By the end of this chapter, you will be able to:
Mastering these skills will help you analyze data clearly and make informed choices that benefit your workplace and career.
Index numbers are powerful quantitative tools that help procurement professionals measure changes in prices, quantities, or values over time. In Kenyan procurement settings, understanding and applying index numbers enables effective cost control, budget adjustments, and supplier performance evaluation. This section focuses on how index numbers guide decision making by quantifying trends and variations in procurement data.
Price index numbers track changes in the prices of goods and services over a period, helping procurement managers adjust budgets to reflect inflation or deflation. The general formula for a price index number is:
$$ \text{Price Index Number} = \frac{\text{Price in current period}}{\text{Price in base period}} \times 100 $$
Example 1: A county government procures cement where the price per bag was Ksh 500 last year and Ksh 550 this year. Calculate the price index number.
Given:
Price in base period = Ksh 500
Price in current period = Ksh 550
$$ \text{Price Index} = \frac{550}{500} \times 100 $$
$$ = 1.1 \times 100 $$
$$ = 110 $$
Answer: 110 (Prices increased by 10%)
Example 2: A hospital bought medical gloves at Ksh 1,200 per box last year. This year, the price dropped to Ksh 1,080. Find the price index number.
Given:
Price in base period = Ksh 1,200
Price in current period = Ksh 1,080
$$ \text{Price Index} = \frac{1080}{1200} \times 100 $$
$$ = 0.9 \times 100 $$
$$ = 90 $$
Answer: 90 (Prices decreased by 10%)
Example 3: A school procures textbooks. In 2019, the average price was Ksh 800 per book; in 2022, it rose to Ksh 1,000. Calculate the price index number.
Given:
Price in base period = Ksh 800
Price in current period = Ksh 1,000
$$ \text{Price Index} = \frac{1000}{800} \times 100 $$
$$ = 1.25 \times 100 $$
$$ = 125 $$
Answer: 125 (Prices increased by 25%)
Example 4: A retail business sees the price of sugar increase from Ksh 90 per kg to Ksh 99 over a year. Calculate the price index.
Given:
Price in base period = Ksh 90
Price in current period = Ksh 99
$$ \text{Price Index} = \frac{99}{90} \times 100 $$
$$ = 1.1 \times 100 $$
$$ = 110 $$
Answer: 110 (10% price increase)
Example 5: A SACCO procures office stationery. The price of paper was Ksh 300 per ream last year and is now Ksh 315. Calculate the price index.
Given:
Price in base period = Ksh 300
Price in current period = Ksh 315
$$ \text{Price Index} = \frac{315}{300} \times 100 $$
$$ = 1.05 \times 100 $$
$$ = 105 $$
Answer: 105 (5% increase)
Quantity index numbers measure changes in quantities purchased or consumed over time, assisting procurement managers in demand forecasting and stock management. The formula is:
$$ \text{Quantity Index Number} = \frac{\text{Quantity in current period}}{\text{Quantity in base period}} \times 100 $$
Example 1: A cooperative bought 1,000 bags of maize last year and 1,200 bags this year. Calculate the quantity index.
Given:
Quantity in base period = 1,000 bags
Quantity in current period = 1,200 bags
$$ \text{Quantity Index} = \frac{1200}{1000} \times 100 $$
$$ = 1.2 \times 100 $$
$$ = 120 $$
Answer: 120 (Quantity increased by 20%)
Example 2: A hotel used 500 litres of cooking oil last quarter and 450 litres this quarter. Calculate the quantity index.
Given:
Quantity in base period = 500 litres
Quantity in current period = 450 litres
$$ \text{Quantity Index} = \frac{450}{500} \times 100 $$
$$ = 0.9 \times 100 $$
$$ = 90 $$
Answer: 90 (Quantity decreased by 10%)
Example 3: A university procured 300 laptops last year and 360 this year. Calculate the quantity index.
Given:
Quantity in base period = 300 laptops
Quantity in current period = 360 laptops
$$ \text{Quantity Index} = \frac{360}{300} \times 100 $$
$$ = 1.2 \times 100 $$
$$ = 120 $$
Answer: 120 (Quantity increased by 20%)
Example 4: A retail store sold 2,000 units of a product last month and 1,800 units this month. Calculate the quantity index.
