Index Numbers
Chapter 5: Index Numbers
University of Botswana
2025/2026 Semester 1
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Index Numbers
5.0 Types of Economic Index Numbers
1
Price indices – measures the change in prices paid and prices
received by producers and consumers.
2
Quantity index – measures changes in production.
3
Value index – measures changes in the value of various
commodities.
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Index Numbers
5.1 Price Indices
Index numbers constructed from a single series such as the
price of a single commodity are called simple price or price
relatives.
For multiple variables (e.g., prices of several commodities), we
use composite indices.
We may construct an aggregate index by summing figures for
the non-base period and comparing with the aggregated
figures for the base period.
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Index Numbers
i. Simple Price Relatives
Measures the change in price in the current period compared to
the price in the base period.
Price relative index =
Pn
price in current period
× 100 =
· 100
price in base period
P0
Where P0 is the price in the base period and Pn is the price in the
non-base period.
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Index Numbers
Example 5.1.1: Price Relatives
Example
See column 3 in Table 1 where 1976 is the base year.
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Index Numbers
ii. Aggregate Price Indexes
We can compute the composite index when there are many items.
a) Unweighted Aggregative Price Index (Composite Index):
P Pn
100
Composite Index = P
P0
Example
Consider Table 1 showing five food items consumed by a typical
family in Gaborone. Base year = 1976.
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Index Numbers
Table 1: Unit Prices and Price Relatives
Food Commodity
Meat
Milk
Eggs
Bread
Coffee
Total
1976 (P0 )
2.12
1.65
0.69
0.63
1.75
6.84
1986 (Pn )
2.59
2.00
1.00
0.90
2.34
8.83
Price Relatives (Pn /P0 · 100)
122.17
121.21
144.93
142.86
133.71
664.88
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Index Numbers
Unweighted Aggregative Index
The base index = 100
P 8.83
Pn
P
100 =
100 ≈ 129.1%
Composite price index =
P0
6.84
Interpretation
It would have cost 29.1% more in 1986 to buy the goods in Table
1.
Limitations
Influenced by high-priced commodities
Easily affected by the nature of units of measurement
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Index Numbers
b) Arithmetic Mean of Price Relatives
Based on the above limitations, it is therefore ideal to consider the
weighted aggregate index.
1 X Pn
· 100
Arithmetic mean of price relatives =
t
P0
Example
Using Table 1:
1
Arithmetic mean of price relatives = (664.88) = 132.98
5
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Index Numbers
c) Weighted Composite Aggregative Index
Weights are assigned to each item to reflect the quantity of
each item that the typical family buys.
General formula:
P
Pn W
Weighted Index = P
· 100
P0 W
where W represents the weight for each item.
If base period weights are Q0 , the weighted aggregative index
is called the Laspeyres index, denoted by Lp :
P
Pn Q 0
P
Lp =
100
P0 Q0
This index measures the change in the total cost of a fixed bill
of goods.
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Index Numbers
Example: Laspeyres Index (Table 2)
Example
Using 1976 as the base year, consider the table below:
Food
Meat
Milk
Eggs
Bread
Coffee
Total
1976 (P0 )
2.12
1.65
0.69
0.63
1.75
6.84
1986 (Pn )
2.59
2.00
1.00
0.90
2.34
8.83
1976 (Q0 )
2
3
1
3
1
P0 Q0
4.24
4.95
0.69
1.89
1.75
13.52
Pn Q0
5.18
6.00
1.00
2.70
2.34
17.22
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Index Numbers
Example: Laspeyres Index (Table 2)
Example
Using 1976 as the base year (Table 2):
P
Pn Q0
17.22
100 =
100 ≈ 127.4%
Lp = P
P0 Q0
13.52
Interpretation
It costs 27.4% more in 1986 to purchase the same basket of goods.
