Index numbers are essential tools in Kenyan professional settings for measuring changes in economic, social, and business variables over time. They help institutions such as banks, retail businesses, county governments, and cooperatives monitor price changes, inflation rates, production levels, and cost variations. Understanding how to compute and interpret index numbers allows professionals across sectors to make informed decisions based on quantitative trends. This chapter introduces the formulae and methods for calculating various index numbers widely used in practice.

8.1 Formulae for computing index numbers

Index numbers summarize changes in a set of data relative to a base period. They quantify relative movements in prices, quantities, or values and are expressed as percentages. Different formulae exist depending on the weights assigned to quantities or prices, leading to distinct types of index numbers.

8.1.1 Price index number formula

A price index number measures the relative change in prices of goods and services between a current period and a base period. The general formula is:

$$ \text{Price Index} = \frac{\text{Price in current period}}{\text{Price in base period}} \times 100 $$

This formula can be applied to single items or aggregated across multiple items using weighted averages.

Worked Examples

Example 1: A retail shop in Nairobi sold sugar at Ksh 100 per kg last year and now sells it at Ksh 110 per kg. Find the price index.

Given: \(P_{base} = 100\), \(P_{current} = 110\)

$$ \text{Price Index} = \frac{110}{100} \times 100 $$$$ = 1.1 \times 100 $$$$ = 110 $$

Answer: 110 (price index)

Example 2: At a county hospital, the price of medicines increased from Ksh 500 to Ksh 575. Calculate the price index.

Given: \(P_{base} = 500\), \(P_{current} = 575\)

$$ \text{Price Index} = \frac{575}{500} \times 100 $$$$ = 1.15 \times 100 $$$$ = 115 $$

Answer: 115 (price index)

Example 3: A cooperative notes that the price of maize changed from Ksh 120 to Ksh 108 per 90 kg bag. Find the price index.

Given: \(P_{base} = 120\), \(P_{current} = 108\)

$$ \text{Price Index} = \frac{108}{120} \times 100 $$$$ = 0.9 \times 100 $$$$ = 90 $$

Answer: 90 (price index)

8.1.2 Quantity index number formula

A quantity index number measures changes in quantities of goods or services consumed or produced between two periods. The formula is:

$$ \text{Quantity Index} = \frac{\text{Quantity in current period}}{\text{Quantity in base period}} \times 100 $$

This helps organizations like farms or manufacturing firms track production changes.

Worked Examples

Example 1: A farm produced 500 bags of maize last season and 600 bags this season. Find the quantity index.

Given: \(Q_{base} = 500\), \(Q_{current} = 600\)

$$ \text{Quantity Index} = \frac{600}{500} \times 100 $$$$ = 1.2 \times 100 $$$$ = 120 $$

Answer: 120 (quantity index)

Example 2: A hotel served 800 customers last month and 720 customers this month. Calculate the quantity index.

Given: \(Q_{base} = 800\), \(Q_{current} = 720\)

$$ \text{Quantity Index} = \frac{720}{800} \times 100 $$$$ = 0.9 \times 100 $$$$ = 90 $$

Answer: 90 (quantity index)

Example 3: A university library lent out 3000 books in 2022 and 3600 books in 2023. Find the quantity index.

Given: \(Q_{base} = 3000\), \(Q_{current} = 3600\)

$$ \text{Quantity Index} = \frac{3600}{3000} \times 100 $$$$ = 1.2 \times 100 $$$$ = 120 $$

Answer: 120 (quantity index)

8.1.3 Value index number formula

A value index number measures changes in the total value of goods or services between two periods. It combines price and quantity changes and is calculated as:

$$ \text{Value Index} = \frac{\text{Value in current period}}{\text{Value in base period}} \times 100 $$

Where value is price multiplied by quantity.

Worked Examples

Example 1: A SACCO recorded loan disbursements worth Ksh 2,000,000 last year and Ksh 2,400,000 this year. Find the value index.

Given: \(V_{base} = 2,000,000\), \(V_{current} = 2,400,000\)

$$ \text{Value Index} = \frac{2,400,000}{2,000,000} \times 100 $$$$ = 1.2 \times 100 $$$$ = 120 $$

Answer: 120 (value index)

Example 2: A county government collected Ksh 1,500,000 in revenue last quarter and Ksh 1,350,000 this quarter. Calculate the value index.