Given:
Quantity in base period = 2,000 units
Quantity in current period = 1,800 units
$$ \text{Quantity Index} = \frac{1800}{2000} \times 100 $$
$$ = 0.9 \times 100 $$
$$ = 90 $$
Answer: 90 (Quantity decreased by 10%)
Example 5: A county government office ordered 5,000 pens last year and 6,250 pens this year. Calculate the quantity index.
Given:
Quantity in base period = 5,000 pens
Quantity in current period = 6,250 pens
$$ \text{Quantity Index} = \frac{6250}{5000} \times 100 $$
$$ = 1.25 \times 100 $$
$$ = 125 $$
Answer: 125 (Quantity increased by 25%)
Value index numbers combine price and quantity changes to measure the overall value change of procurement items over time. This helps procurement managers assess budget variances and supplier cost impacts. The formula is:
$$ \text{Value Index Number} = \frac{\text{Value in current period}}{\text{Value in base period}} \times 100 $$
Where value = price × quantity.
Example 1: A retail store bought 1,000 units at Ksh 50 each last year and 1,100 units at Ksh 55 each this year. Calculate the value index.
Given:
Value in base period = 1,000 × 50 = Ksh 50,000
Value in current period = 1,100 × 55 = Ksh 60,500
$$ \text{Value Index} = \frac{60500}{50000} \times 100 $$
$$ = 1.21 \times 100 $$
$$ = 121 $$
Answer: 121 (Value increased by 21%)
Example 2: A hospital procured 200 units of a drug at Ksh 1,000 each last year and 180 units at Ksh 1,100 each this year. Calculate the value index.
Given:
Value in base period = 200 × 1,000 = Ksh 200,000
Value in current period = 180 × 1,100 = Ksh 198,000
$$ \text{Value Index} = \frac{198000}{200000} \times 100 $$
$$ = 0.99 \times 100 $$
$$ = 99 $$
Answer: 99 (Value decreased by 1%)
Example 3: A SACCO ordered 500 office chairs at Ksh 4,000 each last year and 600 chairs at Ksh 3,800 each this year. Calculate the value index.
Given:
Value in base period = 500 × 4,000 = Ksh 2,000,000
Value in current period = 600 × 3,800 = Ksh 2,280,000
$$ \text{Value Index} = \frac{2280000}{2000000} \times 100 $$
$$ = 1.14 \times 100 $$
$$ = 114 $$
Answer: 114 (Value increased by 14%)
Example 4: A county government bought 1,500 laptops at Ksh 40,000 each last year and 1,400 laptops at Ksh 42,000 each this year. Calculate the value index.
Given:
Value in base period = 1,500 × 40,000 = Ksh 60,000,000
Value in current period = 1,400 × 42,000 = Ksh 58,800,000
$$ \text{Value Index} = \frac{58800000}{60000000} \times 100 $$
$$ = 0.98 \times 100 $$
$$ = 98 $$
Answer: 98 (Value decreased by 2%)
Example 5: A university procured 300 projectors at Ksh 80,000 each last year and 350 projectors at Ksh 85,000 each this year. Calculate the value index.
Given:
Value in base period = 300 × 80,000 = Ksh 24,000,000
Value in current period = 350 × 85,000 = Ksh 29,750,000
$$ \text{Value Index} = \frac{29750000}{24000000} \times 100 $$
$$ = 1.2396 \times 100 $$
$$ = 123.96 $$
Answer: 123.96 (Value increased by approximately 24%)
Index numbers can assess supplier price stability and delivery quantity consistency over contract periods. Procurement managers use these indices to make informed decisions on contract renewals and negotiations.
Example 1: A supplier delivered 1,000 units at Ksh 500 last year and 1,200 units at Ksh 520 this year. Calculate price, quantity, and value indices to evaluate supplier performance.