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Index Numbers
Weighted Aggregative Index: Paasche Index
If weights reflect quantities bought in the non-base period
(Qn ), we get the Paasche index, denoted Pp :
P
Pn Qn
100
Pp = P
P0 Q n
Measures the cost of the basket in the current period relative
to the base period using current period weights.
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Index Numbers
Example: Paasche Index (Table 3)
Example
Using 1978 as the base year, consider the table below:
Item
Apples
Milk
Bread
Eggs
Total
Unit Price
1978(P0 )
0.15
0.30
0.30
0.50
1.25
Quantity
1978 (Q0 )
2
2
3
1
Unit Price
1983 (Pn )
0.25
0.35
0.40
0.65
1.65
Quantity
1983 (Qn )
1
2
3
1
P0 Qn
Pn Qn
0.15
0.60
0.90
0.50
2.15
0.25
0.70
1.20
0.65
2.80
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Index Numbers
Example: Paasche Index (Table 3)
Example
Using Table 3 (1978 as base year):
P
Pn Qn
2.8
100 =
· 100 ≈ 130.23%
Pp = P
P0 Q n
2.15
Interpretation
Prices of the basket of goods have increased by 30.23%.
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Index Numbers
d) Weighted Mean of Price Relatives
Similar to aggregative indexes, we can compute indexes using
averages.
Weighted arithmetic mean of price relatives uses weights
reflecting the value of items consumed, produced, or
purchased.
Formula:
P Pn
Weighted arithmetic mean =
·
100
W
P0
P
W
Weights should ideally be from the base period to allow
period-to-period comparisons.
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Index Numbers
Weighted Arithmetic Mean of Price Relatives (Table 4)
Example
Food
Commodity
Meat
Milk
Eggs
Bread
Coffee
Total
Price
1976
1986
(P0 )
(Pn )
2.12
2.59
1.65
2.00
0.69
1.00
0.63
0.90
1.75
2.34
Price
Relatives
( PPn · 100)
0
122.2
121.2
144.9
142.9
133.7
Quantity
Weight
(Q0 )
2
3
1
3
1
(W )
4.24
4.95
0.69
1.89
1.75
13.52
Weighted
Price Relatives
( PPn · 100 · W )
0
518.13
599.94
99.98
270.08
233.98
1722.11
P Pn
Weighted arithmetic mean =
· 100 W
1722.11
P
=
= 124%
W
13.52
P0
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Index Numbers
Fisher Ideal Index
Defined as the geometric mean of the Laspeyres and Paasche
indexes:
s P
P
p
Pn Qn
Pn Q0
P
P
Fisher = Lp · Pp =
· 100
· 100
P0 Q0
P0 Q n
Example: Given Lp = 145.87 and Pp = 140.05
Example
Fisher =
√
145.87 · 140.05 = 142.94
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Index Numbers
Drobisch Index
This index is preferable when Laspeyres and Paasche indexes
are close.
Formula:
1
(Lp + Pp )
2
Example: Using Lp = 145.87 and Pp = 140.05
Drobisch =
Example
1
Drobisch = (145.87 + 140.05) ≈ 142.96
2
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Index Numbers
Quantity Indexes (Table 5)
n
Quantity Relative Index: Q
Q0 · 100
P
n
Composite Quantity Index: P Q
Q0 · 100
Example
Item
Bread
Meat
Rice
Sorghum meal
Total
Q0
112
2000
32
50
2194
Qn
125
2550
25
80
2780
Quantity Relative
111.61
127.5
78.125
160
477.23
Weight
3
15
2
8
28
Qn
· 100·W
Q0
334.82
1912.5
156.25
1280
3683.57
P
Qn
2780
Composite quantity index = P
· 100 =
· 100 ≈ 126.71%
Q0
2194
Interpretation The quantity of goods consumed increased by 26.71%.
