Kenyan professionals across all sectors rely on index numbers to measure changes in prices, quantities, or values over time. These measurements are crucial for decision-making in finance, economics, and business operations, enabling institutions such as county government offices, banks, and retail businesses to assess inflation, cost trends, and economic performance. Understanding how to construct and interpret index numbers empowers professionals to analyze data effectively and support strategic planning. This chapter focuses on the mathematical foundations and methods for constructing index numbers.

8.1 Construction of index numbers

Index numbers summarize changes in a group of related variables, such as prices or quantities, over different time periods or locations. They are widely used in economics and business to track inflation, cost of living, and productivity. The construction of index numbers involves selecting a base period, choosing the items to include, and calculating the relative changes in values.

8.1.1 Price Index Numbers: Concept and Formula

Price index numbers measure the relative change in the price of a basket of goods and services over time compared to a base period. They indicate how much prices have increased or decreased, which is vital for sectors like county referral hospitals managing procurement budgets.

The general formula for a simple price index number is:

$$ PI = \frac{P_t}{P_0} \times 100 $$

Where:

• \(PI\) = price index number,
• \(P_t\) = price of the item in the current period,
• \(P_0\) = price of the item in the base period.

Worked Examples

Example 1: A retail store in Nairobi sold maize flour at Ksh 120 per kilogram in January 2023. The price in January 2022 was Ksh 100 per kilogram. Calculate the price index number for maize flour in January 2023 using January 2022 as the base period.

Given: \(P_t = 120\), \(P_0 = 100\)

$$ PI = \frac{P_t}{P_0} \times 100 $$

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

$$ PI = 1.2 \times 100 $$

$$ PI = 120 $$

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

Example 2: A county hospital recorded the price of a medical glove at Ksh 150 in 2023, while the price was Ksh 125 in 2020. Calculate the price index for 2023 using 2020 as the base year.

Given: \(P_t = 150\), \(P_0 = 125\)

$$ PI = \frac{150}{125} \times 100 $$

$$ PI = 1.2 \times 100 $$

$$ PI = 120 $$

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

Example 3: A SACCO reports that the price of stationery increased from Ksh 80 in 2019 to Ksh 100 in 2024. Find the price index for 2024 with 2019 as the base year.

Given: \(P_t = 100\), \(P_0 = 80\)

$$ PI = \frac{100}{80} \times 100 $$

$$ PI = 1.25 \times 100 $$

$$ PI = 125 $$

Answer: The price index number is 125, indicating a 25% increase in price.

Example 4: A university cafeteria notes that the price of maize meal was Ksh 90 in 2018 and Ksh 81 in 2022. Calculate the price index for 2022 relative to 2018.

Given: \(P_t = 81\), \(P_0 = 90\)

$$ PI = \frac{81}{90} \times 100 $$

$$ PI = 0.9 \times 100 $$

$$ PI = 90 $$

Answer: The price index number is 90, indicating a 10% decrease in price.

Example 5: A hotel reports that the price of cooking oil was Ksh 180 in 2021 and Ksh 198 in 2023. Calculate the price index for 2023 using 2021 as the base year.

Given: \(P_t = 198\), \(P_0 = 180\)

$$ PI = \frac{198}{180} \times 100 $$

$$ PI = 1.1 \times 100 $$

$$ PI = 110 $$

Answer: The price index number is 110, showing a 10% price increase.

8.1.2 Weighted Price Index Numbers: Laspeyres and Paasche Methods

Weighted price index numbers account for the relative importance or quantity of items in the basket, giving a more accurate reflection of overall price changes. The Laspeyres index uses base period quantities as weights, while the Paasche index uses current period quantities.

The Laspeyres price index formula is:

$$ LPI = \frac{\sum (P_t \times Q_0)}{\sum (P_0 \times Q_0)} \times 100 $$

The Paasche price index formula is:

$$ PPI = \frac{\sum (P_t \times Q_t)}{\sum (P_0 \times Q_t)} \times 100 $$

Where:

• \(P_t\), \(P_0\) are prices in current and base periods,
• \(Q_t\), \(Q_0\) are quantities in current and base periods.

Worked Examples

Example 1: A cooperative sells two products. In the base year, prices and quantities were: maize flour Ksh 100 (50 kg), beans Ksh 150 (30 kg). In the current year, prices are maize flour Ksh 120, beans Ksh 180. Calculate the Laspeyres price index.