Given:
Price base = Ksh 500, price current = Ksh 520
Quantity base = 1,000 units, quantity current = 1,200 units
Value base = 1,000 × 500 = Ksh 500,000
Value current = 1,200 × 520 = Ksh 624,000
Price index:$$ \frac{520}{500} \times 100 = 104 $$
Quantity index:$$ \frac{1200}{1000} \times 100 = 120 $$
Value index:$$ \frac{624000}{500000} \times 100 = 124.8 $$
Answer: Price index = 104 (4% increase), Quantity index = 120 (20% increase), Value index = 124.8 (24.8% increase)
Example 2: A supplier provided 800 units at Ksh 1,000 last year but delivered 700 units at Ksh 1,050 this year. Calculate indices.
Given:
Price base = Ksh 1,000, price current = Ksh 1,050
Quantity base = 800 units, quantity current = 700 units
Value base = 800 × 1,000 = Ksh 800,000
Value current = 700 × 1,050 = Ksh 735,000
Price index:$$ \frac{1050}{1000} \times 100 = 105 $$
Quantity index:$$ \frac{700}{800} \times 100 = 87.5 $$
Value index:$$ \frac{735000}{800000} \times 100 = 91.875 $$
Answer: Price index = 105 (5% increase), Quantity index = 87.5 (12.5% decrease), Value index = 91.875 (8.125% decrease)
Example 3: A supplier's price for steel bars was Ksh 2,000 per ton last year and Ksh 1,900 this year with quantities of 1,500 tons and 1,600 tons respectively. Calculate indices.
Given:
Price base = Ksh 2,000, price current = Ksh 1,900
Quantity base = 1,500 tons, quantity current = 1,600 tons
Value base = 1,500 × 2,000 = Ksh 3,000,000
Value current = 1,600 × 1,900 = Ksh 3,040,000
Price index:$$ \frac{1900}{2000} \times 100 = 95 $$
Quantity index:$$ \frac{1600}{1500} \times 100 = 106.67 $$
Value index:$$ \frac{3040000}{3000000} \times 100 = 101.33 $$
Answer: Price index = 95 (5% decrease), Quantity index = 106.67 (6.67% increase), Value index = 101.33 (1.33% increase)
Example 4: A supplier's price increased from Ksh 150 to Ksh 165 per unit while the quantity supplied decreased from 2,000 to 1,800 units. Calculate indices.
Given:
Price base = Ksh 150, price current = Ksh 165
Quantity base = 2,000 units, quantity current = 1,800 units
Value base = 2,000 × 150 = Ksh 300,000
Value current = 1,800 × 165 = Ksh 297,000
Price index:$$ \frac{165}{150} \times 100 = 110 $$
Quantity index:$$ \frac{1800}{2000} \times 100 = 90 $$
Value index:$$ \frac{297000}{300000} \times 100 = 99 $$
Answer: Price index = 110 (10% increase), Quantity index = 90 (10% decrease), Value index = 99 (1% decrease)
Example 5: A supplier delivered 4,000 units at Ksh 75 each last year and 4,500 units at Ksh 70 this year. Calculate indices.
Given:
Price base = Ksh 75, price current = Ksh 70
Quantity base = 4,000 units, quantity current = 4,500 units
Value base = 4,000 × 75 = Ksh 300,000
Value current = 4,500 × 70 = Ksh 315,000
Price index:$$ \frac{70}{75} \times 100 = 93.33 $$
Quantity index:$$ \frac{4500}{4000} \times 100 = 112.5 $$
Value index:$$ \frac{315000}{300000} \times 100 = 105 $$
Answer: Price index = 93.33 (6.67% decrease), Quantity index = 112.5 (12.5% increase), Value index = 105 (5% increase)
A county government procures 2,000 chairs at Ksh 1,500 each last year and 2,200 chairs at Ksh 1,650 each this year. Calculate the price index, quantity index, and value index. (6 marks)
A hospital bought 1,500 boxes of gloves at Ksh 800 each last year and 1,350 boxes at Ksh 880 each this year. Calculate the price, quantity, and value indices. (6 marks)
A retail store's price for sugar increased from Ksh 90 to Ksh 95 per kg while quantity sold dropped from 3,000 kg to 2,700 kg. Calculate the price, quantity, and value indices. (6 marks)
A university purchased 500 laptops at Ksh 60,000 each last year and 550 laptops at Ksh 58,000 each this year. Calculate the price, quantity, and value indices. (6 marks)
A SACCO procured 1,000 computers at Ksh 40,000 each last year and 950 computers at Ksh 42,000 each this year. Calculate the price, quantity, and value indices. (6 marks)
This chapter introduced the fundamental formulae used to compute index numbers, providing a mathematical basis for their calculation. It then detailed the computation methods for various index numbers, starting with Laspeyre's index which uses base period quantities as weights. Paasche's index was explained next, highlighting the use of current period quantities for weighting. The chapter also covered Fisher's ideal index, which combines both Laspeyre's and Paasche's indices to provide a more accurate measure. Additionally, the Marshal reports method was discussed as another approach to index number calculation. Finally, the chapter explored how index numbers are applied in decision making, emphasizing their role in analyzing economic and financial data for informed choices. Throughout, the focus was on practical computation and interpretation within real-world contexts.