Given: \(V_{base} = 1,500,000\), \(V_{current} = 1,350,000\)

$$ \text{Value Index} = \frac{1,350,000}{1,500,000} \times 100 $$$$ = 0.9 \times 100 $$$$ = 90 $$

Answer: 90 (value index)

Example 3: A retail business sold goods worth Ksh 5,000,000 in 2021 and Ksh 6,250,000 in 2022. Find the value index.

Given: \(V_{base} = 5,000,000\), \(V_{current} = 6,250,000\)

$$ \text{Value Index} = \frac{6,250,000}{5,000,000} \times 100 $$$$ = 1.25 \times 100 $$$$ = 125 $$

Answer: 125 (value index)

8.1.4 Weighted index number formula

Weighted index numbers account for the relative importance of items using weights such as quantities or expenditures. The general form is:

$$ \text{Index Number} = \frac{\sum (P_{current} \times W)}{\sum (P_{base} \times W)} \times 100 $$

Where \(P\) denotes price and \(W\) the weight.

Worked Examples

Example 1: A retail store sells two items: maize (weight 50 units) and beans (weight 30 units). Prices last year were maize Ksh 100, beans Ksh 80. This year, maize Ksh 110, beans Ksh 85. Calculate the weighted price index.

Given: \(P_{base, maize} = 100\), \(P_{current, maize} = 110\), \(W_{maize} = 50\)

\(P_{base, beans} = 80\), \(P_{current, beans} = 85\), \(W_{beans} = 30\)

$$ \text{Index Number} = \frac{(110 \times 50) + (85 \times 30)}{(100 \times 50) + (80 \times 30)} \times 100 $$$$ = \frac{5500 + 2550}{5000 + 2400} \times 100 $$$$ = \frac{8050}{7400} \times 100 $$$$ = 1.0878 \times 100 $$$$ = 108.78 $$

Answer: 108.78 (weighted price index)

Example 2: A hotel monitors prices of food items: rice (weight 40 kg), meat (weight 20 kg). Prices last month rice Ksh 120/kg, meat Ksh 250/kg; this month rice Ksh 130/kg, meat Ksh 270/kg. Find the weighted price index.

Given: \(P_{base, rice} = 120\), \(P_{current, rice} = 130\), \(W_{rice} = 40\)

\(P_{base, meat} = 250\), \(P_{current, meat} = 270\), \(W_{meat} = 20\)

$$ \text{Index Number} = \frac{(130 \times 40) + (270 \times 20)}{(120 \times 40) + (250 \times 20)} \times 100 $$$$ = \frac{5200 + 5400}{4800 + 5000} \times 100 $$$$ = \frac{10600}{9800} \times 100 $$$$ = 1.0816 \times 100 $$$$ = 108.16 $$

Answer: 108.16 (weighted price index)

Example 3: A cooperative tracks fertilizer prices with weights based on usage: fertilizer A (weight 100 kg), fertilizer B (weight 50 kg). Prices last year were Ksh 200/kg and Ksh 150/kg; this year Ksh 220/kg and Ksh 160/kg. Calculate the weighted price index.

Given: \(P_{base,A} = 200\), \(P_{current,A} = 220\), \(W_A = 100\)

\(P_{base,B} = 150\), \(P_{current,B} = 160\), \(W_B = 50\)

$$ \text{Index Number} = \frac{(220 \times 100) + (160 \times 50)}{(200 \times 100) + (150 \times 50)} \times 100 $$$$ = \frac{22000 + 8000}{20000 + 7500} \times 100 $$$$ = \frac{30000}{27500} \times 100 $$$$ = 1.0909 \times 100 $$$$ = 109.09 $$

Answer: 109.09 (weighted price index)

Chapter Summary

This chapter covered the fundamental formulae used to compute various types of index numbers, which are essential tools for measuring changes in economic and statistical data over time. The computation of index numbers was explored in detail through four main methods: Laspeyre's, Paasche's, Fisher's ideal, and Marshal's index numbers, each with its unique approach to weighting and calculation. Laspeyre's index uses base period quantities, while Paasche's index relies on current period quantities. Fisher's ideal index combines both Laspeyre's and Paasche's indices to provide a more balanced measure. Marshal's index was also introduced as another method for computing index numbers. Finally, the chapter demonstrated how these index numbers are applied in decision making, helping businesses and policymakers analyze trends and make informed choices based on quantitative data.