Given:
Base prices \(P_0\): maize 100, beans 150
Current prices \(P_t\): maize 120, beans 180
Base quantities \(Q_0\): maize 50, beans 30

$$ LPI = \frac{(120 \times 50) + (180 \times 30)}{(100 \times 50) + (150 \times 30)} \times 100 $$

$$ LPI = \frac{6000 + 5400}{5000 + 4500} \times 100 $$

$$ LPI = \frac{11400}{9500} \times 100 $$

$$ LPI = 1.2 \times 100 $$

$$ LPI = 120 $$

Answer: The Laspeyres price index is 120, indicating a 20% price increase weighted by base quantities.

Example 2: Using the same data, if current quantities are maize 60 kg and beans 25 kg, calculate the Paasche price index.

Given:
Current quantities \(Q_t\): maize 60, beans 25

$$ PPI = \frac{(120 \times 60) + (180 \times 25)}{(100 \times 60) + (150 \times 25)} \times 100 $$

$$ PPI = \frac{7200 + 4500}{6000 + 3750} \times 100 $$

$$ PPI = \frac{11700}{9750} \times 100 $$

$$ PPI = 1.2 \times 100 $$

$$ PPI = 120 $$

Answer: The Paasche price index is 120, also indicating a 20% price increase weighted by current quantities.

Example 3: A county government office tracks prices of fuel and lubricants. Base year prices and quantities: fuel Ksh 150 (200 litres), lubricants Ksh 300 (50 litres). Current year prices: fuel Ksh 180, lubricants Ksh 270. Current year quantities: fuel 210 litres, lubricants 40 litres. Calculate the Laspeyres and Paasche price indices.

Given:
Base prices \(P_0\): fuel 150, lubricants 300
Current prices \(P_t\): fuel 180, lubricants 270
Base quantities \(Q_0\): fuel 200, lubricants 50
Current quantities \(Q_t\): fuel 210, lubricants 40

Laspeyres index:

$$ LPI = \frac{(180 \times 200) + (270 \times 50)}{(150 \times 200) + (300 \times 50)} \times 100 $$

$$ LPI = \frac{36000 + 13500}{30000 + 15000} \times 100 $$

$$ LPI = \frac{49500}{45000} \times 100 $$

$$ LPI = 1.1 \times 100 $$

$$ LPI = 110 $$

Paasche index:

$$ PPI = \frac{(180 \times 210) + (270 \times 40)}{(150 \times 210) + (300 \times 40)} \times 100 $$

$$ PPI = \frac{37800 + 10800}{31500 + 12000} \times 100 $$

$$ PPI = \frac{48600}{43500} \times 100 $$

$$ PPI = 1.1172 \times 100 $$

$$ PPI = 111.72 $$

Answer: Laspeyres index is 110, Paasche index is 111.72, indicating price increases weighted by base and current quantities respectively.

Example 4: A retail business sells sugar and tea. Base year prices and quantities: sugar Ksh 80 (40 kg), tea Ksh 150 (20 kg). Current prices: sugar Ksh 100, tea Ksh 180. Current quantities: sugar 35 kg, tea 25 kg. Calculate the Laspeyres and Paasche price indices.

Given:
\(P_0\) sugar 80, tea 150;
\(P_t\) sugar 100, tea 180;
\(Q_0\) sugar 40, tea 20;
\(Q_t\) sugar 35, tea 25

Laspeyres index:

$$ LPI = \frac{(100 \times 40) + (180 \times 20)}{(80 \times 40) + (150 \times 20)} \times 100 $$

$$ LPI = \frac{4000 + 3600}{3200 + 3000} \times 100 $$

$$ LPI = \frac{7600}{6200} \times 100 $$

$$ LPI = 1.2258 \times 100 $$

$$ LPI = 122.58 $$

Paasche index:

$$ PPI = \frac{(100 \times 35) + (180 \times 25)}{(80 \times 35) + (150 \times 25)} \times 100 $$

$$ PPI = \frac{3500 + 4500}{2800 + 3750} \times 100 $$

$$ PPI = \frac{8000}{6550} \times 100 $$

$$ PPI = 1.2214 \times 100 $$

$$ PPI = 122.14 $$

Answer: Laspeyres index is 122.58, Paasche index is 122.14, showing similar price increase weights.

Example 5: A hotel monitors prices of beef and vegetables. Base year prices and quantities: beef Ksh 400 (60 kg), vegetables Ksh 100 (80 kg). Current year prices: beef Ksh 420, vegetables Ksh 110. Current year quantities: beef 65 kg, vegetables 75 kg. Calculate the Laspeyres and Paasche price indices.