Example 1: Laspeyres Price Index
A procurement manager tracks the price changes of two items:
| Item | Price in Base Year (Ksh) | Price in Current Year (Ksh) | Quantity in Base Year |
|---|---|---|---|
| A | 100 | 120 | 50 |
| B | 200 | 180 | 30 |
Laspeyres Price Index formula:$$ I_L = \frac{\sum P_1 Q_0}{\sum P_0 Q_0} \times 100 $$
Calculate numerator:$$ \sum P_1 Q_0 = (120 \times 50) + (180 \times 30) = 6000 + 5400 = 11400 $$
Calculate denominator:$$ \sum P_0 Q_0 = (100 \times 50) + (200 \times 30) = 5000 + 6000 = 11000 $$
Calculate index:$$ I_L = \frac{11400}{11000} \times 100 = 103.64 $$
Answer: 103.64 (Prices increased by 3.64%)
Example 2: Paasche Price Index
Suppose current year quantities are:
| Item | Price in Base Year (Ksh) | Price in Current Year (Ksh) | Quantity in Current Year |
|---|---|---|---|
| A | 100 | 120 | 60 |
| B | 200 | 180 | 25 |
Paasche Price Index formula:$$ I_P = \frac{\sum P_1 Q_1}{\sum P_0 Q_1} \times 100 $$
Calculate numerator:$$ \sum P_1 Q_1 = (120 \times 60) + (180 \times 25) = 7200 + 4500 = 11700 $$
Calculate denominator:$$ \sum P_0 Q_1 = (100 \times 60) + (200 \times 25) = 6000 + 5000 = 11000 $$
Calculate index:$$ I_P = \frac{11700}{11000} \times 100 = 106.36 $$
Answer: 106.36 (Prices increased by 6.36%)
Example 3: Fisher's Ideal Index
Given Laspeyres Index \(I_L = 103.64\) and Paasche Index \(I_P = 106.36\):
Fisher's Ideal Index formula:$$ I_F = \sqrt{I_L \times I_P} $$
Calculate:$$ I_F = \sqrt{103.64 \times 106.36} = \sqrt{11027.02} = 105.03 $$
Answer: 105.03 (Average price increase of 5.03%)
Example 4: Marshal-Edgeworth Price Index
Suppose total expenditure in base year is Ksh 50,000 and in current year is Ksh 55,000, and quantities are equal.
