Kenyan professionals across diverse sectors rely heavily on index numbers to analyze economic performance, price changes, and cost variations over time. Index numbers provide a simplified way to compare data sets and identify trends essential for decision-making in businesses, government offices, hospitals, and cooperatives. Mastery of constructing and interpreting index numbers enables accurate financial planning and policy formulation in Kenya’s dynamic economic environment.

8.1 Construction of index numbers

Index numbers measure relative changes in variables such as prices, quantities, or values over time or between locations. The construction involves selecting a base period, choosing items to include, and applying formulas to compute the indices. This topic covers the main types of index numbers and step-by-step methods to calculate them.

8.1.1 Price index numbers: Laspeyres, Paasche, and Fisher methods

Price index numbers compare the cost of a fixed basket of goods and services between periods. The three main formulae differ by how they weight prices and quantities.

The Laspeyres price index uses base period quantities as weights, showing how prices have changed relative to the base period.

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

where \(p_t\) = price in current period, \(p_0\) = price in base period, \(q_0\) = quantity in base period.

The Paasche price index uses current period quantities as weights:

$$ I_P = \frac{\sum p_t q_t}{\sum p_0 q_t} \times 100 $$

The Fisher price index is the geometric mean of Laspeyres and Paasche:

$$ I_F = \sqrt{I_L \times I_P} $$

Worked Examples

Example 1: A county hospital tracks prices of medicines. Base year prices and quantities are: Medicine A: Ksh 100 (50 units), Medicine B: Ksh 200 (30 units). Current year prices: Medicine A: Ksh 120, Medicine B: Ksh 180. Calculate Laspeyres price index.

Given: \(p_0 = [100, 200]\), \(q_0 = [50, 30]\), \(p_t = [120, 180]\)

$$ I_L = \frac{(120 \times 50) + (180 \times 30)}{(100 \times 50) + (200 \times 30)} \times 100 $$

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

$$ = \frac{11400}{11000} \times 100 = 103.64 $$

Answer: 103.64

Example 2: Using the data from Example 1, current quantities are Medicine A: 40 units, Medicine B: 35 units. Calculate the Paasche price index.

Given: \(p_0 = [100, 200]\), \(q_t = [40, 35]\), \(p_t = [120, 180]\)

$$ I_P = \frac{(120 \times 40) + (180 \times 35)}{(100 \times 40) + (200 \times 35)} \times 100 $$

$$ = \frac{4800 + 6300}{4000 + 7000} \times 100 $$

$$ = \frac{11100}{11000} \times 100 = 100.91 $$

Answer: 100.91

Example 3: Calculate the Fisher price index from Examples 1 and 2.

Given: \(I_L = 103.64\), \(I_P = 100.91\)

$$ I_F = \sqrt{103.64 \times 100.91} $$

$$ = \sqrt{10457.97} = 102.27 $$

Answer: 102.27

Example 4: A retail business in Nairobi has base year prices and quantities for three products: P1: Ksh 50 (100 units), P2: Ksh 80 (60 units), P3: Ksh 30 (150 units). Current year prices are P1: Ksh 55, P2: Ksh 85, P3: Ksh 35. Calculate Laspeyres price index.

Given: \(p_0 = [50, 80, 30]\), \(q_0 = [100, 60, 150]\), \(p_t = [55, 85, 35]\)

$$ I_L = \frac{(55 \times 100) + (85 \times 60) + (35 \times 150)}{(50 \times 100) + (80 \times 60) + (30 \times 150)} \times 100 $$

$$ = \frac{5500 + 5100 + 5250}{5000 + 4800 + 4500} \times 100 $$

$$ = \frac{15850}{14300} \times 100 = 110.84 $$

Answer: 110.84

Example 5: Using the data in Example 4, current quantities are P1: 90, P2: 70, P3: 140. Calculate Paasche price index.

Given: \(p_0 = [50, 80, 30]\), \(q_t = [90, 70, 140]\), \(p_t = [55, 85, 35]\)

$$ I_P = \frac{(55 \times 90) + (85 \times 70) + (35 \times 140)}{(50 \times 90) + (80 \times 70) + (30 \times 140)} \times 100 $$

$$ = \frac{4950 + 5950 + 4900}{4500 + 5600 + 4200} \times 100 $$

$$ = \frac{15800}{14300} \times 100 = 110.49 $$

Answer: 110.49

8.1.2 Quantity index numbers and value index numbers

Quantity index numbers measure changes in quantities consumed or produced, while value index numbers track changes in total value (price × quantity).

The Laspeyres quantity index uses base period prices:

$$ I_{Q_L} = \frac{\sum p_0 q_t}{\sum p_0 q_0} \times 100 $$

The Paasche quantity index uses current period prices:

$$ I_{Q_P} = \frac{\sum p_t q_t}{\sum p_t q_0} \times 100 $$

The value index compares total current value to base value:

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

Worked Examples

Example 1: A cooperative sells two crops. Base year prices and quantities: Maize: Ksh 40 (100 bags), Beans: Ksh 60 (50 bags). Current year quantities: Maize: 120 bags, Beans: 45 bags. Calculate Laspeyres quantity index.