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Index Numbers
Weighted Composite Quantity Index
Weighted formula:
P
Q W
P n · 100
Q0 W
If W = P0 (base period price):
P
Q P
P n 0 · 100
Q0 P0
If W = Pn (current period price):
P
Q P
P n n · 100
Q0 Pn
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Index Numbers
Weighted Average of Quantity Relative Index
Definition
The weighted average of quantity relative index is given by
P Qn
Q0 · W
Iq = P
× 100
W
where:
Qn = current year quantity
Q0 = base year quantity
W = weight assigned to each commodity
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Index Numbers
Example
Using the example in Table 5, the weighted average of quantity
index is computed as
Iq =
3683.57
= 131.56%
28
If W = P0 Q0
Then the weighted average quantity relative index becomes
P Qn
P
P0 Q n
Q0 × 100 × P0 Q0
P
Iq =
=P
× 100
P0 Q0
P0 Q0
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Index Numbers
Example
Using the example in Table 5, the weighted average of quantity
index is computed as
Iq =
3683.57
= 131.56%
28
If W = P0 Q0
Then the weighted average quantity relative index becomes
P Qn
P
P0 Q n
Q0 × 100 × P0 Q0
P
Iq =
=P
× 100
P0 Q0
P0 Q0
Interpretation
This is the Laspeyres Quantity Index, which uses base-year
prices as weights.
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Index Numbers
EXERCISE
Example
Table 6 shows the average prices and quantities of the four raw
materials used in 1978 and 1983.
Use the table to construct the following:
a)
Price relatives of each raw material.
b)
Unweighted aggregative index for 1983 using 1978 as base.
c)
Weighted aggregative index using base-period weights.
d)
Weighted aggregative index with nonbase-period weights.
e)
Weighted arithmetic mean of relative index.
f)
Interpret each index.
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Index Numbers
Table 6
Table 6
Raw Material
A
B
C
D
1978
Price Quantity
10
10
3
20
5
50
2
30
1983
Price Quantity
12
15
5
25
10
60
5
40
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Index Numbers
(a) Price relatives
Computation
Price relative for item i: Ri =
A:
B:
C:
D:
Pn
× 100.
P0
12
× 100 = 120%
10
5
× 100 = 166.6667%
3
10
× 100 = 200%
5
5
× 100 = 250%
2
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Index Numbers
(b) Unweighted aggregative index
Formula & calculation
Simple aggregative index:
P
Pn
× 100.
Unweighted aggregative index = P
P0
Compute sums:
X
P0 = 10 + 3 + 5 + 2 = 20,
X
Pn = 12 + 5 + 10 + 5 = 32.
Unweighted aggregative index =
32
× 100 = 160%.
20
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Index Numbers
(b) Unweighted aggregative index
Formula & calculation
Simple aggregative index:
P
Pn
× 100.
Unweighted aggregative index = P
P0
Compute sums:
X
P0 = 10 + 3 + 5 + 2 = 20,
X
Pn = 12 + 5 + 10 + 5 = 32.
Unweighted aggregative index =
32
× 100 = 160%.
20
Answer / Interpretation
Simple (unweighted) price index = 160%. The (unweighted)
average price increased by 60%.
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Index Numbers
(c) Weighted aggregative (Laspeyres)
Formula
Laspeyres price index:
P
Pn Q0
× 100.
Lp = P
P0 Q0
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Index Numbers
(c) Weighted aggregative (Laspeyres)
Formula
Laspeyres price index:
P
Pn Q0
× 100.
Lp = P
P0 Q0
Calculation
X
X
P0 Q0 = 10 · 10 + 3 · 20 + 5 · 50 + 2 · 30 = 470.
Pn Q0 = 12 · 10 + 5 · 20 + 10 · 50 + 5 · 30 = 870.
Lp =
870
× 100 = 185.106%.
470
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Index Numbers
(c) Weighted aggregative (Laspeyres)
Formula
Laspeyres price index:
P
Pn Q0
× 100.
Lp = P
P0 Q0
Calculation
X
X
P0 Q0 = 10 · 10 + 3 · 20 + 5 · 50 + 2 · 30 = 470.