Marshal-Edgeworth formula:$$ I_{ME} = \frac{E_1 + E_0}{2 E_0} \times 100 $$
Calculate:$$ I_{ME} = \frac{55,000 + 50,000}{2 \times 50,000} \times 100 = \frac{105,000}{100,000} \times 100 = 105 $$
Answer: 105 (Prices increased by 5%)
A procurement officer recorded the price of a commodity as Ksh 120 in the base year and Ksh 150 in the current year. Calculate the Laspeyre’s Price Index. (2 marks)
The quantity of an item purchased in the base year was 200 units at Ksh 50 per unit, and in the current year, 180 units at Ksh 60 per unit. Calculate the Paasche’s Price Index. (3 marks)
Given the following data for an item: Base year price Ksh 80, current year price Ksh 100, base year quantity 150 units, current year quantity 120 units. Compute the Fisher’s Ideal Price Index. (4 marks)
A procurement manager wants to calculate the Laspeyre’s Quantity Index. The base year quantity is 500 units, current year quantity is 600 units, base year price Ksh 40, and current year price Ksh 50. Calculate the index. (3 marks)
For two commodities, the following data is available:
| Commodity | Base Year Price (Ksh) | Current Year Price (Ksh) | Base Year Quantity | Current Year Quantity |
|---|---|---|---|---|
| A | 100 | 120 | 300 | 250 |
| B | 200 | 220 | 150 | 180 |
Calculate the Paasche’s Price Index for the combined commodities. (5 marks)
A procurement officer uses data to compute the Marshal-Edgeworth Price Index. The total expenditure in the base year is Ksh 50,000 and in the current year Ksh 55,000. Given that the base year quantity and current year quantity are equal, calculate the index. (3 marks)
A procurement department noted the following prices and quantities for three items:
| Item | Base Year Price (Ksh) | Current Year Price (Ksh) | Base Year Quantity | Current Year Quantity |
|---|---|---|---|---|
| 1 | 60 | 75 | 400 | 350 |
| 2 | 100 | 110 | 300 | 320 |
| 3 | 80 | 90 | 200 | 210 |
Calculate the Fisher’s Ideal Quantity Index. (6 marks)
A procurement manager wants to understand the effect of price changes on total expenditure. The base year total expenditure was Ksh 1,200,000 and the current year total expenditure is Ksh 1,380,000. Calculate the Laspeyre’s Price Index given base year quantities and prices. (4 marks)
Using the following data, compute the Paasche’s Quantity Index:
| Commodity | Base Year Price (Ksh) | Current Year Price (Ksh) | Base Year Quantity | Current Year Quantity |
|---|---|---|---|---|
| X | 150 | 160 | 500 | 550 |
| Y | 100 | 110 | 400 | 390 |
(5 marks)
| Item | Base Year Price (Ksh) | Base Year Quantity |
|---|---|---|
| A | 200 | 300 |
| B | 150 | 400 |
And current year prices and quantities:
| Item | Current Year Price (Ksh) | Current Year Quantity |
|---|---|---|
| A | 220 | 320 |
| B | 180 | 390 |
Calculate the Fisher’s Ideal Price Index and interpret the result. (7 marks)
Given:
Base year price \(P_0 = 120\), Current year price \(P_1 = 150\)
Formula:$$I_L = \frac{P_1}{P_0} \times 100$$
Calculation:$$I_L = \frac{150}{120} \times 100 = 125$$
Answer: The Laspeyre’s Price Index is 125.
Given:
Base year quantity \(Q_0 = 200\), Base year price \(P_0 = 50\)
Current year quantity \(Q_1 = 180\), Current year price \(P_1 = 60\)
Formula:$$I_P = \frac{\sum P_1 Q_1}{\sum P_0 Q_1} \times 100$$
Calculate numerator:$$P_1 Q_1 = 60 \times 180 = 10,800$$
Calculate denominator:$$P_0 Q_1 = 50 \times 180 = 9,000$$
Calculate index:$$I_P = \frac{10,800}{9,000} \times 100 = 120$$
Answer: The Paasche’s Price Index is 120.
Given:
\(P_0 = 80\), \(P_1 = 100\), \(Q_0 = 150\), \(Q_1 = 120\)
Calculate Laspeyre’s Price Index:$$I_L = \frac{P_1 Q_0}{P_0 Q_0} \times 100 = \frac{100 \times 150}{80 \times 150} \times 100 = \frac{15,000}{12,000} \times 100 = 125$$
Calculate Paasche’s Price Index:$$I_P = \frac{P_1 Q_1}{P_0 Q_1} \times 100 = \frac{100 \times 120}{80 \times 120} \times 100 = \frac{12,000}{9,600} \times 100 = 125$$
Fisher’s Ideal Index:$$I_F = \sqrt{I_L \times I_P} = \sqrt{125 \times 125} = 125$$
Answer: The Fisher’s Ideal Price Index is 125.