Given: \(p_0 = [40, 60]\), \(q_0 = [100, 50]\), \(q_t = [120, 45]\)

$$ I_{Q_L} = \frac{(40 \times 120) + (60 \times 45)}{(40 \times 100) + (60 \times 50)} \times 100 $$

$$ = \frac{4800 + 2700}{4000 + 3000} \times 100 $$

$$ = \frac{7500}{7000} \times 100 = 107.14 $$

Answer: 107.14

Example 2: Using Example 1 data, current prices are Maize: Ksh 45, Beans: Ksh 55. Calculate Paasche quantity index.

Given: \(p_t = [45, 55]\), \(q_t = [120, 45]\), \(q_0 = [100, 50]\)

$$ I_{Q_P} = \frac{(45 \times 120) + (55 \times 45)}{(45 \times 100) + (55 \times 50)} \times 100 $$

$$ = \frac{5400 + 2475}{4500 + 2750} \times 100 $$

$$ = \frac{7875}{7250} \times 100 = 108.62 $$

Answer: 108.62

Example 3: Calculate value index using data from Example 1 and 2.

Given: \(p_0 = [40, 60]\), \(q_0 = [100, 50]\), \(p_t = [45, 55]\), \(q_t = [120, 45]\)

$$ I_V = \frac{(45 \times 120) + (55 \times 45)}{(40 \times 100) + (60 \times 50)} \times 100 $$

$$ = \frac{5400 + 2475}{4000 + 3000} \times 100 $$

$$ = \frac{7875}{7000} \times 100 = 112.50 $$

Answer: 112.50

Example 4: A retail shop in Mombasa tracks quantity changes for three items. Base prices and quantities: Item1: Ksh 20 (150 units), Item2: Ksh 35 (80 units), Item3: Ksh 15 (200 units). Current quantities: Item1: 160, Item2: 75, Item3: 190. Calculate Laspeyres quantity index.

Given: \(p_0 = [20, 35, 15]\), \(q_0 = [150, 80, 200]\), \(q_t = [160, 75, 190]\)

$$ I_{Q_L} = \frac{(20 \times 160) + (35 \times 75) + (15 \times 190)}{(20 \times 150) + (35 \times 80) + (15 \times 200)} \times 100 $$

$$ = \frac{3200 + 2625 + 2850}{3000 + 2800 + 3000} \times 100 $$

$$ = \frac{8675}{8800} \times 100 = 98.58 $$

Answer: 98.58

Example 5: Using Example 4 data, current prices are Item1: Ksh 22, Item2: Ksh 33, Item3: Ksh 18. Calculate Paasche quantity index.

Given: \(p_t = [22, 33, 18]\), \(q_0 = [150, 80, 200]\), \(q_t = [160, 75, 190]\)

$$ I_{Q_P} = \frac{(22 \times 160) + (33 \times 75) + (18 \times 190)}{(22 \times 150) + (33 \times 80) + (18 \times 200)} \times 100 $$

$$ = \frac{3520 + 2475 + 3420}{3300 + 2640 + 3600} \times 100 $$

$$ = \frac{9415}{9540} \times 100 = 98.69 $$

Answer: 98.69

8.1.3 Steps in constructing an index number

Constructing an index number requires a systematic approach to ensure accuracy and relevance. The process involves selecting base period, items, data collection, choosing formula, and computing the index.

Steps for constructing an index number

1. Define the purpose of the index: Identify whether the index is for prices, quantities, or values to guide the data selection and method.

2. Select the base period: Choose a year or time period as the benchmark against which changes will be measured; it should be representative and stable.

3. Select items to include: Choose a basket of goods or services relevant to the purpose, ensuring they reflect typical consumption or production patterns.

4. Collect data: Gather accurate and consistent data on prices and quantities for both base and current periods from reliable sources such as county government reports or retail records.

5. Choose the appropriate formula: Decide on Laspeyres, Paasche, Fisher, or other index formula based on the data available and analysis requirements.

6. Calculate the index number: Apply the formula by substituting data and perform arithmetic operations to compute the index value.

7. Interpret and present results: Analyze the index in context and present clearly to stakeholders for informed decision-making.

Worked Examples

Example 1: A county government office wants to construct a price index for office supplies using base year 2022 and current year 2023 data. Prices and quantities for pens and paper are given. Follow the steps to compute the Laspeyres price index.

Given: Base prices \(p_0 = [Ksh 15, Ksh 50]\), base quantities \(q_0 = [200, 100]\), current prices \(p_t = [Ksh 18, Ksh 45]\).

Step 1: Define purpose, price index of office supplies.

Step 2: Base period - 2022.

Step 3: Items, pens and paper.

Step 4: Data collected as above.

