Use index numbers to measure changes in quantities such as prices, wages, or quantities over time, critical for human resource professionals tracking salary trends, inflation effects, or cost-of-living adjustments. This chapter focuses on the mathematical computation of index numbers using different formulae, enabling HR practitioners to analyze economic data relevant to workforce management in Kenya. Understanding these calculations helps in making informed decisions on wage adjustments, budgeting, and policy formulation.

8.1 Formulae for computing index numbers

Index numbers quantify changes in variables over time or between groups, typically expressed as percentages relative to a base period or base group. The primary formulae used in computing index numbers include Laspeyres, Paasche, Fisher’s ideal, and Marshall-Edgeworth indices, each differing in how quantities and prices are weighted. These formulae help HR professionals assess cost changes affecting employee compensation, benefits, and organizational budgeting.

8.1.1 Laspeyres Index Formula

The Laspeyres index uses base period quantities as weights to measure price changes over time. It is suitable when the base period consumption pattern is stable and is commonly used to calculate consumer price indices affecting wage adjustments.

The formula is:

$$ L = \frac{\sum p_{t} q_{0}}{\sum p_{0} q_{0}} \times 100 $$

Where:

• \(p_t\) = price in the current period
• \(p_0\) = price in the base period
• \(q_0\) = quantity in the base period

Worked Examples

Example 1: A SACCO wants to measure the price change of office supplies between 2022 (base year) and 2023 (current year). Given base year quantities and prices:

Calculate the Laspeyres price index.

Given: \(q_0\), \(p_0\), \(p_t\)

$$ L = \frac{\sum p_t q_0}{\sum p_0 q_0} \times 100 $$

Calculate numerator:

$$ (55 \times 100) + (850 \times 20) = 5500 + 17000 = 22500 $$

Calculate denominator:

$$ (50 \times 100) + (800 \times 20) = 5000 + 16000 = 21000 $$

Calculate index:

$$ L = \frac{22500}{21000} \times 100 = 107.14 $$

Answer: 107.14 (Prices increased by 7.14%)

Example 2: A university HR department tracks changes in cafeteria meal prices using base quantities from 2021 and prices from 2021 and 2023:

Calculate the Laspeyres index.

Given: \(q_0\), \(p_0\), \(p_t\)

$$ L = \frac{(165 \times 300) + (200 \times 200)}{(150 \times 300) + (180 \times 200)} \times 100 $$

Calculate numerator:

$$ (165 \times 300) + (200 \times 200) = 49500 + 40000 = 89500 $$

Calculate denominator:

$$ (150 \times 300) + (180 \times 200) = 45000 + 36000 = 81000 $$

Calculate index:

$$ L = \frac{89500}{81000} \times 100 = 110.49 $$

Answer: 110.49 (Prices increased by 10.49%)

Example 3: A county government office is checking changes in stationery costs between 2020 and 2022. Given:

Calculate the Laspeyres index.

Given: \(q_0\), \(p_0\), \(p_t\)

$$ L = \frac{(22 \times 500) + (110 \times 200) + (165 \times 100)}{(20 \times 500) + (100 \times 200) + (150 \times 100)} \times 100 $$

Calculate numerator:

$$ 11000 + 22000 + 16500 = 49500 $$

Calculate denominator:

$$ 10000 + 20000 + 15000 = 45000 $$

Calculate index:

$$ L = \frac{49500}{45000} \times 100 = 110 $$

Answer: 110 (Prices increased by 10%)

8.1.2 Paasche Index Formula

The Paasche index uses current period quantities as weights, reflecting current consumption patterns and is useful when the focus is on recent trends in resource use or costs.

The formula is:

$$ P = \frac{\sum p_{t} q_{t}}{\sum p_{0} q_{t}} \times 100 $$

Where:

• \(p_t\) = price in the current period
• \(p_0\) = price in the base period
• \(q_t\) = quantity in the current period

Worked Examples

Example 1: A retail business tracks changes in the price of employee uniforms. Given:

Calculate the Paasche price index.

Given: \(q_t\), \(p_0\), \(p_t\)

$$ P = \frac{(130 \times 150) + (160 \times 100)}{(120 \times 150) + (150 \times 100)} \times 100 $$

Calculate numerator:

$$ 19500 + 16000 = 35500 $$

Calculate denominator:

$$ 18000 + 15000 = 33000 $$

Calculate index:

$$ P = \frac{35500}{33000} \times 100 = 107.58 $$

Answer: 107.58 (Prices increased by 7.58%)

Example 2: A hospital HR department examines changes in medical supplies prices using quantities and prices from 2022 and 2023:

Calculate the Paasche index.

Given: \(q_t\), \(p_0\), \(p_t\)

$$ P = \frac{(55 \times 500) + (33 \times 400) + (110 \times 300)}{(50 \times 500) + (30 \times 400) + (100 \times 300)} \times 100 $$

Calculate numerator:

$$ 27500 + 13200 + 33000 = 73700 $$

Calculate denominator:

$$ 25000 + 12000 + 30000 = 67000 $$

Calculate index:

$$ P = \frac{73700}{67000} \times 100 = 110.00 $$

Answer: 110.00 (Prices increased by 10%)

Example 3: A hotel HR manager tracks changes in food ingredient prices with quantities and prices:

Calculate the Paasche index.