Given:
\(Q_0 = 500\), \(Q_1 = 600\), \(P_0 = 40\), \(P_1 = 50\)
Formula:$$I_{QL} = \frac{\sum Q_1 P_0}{\sum Q_0 P_0} \times 100 = \frac{Q_1}{Q_0} \times 100$$
Calculate index:$$I_{QL} = \frac{600}{500} \times 100 = 120$$
Answer: The Laspeyre’s Quantity Index is 120.
Given:
| Commodity | \(P_0\) | \(P_1\) | \(Q_0\) | \(Q_1\) |
|---|---|---|---|---|
| A | 100 | 120 | 300 | 250 |
| B | 200 | 220 | 150 | 180 |
Calculate numerator \(\sum P_1 Q_1\):
For A: \(120 \times 250 = 30,000\)
For B: \(220 \times 180 = 39,600\)
Total numerator: \(30,000 + 39,600 = 69,600\)
Calculate denominator \(\sum P_0 Q_1\):
For A: \(100 \times 250 = 25,000\)
For B: \(200 \times 180 = 36,000\)
Total denominator: \(25,000 + 36,000 = 61,000\)
Compute index:$$I_P = \frac{69,600}{61,000} \times 100 = 114.1$$
Answer: The Paasche’s Price Index is approximately 114.1.
Given:
Base year total expenditure \(E_0 = 50,000\)
Current year total expenditure \(E_1 = 55,000\)
Since quantities are equal, the index is:$$I_{ME} = \frac{E_1 + E_0}{2 E_0} \times 100 = \frac{55,000 + 50,000}{2 \times 50,000} \times 100 = \frac{105,000}{100,000} \times 100 = 105$$
Answer: The Marshal-Edgeworth Price Index is 105.
Given:
| Item | \(P_0\) | \(P_1\) | \(Q_0\) | \(Q_1\) |
|---|---|---|---|---|
| 1 | 60 | 75 | 400 | 350 |
| 2 | 100 | 110 | 300 | 320 |
| 3 | 80 | 90 | 200 | 210 |
Calculate Laspeyre’s Quantity Index:
Numerator:$$\sum Q_1 P_0 = (350 \times 60) + (320 \times 100) + (210 \times 80) = 21,000 + 32,000 + 16,800 = 69,800$$
Denominator:$$\sum Q_0 P_0 = (400 \times 60) + (300 \times 100) + (200 \times 80) = 24,000 + 30,000 + 16,000 = 70,000$$
Laspeyre’s Quantity Index:$$I_{QL} = \frac{69,800}{70,000} \times 100 = 99.7$$
Calculate Paasche’s Quantity Index:
Numerator:$$\sum Q_1 P_1 = (350 \times 75) + (320 \times 110) + (210 \times 90) = 26,250 + 35,200 + 18,900 = 80,350$$
Denominator:$$\sum Q_0 P_1 = (400 \times 75) + (300 \times 110) + (200 \times 90) = 30,000 + 33,000 + 18,000 = 81,000$$
Paasche’s Quantity Index:$$I_{QP} = \frac{80,350}{81,000} \times 100 = 99.2$$
Fisher’s Ideal Quantity Index:$$I_{QF} = \sqrt{I_{QL} \times I_{QP}} = \sqrt{99.7 \times 99.2} = \sqrt{9890.24} = 99.45$$
Answer: The Fisher’s Ideal Quantity Index is approximately 99.45.
Given:
Base year total expenditure \(E_0 = 1,200,000\)
Current year total expenditure \(E_1 = 1,380,000\)
If quantities remain constant,$$I_L = \frac{E_1}{E_0} \times 100 = \frac{1,380,000}{1,200,000} \times 100 = 115$$
Answer: The Laspeyre’s Price Index is 115.