Given:
\(P_0\) beef 400, vegetables 100;
\(P_t\) beef 420, vegetables 110;
\(Q_0\) beef 60, vegetables 80;
\(Q_t\) beef 65, vegetables 75

Laspeyres index:

$$ LPI = \frac{(420 \times 60) + (110 \times 80)}{(400 \times 60) + (100 \times 80)} \times 100 $$

$$ LPI = \frac{25200 + 8800}{24000 + 8000} \times 100 $$

$$ LPI = \frac{34000}{32000} \times 100 $$

$$ LPI = 1.0625 \times 100 $$

$$ LPI = 106.25 $$

Paasche index:

$$ PPI = \frac{(420 \times 65) + (110 \times 75)}{(400 \times 65) + (100 \times 75)} \times 100 $$

$$ PPI = \frac{27300 + 8250}{26000 + 7500} \times 100 $$

$$ PPI = \frac{35550}{33500} \times 100 $$

$$ PPI = 1.0604 \times 100 $$

$$ PPI = 106.04 $$

Answer: Laspeyres index is 106.25, Paasche index is 106.04, indicating a modest price increase.

8.1.3 Construction of Quantity Index Numbers

Quantity index numbers measure the relative change in quantities of goods and services over time, using prices as weights. They help organizations like farms or cooperatives assess production trends.

The Laspeyres quantity index formula is:

$$ LQI = \frac{\sum (Q_t \times P_0)}{\sum (Q_0 \times P_0)} \times 100 $$

The Paasche quantity index formula is:

$$ PQI = \frac{\sum (Q_t \times P_t)}{\sum (Q_0 \times P_t)} \times 100 $$

Where:

• \(Q_t\), \(Q_0\) are quantities in current and base periods,
• \(P_t\), \(P_0\) are prices in current and base periods.

Worked Examples

Example 1: A farm produced 1000 kg of maize in 2021 and 1200 kg in 2023. The price of maize was Ksh 80 in 2021 and Ksh 100 in 2023. Calculate the Laspeyres quantity index.

Given:
\(Q_0 = 1000\), \(Q_t = 1200\), \(P_0 = 80\)

$$ LQI = \frac{1200 \times 80}{1000 \times 80} \times 100 $$

$$ LQI = \frac{96000}{80000} \times 100 $$

$$ LQI = 1.2 \times 100 $$

$$ LQI = 120 $$

Answer: The Laspeyres quantity index is 120, indicating a 20% increase in quantity produced.

Example 2: A cooperative produced 500 kg of coffee in 2019 and 450 kg in 2022. Prices were Ksh 250 in 2019 and Ksh 300 in 2022. Calculate the Paasche quantity index.

Given:
\(Q_0 = 500\), \(Q_t = 450\), \(P_t = 300\)

$$ PQI = \frac{450 \times 300}{500 \times 300} \times 100 $$

$$ PQI = \frac{135000}{150000} \times 100 $$

$$ PQI = 0.9 \times 100 $$

$$ PQI = 90 $$

Answer: The Paasche quantity index is 90, showing a 10% decrease in quantity.

Example 3: A retail business sold 100 units of soap in 2020 and 120 units in 2023. Prices were Ksh 50 in 2020 and Ksh 55 in 2023. Calculate the Laspeyres quantity index.

Given:
\(Q_0=100\), \(Q_t=120\), \(P_0=50\)

$$ LQI = \frac{120 \times 50}{100 \times 50} \times 100 $$

$$ LQI = \frac{6000}{5000} \times 100 $$

$$ LQI = 1.2 \times 100 $$

$$ LQI = 120 $$

Answer: The Laspeyres quantity index is 120, indicating a 20% increase in quantity sold.

Example 4: A county government purchased 300 laptops in 2021 and 280 laptops in 2024. Prices were Ksh 50,000 in 2021 and Ksh 55,000 in 2024. Calculate the Paasche quantity index.

Given:
\(Q_0=300\), \(Q_t=280\), \(P_t=55000\)

$$ PQI = \frac{280 \times 55000}{300 \times 55000} \times 100 $$

$$ PQI = \frac{15400000}{16500000} \times 100 $$

$$ PQI = 0.9333 \times 100 $$

$$ PQI = 93.33 $$

Answer: The Paasche quantity index is 93.33, indicating a 6.67% decrease in quantity purchased.

Example 5: A university library acquired 2000 books in 2018 and 2200 books in 2023. Prices were Ksh 1500 in 2018 and Ksh 1800 in 2023. Compute the Laspeyres and Paasche quantity indices.

Given:
\(Q_0=2000\), \(Q_t=2200\), \(P_0=1500\), \(P_t=1800\)

Laspeyres quantity index:

$$ LQI = \frac{2200 \times 1500}{2000 \times 1500} \times 100 $$

$$ LQI = \frac{3300000}{3000000} \times 100 $$

$$ LQI = 1.1 \times 100 $$

$$ LQI = 110 $$

Paasche quantity index:

$$ PQI = \frac{2200 \times 1800}{2000 \times 1800} \times 100 $$

$$ PQI = \frac{3960000}{3600000} \times 100 $$

$$ PQI = 1.1 \times 100 $$

$$ PQI = 110 $$

Answer: Both Laspeyres and Paasche quantity indices are 110, indicating a 10% increase in quantity acquired.

