Index numbers are crucial tools for cooperative managers in Kenya to measure changes in economic variables such as prices, quantities, and costs over time. They help monitor inflation, cost of living adjustments, and productivity trends which directly affect cooperative operations and member welfare. This chapter focuses on the formulae and computation methods of index numbers, equipping cooperative professionals with quantitative skills to analyze economic data effectively and make informed decisions.

8.1 Formulae for computing index numbers

Index numbers summarize changes in a variable or group of variables relative to a base period or base group. The formulas differ based on the type of index number and the weighting method used. Understanding these formulas enables cooperative managers to select the appropriate index number for price analysis, cost control, or performance measurement.

8.1.1 Laspeyres’ Price Index Formula

The Laspeyres’ index uses base period quantities as weights, measuring how much the cost of purchasing the base period basket of goods changes over time. It is expressed as:

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

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

Worked Examples

Example 1: A cooperative sells maize. Base year price \(p_0 = 40\) Ksh/kg, quantity \(q_0 = 1000\) kg. Current year price \(p_t = 50\) Ksh/kg. Calculate Laspeyres’ Price Index.

Given: \(p_0 = 40\), \(q_0 = 1000\), \(p_t = 50\)

$$ LPI = \frac{p_t \times q_0}{p_0 \times q_0} \times 100 $$

$$ = \frac{50 \times 1000}{40 \times 1000} \times 100 $$

$$ = \frac{50,000}{40,000} \times 100 = 1.25 \times 100 $$

Answer: 125 (Prices increased by 25% compared to base period)

Example 2: For two products in a cooperative store:

Calculate Laspeyres’ Price Index.

Given:

\[ \sum p_t q_0 = (100 \times 200) + (180 \times 100) = 20,000 + 18,000 = 38,000 \]

\[ \sum p_0 q_0 = (80 \times 200) + (150 \times 100) = 16,000 + 15,000 = 31,000 \]

$$ LPI = \frac{38,000}{31,000} \times 100 = 1.2258 \times 100 $$

Answer: 122.58 (Prices rose by 22.58%)

Example 3: A cooperative’s base period basket has three products with prices and quantities:

Calculate Laspeyres’ Price Index.

Given:

\[ \sum p_t q_0 = (72 \times 150) + (110 \times 120) + (55 \times 200) = 10,800 + 13,200 + 11,000 = 35,000 \]

\[ \sum p_0 q_0 = (60 \times 150) + (100 \times 120) + (50 \times 200) = 9,000 + 12,000 + 10,000 = 31,000 \]

$$ LPI = \frac{35,000}{31,000} \times 100 = 1.129 \times 100 $$

Answer: 112.9 (Price increased by 12.9%)

8.1.2 Paasche’s Price Index Formula

Paasche’s index uses current period quantities as weights, showing the cost change of buying the current basket of goods at base and current period prices. The formula is:

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

where \(p_t\) and \(q_t\) are current period price and quantity, and \(p_0\) is base period price.

Worked Examples

Example 1: A SACCO tracks two products:

Calculate Paasche’s Price Index.

Given:

\[ \sum p_t q_t = (90 \times 150) + (55 \times 200) = 13,500 + 11,000 = 24,500 \]

\[ \sum p_0 q_t = (80 \times 150) + (50 \times 200) = 12,000 + 10,000 = 22,000 \]

$$ PPI = \frac{24,500}{22,000} \times 100 = 1.1136 \times 100 $$

Answer: 111.36 (Prices increased by 11.36%)

Example 2: For three products in a cooperative shop:

Calculate Paasche’s Price Index.

Given:

\[ \sum p_t q_t = (95 \times 100) + (160 \times 80) + (130 \times 60) = 9,500 + 12,800 + 7,800 = 30,100 \]

\[ \sum p_0 q_t = (90 \times 100) + (150 \times 80) + (120 \times 60) = 9,000 + 12,000 + 7,200 = 28,200 \]

$$ PPI = \frac{30,100}{28,200} \times 100 = 1.0677 \times 100 $$

Answer: 106.77 (Price increase of 6.77%)

Example 3: A cooperative’s current quantities and prices for maize and beans:

Calculate Paasche’s Price Index.