Given: \(q_t\), \(p_0\), \(p_t\)

$$ P = \frac{(90 \times 1000) + (65 \times 500) + (220 \times 300)}{(80 \times 1000) + (60 \times 500) + (200 \times 300)} \times 100 $$

Calculate numerator:

$$ 90000 + 32500 + 66000 = 188500 $$

Calculate denominator:

$$ 80000 + 30000 + 60000 = 170000 $$

Calculate index:

$$ P = \frac{188500}{170000} \times 100 = 110.88 $$

Answer: 110.88 (Prices increased by 10.88%)

8.1.3 Fisher’s Ideal Index Formula

Fisher’s ideal index is the geometric mean of Laspeyres and Paasche indices, providing a more accurate measure by balancing base and current period weights. It is considered the best measure for price and quantity changes.

The formula is:

$$ F = \sqrt{L \times P} $$

Where:

• \(L\) = Laspeyres index
• \(P\) = Paasche index

Worked Examples

Example 1: Given a Laspeyres index of 105 and Paasche index of 110, calculate Fisher’s ideal index.

$$ F = \sqrt{105 \times 110} $$

Calculate product:

$$ 105 \times 110 = 11550 $$

Calculate square root:

$$ \sqrt{11550} = 107.47 $$

Answer: 107.47

Example 2: A cooperative measures price changes with Laspeyres index 108.5 and Paasche index 112. Calculate Fisher’s ideal index.

$$ F = \sqrt{108.5 \times 112} $$

Calculate product:

$$ 108.5 \times 112 = 12152 $$

Calculate square root:

$$ \sqrt{12152} = 110.26 $$

Answer: 110.26

Example 3: A bank HR department has Laspeyres index of 102 and Paasche index of 104. Calculate Fisher’s ideal index.

$$ F = \sqrt{102 \times 104} $$

Calculate product:

$$ 102 \times 104 = 10608 $$

Calculate square root:

$$ \sqrt{10608} = 103.00 $$

Answer: 103.00

8.1.4 Marshall-Edgeworth Index Formula

The Marshall-Edgeworth index uses the average of base and current period quantities as weights, balancing the influence of both periods. It is useful when changes in quantities are significant.

The formula is:

$$ M = \frac{\sum p_t (q_0 + q_t)}{\sum p_0 (q_0 + q_t)} \times 100 $$

Where:

• \(p_t\) = price in the current period
• \(p_0\) = price in the base period
• \(q_0\) = quantity in the base period
• \(q_t\) = quantity in the current period

Worked Examples

Example 1: A retail business compares prices of items with given data:

Calculate the Marshall-Edgeworth index.

$$ M = \frac{(2100 \times (100 + 120)) + (3200 \times (80 + 90))}{(2000 \times (100 + 120)) + (3000 \times (80 + 90))} \times 100 $$

Calculate numerator:

$$ 2100 \times 220 = 462000 $$

$$ 3200 \times 170 = 544000 $$

Sum numerator:

$$ 462000 + 544000 = 1006000 $$

Calculate denominator:

$$ 2000 \times 220 = 440000 $$

$$ 3000 \times 170 = 510000 $$

Sum denominator:

$$ 440000 + 510000 = 950000 $$

Calculate index:

$$ M = \frac{1006000}{950000} \times 100 = 105.89 $$

Answer: 105.89 (Prices increased by 5.89%)

Example 2: A hotel HR department compares ingredient prices:

Calculate Marshall-Edgeworth index.

$$ M = \frac{(110 \times (500 + 550)) + (160 \times (300 + 350))}{(100 \times (500 + 550)) + (150 \times (300 + 350))} \times 100 $$

Calculate numerator:

$$ 110 \times 1050 = 115500 $$

$$ 160 \times 650 = 104000 $$

Sum numerator:

$$ 115500 + 104000 = 219500 $$

Calculate denominator:

$$ 100 \times 1050 = 105000 $$

$$ 150 \times 650 = 97500 $$

Sum denominator:

$$ 105000 + 97500 = 202500 $$

Calculate index:

$$ M = \frac{219500}{202500} \times 100 = 108.39 $$

Answer: 108.39

Example 3: A county government office compares stationery prices:

Calculate Marshall-Edgeworth index.

$$ M = \frac{(22 \times (400 + 450)) + (110 \times (250 + 300))}{(20 \times (400 + 450)) + (100 \times (250 + 300))} \times 100 $$

Calculate numerator:

$$ 22 \times 850 = 18700 $$

$$ 110 \times 550 = 60500 $$

Sum numerator:

$$ 18700 + 60500 = 79200 $$

Calculate denominator:

$$ 20 \times 850 = 17000 $$

$$ 100 \times 550 = 55000 $$

Sum denominator:

$$ 17000 + 55000 = 72000 $$

Calculate index:

$$ M = \frac{79200}{72000} \times 100 = 110 $$

Answer: 110

Chapter Summary

This chapter focused on the formulae used for computing various types of index numbers, essential tools in business mathematics and statistics. It detailed the computation methods for key index numbers including Laspeyres, Paasche, Fisher's ideal, and Marshall indexes, explaining how each is calculated and applied. The chapter highlighted the differences between these index numbers based on their weighting systems and base periods. Practical computation examples demonstrated the use of these formulae in real-world scenarios. Additionally, the chapter emphasized the importance of index numbers in decision making, showing how they assist managers and analysts in tracking price changes, inflation, and economic trends. Trainees engaged in practical work to reinforce their understanding of calculating and interpreting index numbers effectively. This knowledge equips students with the skills to apply index numbers confidently in business and economic analysis.