8.1.4 Construction of Composite Index Numbers

Composite index numbers combine price or quantity indices of several items into a single measure, reflecting overall changes in a group of goods or services. They use weights to reflect the relative importance of each item in the total basket.

The general formula for a weighted composite index number is:

$$ CI = \frac{\sum (Index_i \times Weight_i)}{\sum Weight_i} $$

Where:

• \(Index_i\) = index number of item \(i\),
• \(Weight_i\) = weight of item \(i\).

Worked Examples

Example 1: A retail business tracks price indices for maize (120), beans (110), and sugar (130). The weights based on expenditure are maize 50, beans 30, sugar 20. Calculate the composite price index.

Given:
Indices: maize 120, beans 110, sugar 130
Weights: maize 50, beans 30, sugar 20

$$ CI = \frac{(120 \times 50) + (110 \times 30) + (130 \times 20)}{50 + 30 + 20} $$

$$ CI = \frac{6000 + 3300 + 2600}{100} $$

$$ CI = \frac{11900}{100} $$

$$ CI = 119 $$

Answer: The composite price index is 119, indicating a 19% overall price increase.

Example 2: A county office monitors quantity indices for fuel (105), lubricants (110), and tyres (95). Weights based on usage are fuel 60, lubricants 25, tyres 15. Calculate the composite quantity index.

Given:
Indices: fuel 105, lubricants 110, tyres 95
Weights: fuel 60, lubricants 25, tyres 15

$$ CI = \frac{(105 \times 60) + (110 \times 25) + (95 \times 15)}{60 + 25 + 15} $$

$$ CI = \frac{6300 + 2750 + 1425}{100} $$

$$ CI = \frac{10475}{100} $$

$$ CI = 104.75 $$

Answer: The composite quantity index is 104.75, indicating a 4.75% increase in quantities.

Example 3: A hotel tracks price indices for food (115), beverages (120), and cleaning supplies (110). Weights by expenditure are food 70, beverages 20, cleaning supplies 10. Find the composite price index.

Given:
Indices: food 115, beverages 120, cleaning supplies 110
Weights: food 70, beverages 20, cleaning supplies 10

$$ CI = \frac{(115 \times 70) + (120 \times 20) + (110 \times 10)}{70 + 20 + 10} $$

$$ CI = \frac{8050 + 2400 + 1100}{100} $$

$$ CI = \frac{11550}{100} $$

$$ CI = 115.5 $$

Answer: The composite price index is 115.5, indicating a 15.5% overall price increase.

Example 4: A cooperative monitors quantity indices for tea (90), coffee (100), and sugar (105) with weights 40, 35, and 25 respectively. Calculate the composite quantity index.

Given:
Indices: tea 90, coffee 100, sugar 105
Weights: tea 40, coffee 35, sugar 25

$$ CI = \frac{(90 \times 40) + (100 \times 35) + (105 \times 25)}{40 + 35 + 25} $$

$$ CI = \frac{3600 + 3500 + 2625}{100} $$

$$ CI = \frac{9725}{100} $$

$$ CI = 97.25 $$

Answer: The composite quantity index is 97.25, indicating a 2.75% decrease overall.

Example 5: A university tracks price indices for books (125), stationery (115), and equipment (130) with weights 50, 30, and 20. Calculate the composite price index.

Given:
Indices: books 125, stationery 115, equipment 130
Weights: 50, 30, 20

$$ CI = \frac{(125 \times 50) + (115 \times 30) + (130 \times 20)}{100} $$

$$ CI = \frac{6250 + 3450 + 2600}{100} $$

$$ CI = \frac{12300}{100} $$

$$ CI = 123 $$

Answer: The composite price index is 123, indicating a 23% increase overall.

Chapter Summary

This chapter focused on the construction of index numbers, explaining the step-by-step process involved in compiling them using relevant data. It highlighted the purpose of index numbers in measuring changes in economic variables such as prices and quantities over time. The chapter covered simple index numbers, specifically the fixed base and chain base methods, showing how they track changes relative to a base period. A detailed explanation of the Consumer Price Index (CPI) was provided as a key application of index numbers in monitoring inflation. The chapter then introduced weighted index numbers, including Laspeyre's, Paasche's, and Fisher's ideal indices, which incorporate the relative importance of items in the calculation. Marshall-Edgeworth's method was discussed as an alternative approach for calculating both price and quantity index numbers. Finally, the chapter addressed the limitations of index numbers and explored emerging issues and trends affecting their use in modern economic analysis.