 \times 450 = 63,000 $$

Value index:

$$ VI = \frac{75,000}{63,000} \times 100 = 1.1905 \times 100 = 119.05 $$

Answer: The value index is 119.05, indicating a 19.05% increase in sales value.

Example 3: A hospital consumed 1,000 units of medicine at Ksh 50 per unit in 2025, and 900 units at Ksh 45 per unit in 2020. Calculate the value index.

Given:

$$ \Sigma (P_n \times Q_n) = 50 \times 1000 = 50,000 $$

$$ \Sigma (P_0 \times Q_0) = 45 \times 900 = 40,500 $$

Value index:

$$ VI = \frac{50,000}{40,500} \times 100 = 1.2346 \times 100 = 123.46 $$

Answer: The value index is 123.46, showing a 23.46% increase in value.

Example 4: A cooperative sold 200 units of fertilizer at Ksh 1,200 each in 2023, and 250 units at Ksh 1,000 each in 2020. Calculate the value index.

Given:

$$ \Sigma (P_n \times Q_n) = 1,200 \times 200 = 240,000 $$

$$ \Sigma (P_0 \times Q_0) = 1,000 \times 250 = 250,000 $$

Value index:

$$ VI = \frac{240,000}{250,000} \times 100 = 0.96 \times 100 = 96 $$

Answer: The value index is 96, indicating a 4% decrease in sales value.

Example 5: A SACCO disbursed 1,500 loans at Ksh 10,000 each in 2024, and 1,200 loans at Ksh 9,000 each in 2020. Calculate the value index.

Given:

$$ \Sigma (P_n \times Q_n) = 10,000 \times 1,500 = 15,000,000 $$

$$ \Sigma (P_0 \times Q_0) = 9,000 \times 1,200 = 10,800,000 $$

Value index:

$$ VI = \frac{15,000,000}{10,800,000} \times 100 = 1.3889 \times 100 = 138.89 $$

Answer: The value index is 138.89, reflecting a 38.89% increase in loan disbursement value.

Chapter Summary

This chapter focused on the determination of index numbers, beginning with their construction as a tool for measuring changes in economic data over time. It highlighted the purpose of index numbers in analyzing trends such as inflation, cost of living, and production levels. The chapter then explored simple index numbers, explaining the fixed base method and the chain base method as fundamental approaches to comparison. The Consumer Price Index was discussed as a key example of an index used to monitor price changes affecting consumers. Weighted index numbers were introduced next, with detailed examination of Laspeyre's, Paasche's, and Fisher's methods, each incorporating different weighting techniques to improve accuracy. Finally, the chapter covered Marshall-Edgeworth's method for calculating both price and quantity index numbers, emphasizing its application in producing comprehensive reports. Throughout, the chapter provided the mathematical foundation and practical context necessary for constructing and interpreting various types of index numbers.