Given:

\[ \sum p_t q_t = (55 \times 300) + (90 \times 150) = 16,500 + 13,500 = 30,000 \]

\[ \sum p_0 q_t = (50 \times 300) + (80 \times 150) = 15,000 + 12,000 = 27,000 \]

$$ PPI = \frac{30,000}{27,000} \times 100 = 1.1111 \times 100 $$

Answer: 111.11 (Prices increased by 11.11%)

8.1.3 Fisher’s Ideal Price Index Formula

Fisher’s index is the geometric mean of Laspeyres’ and Paasche’s indices, providing a balanced measure that compensates for their biases. It is expressed as:

$$ FPI = \sqrt{LPI \times PPI} $$

where \(LPI\) and \(PPI\) are Laspeyres’ and Paasche’s price indices respectively.

Worked Examples

Example 1: Given \(LPI = 125\) and \(PPI = 111.36\), calculate Fisher’s Price Index.

$$ FPI = \sqrt{125 \times 111.36} $$

$$ = \sqrt{13,920} = 118.0 $$

Answer: 118.0 (Balanced price increase of 18%)

Example 2: \(LPI = 122.58\), \(PPI = 106.77\). Find Fisher’s Price Index.

$$ FPI = \sqrt{122.58 \times 106.77} $$

$$ = \sqrt{13,086.8} = 114.4 $$

Answer: 114.4

Example 3: \(LPI = 112.9\), \(PPI = 111.11\). Compute Fisher’s Price Index.

$$ FPI = \sqrt{112.9 \times 111.11} $$

$$ = \sqrt{12,537.5} = 111.9 $$

Answer: 111.9

8.1.4 Marshall-Edgeworth Price Index Formula

Marshall-Edgeworth index uses average quantities of base and current periods as weights, aiming to reduce bias by averaging quantities. The formula is:

$$ MEPI = \frac{\sum p_t \frac{(q_0 + q_t)}{2}}{\sum p_0 \frac{(q_0 + q_t)}{2}} \times 100 $$

where \(q_0\) and \(q_t\) are base and current period quantities.

Worked Examples

Example 1: For a cooperative with one product:

\(p_0 = 40\), \(q_0 = 1000\), \(p_t = 50\), \(q_t = 1200\).

Calculate Marshall-Edgeworth Price Index.

Calculate average quantity:

$$ \frac{q_0 + q_t}{2} = \frac{1000 + 1200}{2} = 1100 $$

Calculate numerator:

$$ p_t \times \text{avg quantity} = 50 \times 1100 = 55,000 $$

Calculate denominator:

$$ p_0 \times \text{avg quantity} = 40 \times 1100 = 44,000 $$

Calculate index:

$$ MEPI = \frac{55,000}{44,000} \times 100 = 1.25 \times 100 = 125 $$

Answer: 125

Example 2: Two products:

Calculate Marshall-Edgeworth Price Index.

Average quantities:

\[ \text{Sugar} = \frac{200 + 220}{2} = 210 \]

\[ \text{Tea} = \frac{100 + 130}{2} = 115 \]

Numerator:

\[ (100 \times 210) + (180 \times 115) = 21,000 + 20,700 = 41,700 \]

Denominator:

\[ (80 \times 210) + (150 \times 115) = 16,800 + 17,250 = 34,050 \]

Calculate index:

$$ MEPI = \frac{41,700}{34,050} \times 100 = 1.225 \times 100 = 122.5 $$

Answer: 122.5

Example 3: Three products:

Average quantities:

\[ \text{Beans} = \frac{150 + 180}{2} = 165 \]

\[ \text{Rice} = \frac{120 + 130}{2} = 125 \]

\[ \text{Maize} = \frac{200 + 210}{2} = 205 \]

Numerator:

\[ (72 \times 165) + (110 \times 125) + (55 \times 205) = 11,880 + 13,750 + 11,275 = 36,905 \]

Denominator:

\[ (60 \times 165) + (100 \times 125) + (50 \times 205) = 9,900 + 12,500 + 10,250 = 32,650 \]

Calculate index:

$$ MEPI = \frac{36,905}{32,650} \times 100 = 1.13 \times 100 = 113 $$

Answer: 113

Chapter Summary

This chapter covered the essential formulae used to compute various types of index numbers, providing a foundation for understanding price and quantity changes over time. It detailed the computation methods for key index numbers including Laspeyre's, which uses base period quantities, Paasche's, which relies on current period quantities, Fisher's ideal index that combines both Laspeyre's and Paasche's to provide a more accurate measure, and Marshal's method. Each computation technique was explained with its formula and significance in measuring economic trends. The chapter concluded by discussing how index numbers are applied in decision making, highlighting their role in analyzing inflation, cost of living, and business performance to guide strategic planning and policy formulation. Understanding these concepts equips learners with tools to interpret economic data effectively.