Step 5: Formula - Laspeyres price index:

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

Step 6: Calculate numerator:

$$ (18 \times 200) + (45 \times 100) = 3600 + 4500 = 8100 $$

Calculate denominator:

$$ (15 \times 200) + (50 \times 100) = 3000 + 5000 = 8000 $$

Compute index:

$$ I_L = \frac{8100}{8000} \times 100 = 101.25 $$

Step 7: Interpretation, prices increased by 1.25% from 2022 to 2023.

Answer: 101.25

Example 2: A retail business wants to construct a quantity index for three products. Base prices and quantities are: Product A: Ksh 100 (50 units), Product B: Ksh 150 (30 units), Product C: Ksh 75 (40 units). Current quantities are 55, 35, and 45 units respectively. Calculate Laspeyres quantity index.

Step 1: Purpose, quantity index.

Step 2: Base period, current prices used.

Step 3: Items - Products A, B, C.

Step 4: Data collected.

Step 5: Formula:

$$ I_{Q_L} = \frac{\sum p_0 q_t}{\sum p_0 q_0} \times 100 $$

Step 6: Calculate numerator:

$$ (100 \times 55) + (150 \times 35) + (75 \times 45) = 5500 + 5250 + 3375 = 14125 $$

Calculate denominator:

$$ (100 \times 50) + (150 \times 30) + (75 \times 40) = 5000 + 4500 + 3000 = 12500 $$

Compute index:

$$ I_{Q_L} = \frac{14125}{12500} \times 100 = 113.00 $$

Step 7: Interpretation, quantity increased by 13%.

Answer: 113.00

Example 3: A cooperative wants to calculate a value index for two crops. Base prices and quantities: Crop X: Ksh 60 (100 bags), Crop Y: Ksh 80 (50 bags). Current prices and quantities: Crop X: Ksh 65 (110 bags), Crop Y: Ksh 85 (55 bags).

Step 1: Purpose, value index.

Step 2: Base period selected.

Step 3: Items - Crop X and Y.

Step 4: Data gathered.

Step 5: Formula:

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

Step 6: Calculate numerator:

$$ (65 \times 110) + (85 \times 55) = 7150 + 4675 = 11825 $$

Calculate denominator:

$$ (60 \times 100) + (80 \times 50) = 6000 + 4000 = 10000 $$

Compute index:

$$ I_V = \frac{11825}{10000} \times 100 = 118.25 $$

Step 7: Interpretation, total value increased by 18.25%.

Answer: 118.25

Example 4: A hotel tracks price changes for three services. Base prices and quantities: Room service: Ksh 2000 (100 services), Laundry: Ksh 500 (150 services), Food: Ksh 1000 (200 services). Current prices: Room service: Ksh 2100, Laundry: Ksh 450, Food: Ksh 1100. Calculate the Laspeyres price index.

Step 1: Purpose, price index.

Step 2: Base period selected.

Step 3: Items, services.

Step 4: Data collected.

Step 5: Formula:

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

Step 6: Numerator:

$$ (2100 \times 100) + (450 \times 150) + (1100 \times 200) = 210000 + 67500 + 220000 = 497500 $$

Denominator:

$$ (2000 \times 100) + (500 \times 150) + (1000 \times 200) = 200000 + 75000 + 200000 = 475000 $$

Compute index:

$$ I_L = \frac{497500}{475000} \times 100 = 104.74 $$

Step 7: Interpretation, prices increased by 4.74%.

Answer: 104.74

Example 5: Using Example 4 data, current quantities are Room service: 110, Laundry: 140, Food: 210. Calculate Paasche price index.

Step 1: Purpose, price index.

Step 2: Base period selected.

Step 3: Items, services.

Step 4: Data gathered.

Step 5: Formula:

$$ I_P = \frac{\sum p_t q_t}{\sum p_0 q_t} \times 100 $$

Step 6: Numerator:

$$ (2100 \times 110) + (450 \times 140) + (1100 \times 210) = 231000 + 63000 + 231000 = 525000 $$

Denominator:

$$ (2000 \times 110) + (500 \times 140) + (1000 \times 210) = 220000 + 70000 + 210000 = 500000 $$

Compute index:

$$ I_P = \frac{525000}{500000} \times 100 = 105.00 $$

Step 7: Interpretation, prices increased by 5%.

Answer: 105.00

Chapter Summary

This chapter explored the construction of index numbers, explaining how they are formulated to measure changes in economic variables over time. It highlighted the purpose of index numbers in tracking price movements and economic trends. The chapter detailed simple index numbers using both fixed base and chain base methods, providing foundational techniques for analysis. The Consumer Price Index was examined as a key example of an index number used to monitor inflation. Further, weighted index numbers were introduced, with emphasis on Laspeyre's, Paasche's, Fisher's ideal, and Marshall-Edgeworth's methods for both price and quantity indexes. The chapter also discussed the limitations of index numbers, including issues related to accuracy and representativeness. Finally, emerging issues and trends in the use of index numbers were considered, reflecting ongoing developments in economic measurement.