Given:
| Commodity | \(P_0\) | \(P_1\) | \(Q_0\) | \(Q_1\) |
|---|---|---|---|---|
| X | 150 | 160 | 500 | 550 |
| Y | 100 | 110 | 400 | 390 |
Calculate numerator \(\sum Q_1 P_1\):
For X: \(550 \times 160 = 88,000\)
For Y: \(390 \times 110 = 42,900\)
Total numerator: \(88,000 + 42,900 = 130,900\)
Calculate denominator \(\sum Q_0 P_1\):
For X: \(500 \times 160 = 80,000\)
For Y: \(400 \times 110 = 44,000\)
Total denominator: \(80,000 + 44,000 = 124,000\)
Calculate index:$$I_{QP} = \frac{130,900}{124,000} \times 100 = 105.56$$
Answer: The Paasche’s Quantity Index is approximately 105.56.
Given:
| Item | \(P_0\) | \(Q_0\) | \(P_1\) | \(Q_1\) |
|---|---|---|---|---|
| A | 200 | 300 | 220 | 320 |
| B | 150 | 400 | 180 | 390 |
Calculate Laspeyre’s Price Index:
Numerator:$$\sum P_1 Q_0 = (220 \times 300) + (180 \times 400) = 66,000 + 72,000 = 138,000$$
Denominator:$$\sum P_0 Q_0 = (200 \times 300) + (150 \times 400) = 60,000 + 60,000 = 120,000$$
Laspeyre’s Price Index:$$I_L = \frac{138,000}{120,000} \times 100 = 115$$
Calculate Paasche’s Price Index:
Numerator:$$\sum P_1 Q_1 = (220 \times 320) + (180 \times 390) = 70,400 + 70,200 = 140,600$$
Denominator:$$\sum P_0 Q_1 = (200 \times 320) + (150 \times 390) = 64,000 + 58,500 = 122,500$$
Paasche’s Price Index:$$I_P = \frac{140,600}{122,500} \times 100 = 114.78$$
Calculate Fisher’s Ideal Price Index:$$I_F = \sqrt{I_L \times I_P} = \sqrt{115 \times 114.78} = \sqrt{13299.7} = 115.35$$
Interpretation: The Fisher’s Ideal Price Index of approximately 115.35 indicates that, on average, prices have increased by about 15.35% from the base year to the current year.
Answer: The Fisher’s Ideal Price Index is approximately 115.35.
(4 marks)
7. Define an index number and explain its role in procurement management price monitoring. (4 marks)
8. Calculate the Laspeyres quantity index for the following data: Base year prices: Item X Ksh 300, Item Y Ksh 150; Base year quantities: X 80, Y 60; Current year quantities: X 100, Y 50. (4 marks)
9. How can index numbers assist a county government procurement officer in budget forecasting? (4 marks)
10. A procurement analyst observes that the Paasche price index is lower than the Laspeyres price index for a commodity basket. What does this indicate about quantity changes and price trends? (4 marks)
Question 11 (Compulsory - 20 marks)
The Kenya Medical Supplies Authority (KEMSA) wants to analyze the price changes of essential drugs over the last year to adjust their procurement budget. They have the following data:
| Drug | Base Year Price (Ksh) | Base Year Quantity | Current Year Price (Ksh) | Current Year Quantity |
|---|---|---|---|---|
| Paracetamol | 150 | 1000 | 165 | 1100 |
| Amoxicillin | 200 | 800 | 190 | 900 |
| Cough Syrup | 120 | 600 | 130 | 550 |
a) Compute the Laspeyres price index and interpret the result. (10 marks)
b) Compute the Fisher’s ideal price index and explain why it might be preferred for KEMSA’s decision making. (10 marks)
Question 12 (20 marks)
A procurement officer at a county government is tracking the price and quantity changes of stationery over two years. Using the following data, calculate the Paasche quantity index and discuss its usefulness in procurement planning.
| Item | Base Year Price (Ksh) | Base Year Quantity | Current Year Price (Ksh) | Current Year Quantity |
|---|---|---|---|---|
| Notebooks | 80 | 500 | 85 | 600 |
| Pens | 40 | 1000 | 42 | 950 |
| Folders | 100 | 300 | 110 | 280 |
Question 13 (20 marks)
Explain how index numbers can be applied in supplier price comparison and contract evaluation in procurement management. Support your explanation with a numerical example using Laspeyres price index for three suppliers offering different prices and quantities of a commodity.
Question 14 (20 marks)
A procurement analyst at a manufacturing firm uses Marshal’s report to analyze price and quantity changes for raw materials. Given the following data, prepare a Marshal’s report and interpret the findings for procurement decision making:
| Material | Base Year Price (Ksh) | Base Year Quantity | Current Year Price (Ksh) | Current Year Quantity |
|---|---|---|---|---|
| Steel | 500 | 200 | 550 | 180 |
| Cement | 400 | 300 | 420 | 320 |
Question 11
a) Laspeyres price index \(I_L = \frac{\sum P_1 Q_0}{\sum P_0 Q_0} \times 100\)
\(\sum P_1 Q_0 = (165 \times 1000) + (190 \times 800) + (130 \times 600) = 165000 + 152000 + 78000 = 395000\)
\(\sum P_0 Q_0 = (150 \times 1000) + (200 \times 800) + (120 \times 600) = 150000 + 160000 + 72000 = 382000\)
\(I_L = \frac{395000}{382000} \times 100 = 103.40\)
Interpretation: Prices increased by 3.40% compared to the base year.
b) Compute Paasche price index \(I_P = \frac{\sum P_1 Q_1}{\sum P_0 Q_1} \times 100\)
\(\sum P_1 Q_1 = (165 \times 1100) + (190 \times 900) + (130 \times 550) = 181500 + 171000 + 71500 = 424000\)
\(\sum P_0 Q_1 = (150 \times 1100) + (200 \times 900) + (120 \times 550) = 165000 + 180000 + 66000 = 411000\)
\(I_P = \frac{424000}{411000} \times 100 = 103.17\)
Fisher’s ideal index \(I_F = \sqrt{I_L \times I_P} = \sqrt{103.40 \times 103.17} = \sqrt{10667.98} = 103.31\)
Preferred because it balances the base and current year quantities weighting, providing a more accurate price change measure for budgeting.
Question 12
Paasche quantity index \(I_{PQ} = \frac{\sum P_1 Q_1}{\sum P_1 Q_0} \times 100\)
\(\sum P_1 Q_1 = (85 \times 600) + (42 \times 950) + (110 \times 280) = 51000 + 39900 + 30800 = 121700\)
\(\sum P_1 Q_0 = (85 \times 500) + (42 \times 1000) + (110 \times 300) = 42500 + 42000 + 33000 = 117500\)
\(I_{PQ} = \frac{121700}{117500} \times 100 = 103.62\)
Usefulness: Shows increase in quantities purchased weighted by current prices, helping plan procurement volumes and costs in response to demand changes.
Question 13
Index numbers enable comparison of supplier prices accounting for quantities offered. For example:
| Supplier | Price per Unit (Ksh) | Quantity |
|---|---|---|
| A | 150 | 100 |
| B | 140 | 120 |
| C | 160 | 90 |
Using Supplier A as base:
Laspeyres price index for B:
\(I_L = \frac{140 \times 100}{150 \times 100} \times 100 = 93.33\)
For C:
\(I_L = \frac{160 \times 100}{150 \times 100} \times 100 = 106.67\)
Interpretation: Supplier B offers 6.67% lower price than A, C offers 6.67% higher. Procurement can evaluate contracts based on cost-effectiveness considering quantities required.
Question 14
Marshal’s report separates price and quantity effects:
Price effect \(= \sum (P_1 - P_0) Q_0\)
Quantity effect \(= \sum P_0 (Q_1 - Q_0)\)
Price effect:
Steel: \((550 - 500) \times 200 = 50 \times 200 = 10000\)
Cement: \((420 - 400) \times 300 = 20 \times 300 = 6000\)
Total price effect = 16000
Quantity effect:
Steel: \(500 \times (180 - 200) = 500 \times (-20) = -10000\)
Cement: \(400 \times (320 - 300) = 400 \times 20 = 8000\)
Total quantity effect = -2000
Interpretation: Price increases added Ksh 16,000 to procurement cost; quantity changes reduced cost by Ksh 2,000. Procurement decisions can focus on negotiating better prices or adjusting quantities to optimize costs.
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