Pn Q0 = 12 · 10 + 5 · 20 + 10 · 50 + 5 · 30 = 870.
Lp =
870
× 100 = 185.106%.
470
Interpretation
It costs 85.11% more in 1983 to purchase the same basket of raw
materials as in 1978.
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Index Numbers
(d) Weighted aggregative (Paasche)
Formula
Paasche price index:
P
Pn Q n
× 100.
PP = P
P0 Qn
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Index Numbers
(d) Weighted aggregative (Paasche)
Formula
Paasche price index:
P
Pn Q n
× 100.
PP = P
P0 Qn
Calculation
X
X
P0 Qn = 10 · 15 + 3 · 25 + 5 · 60 + 2 · 40 = 605.
Pn Qn = 12 · 15 + 5 · 25 + 10 · 60 + 5 · 40 = 1105.
PP =
1105
× 100 = 182.6446%.
605
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Index Numbers
(d) Weighted aggregative (Paasche)
Formula
Paasche price index:
P
Pn Q n
× 100.
PP = P
P0 Qn
Calculation
X
X
P0 Qn = 10 · 15 + 3 · 25 + 5 · 60 + 2 · 40 = 605.
Pn Qn = 12 · 15 + 5 · 25 + 10 · 60 + 5 · 40 = 1105.
PP =
1105
× 100 = 182.6446%.
605
Interpretation
“It costs 82.64% more in 1983 to purchase the current (1983)
basket of raw materials than it would have cost in 1978.”
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Index Numbers
Example: Weighted Arithmetic Mean of Price Relatives
Formula:
P Pn
Weighted arithmetic mean =
×
100
W
P0
P
W
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Index Numbers
Example: Weighted Arithmetic Mean of Price Relatives
Formula:
P Pn
Weighted arithmetic mean =
Commodity
P0
Pn
A
B
C
D
Total
10
3
5
2
12
5
10
5
Pn
P0 × 100
120.00
166.67
200.00
250.00
×
100
W
P0
P
W
Pn
P0 × 100
W
120 × 100 = 12000
166.67 × 60 = 10000.20
200 × 250 = 50000
250 × 60 = 15000
87000.20
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Index Numbers
Example: Weighted Arithmetic Mean of Price Relatives
Formula:
P Pn
Weighted arithmetic mean =
Commodity
P0
Pn
A
B
C
D
Total
10
3
5
2
12
5
10
5
X
Pn
P0 × 100
120.00
166.67
200.00
250.00
×
100
W
P0
P
W
Pn
P0 × 100
W
120 × 100 = 12000
166.67 × 60 = 10000.20
200 × 250 = 50000
250 × 60 = 15000
87000.20
W = 100 + 60 + 250 + 60 = 470
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Index Numbers
Weighted Arithmetic Mean of Price Relatives
Index =
87000.20
= 185.11%
470
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Index Numbers
Weighted Arithmetic Mean of Price Relatives
Index =
87000.20
= 185.11%
470
Interpretation
It costs 85.11% more in 1983 to purchase the same basket of
raw materials as in 1978.
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Index Numbers
5.1 Consumer Price Index (CPI)
CPI – Botswana Case
The Prices Statistics Unit under Statistics Botswana is
responsible for the collection, compilation, and production of
price statistics.
While price statistics cover a wide area, the unit mainly deals
with:
Retail prices
Wholesale prices
Construction prices
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Index Numbers
5.1 Consumer Price Index (CPI)
CPI – Botswana Case
The Prices Statistics Unit under Statistics Botswana is
responsible for the collection, compilation, and production of
price statistics.
While price statistics cover a wide area, the unit mainly deals
with:
Retail prices
Wholesale prices
Construction prices
Purpose of Price Statistics
Central to price statistics is the concept of indices.
Key indices produced include:
Consumer Price Index (CPI)
Wholesale Price Index (WPI)
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Index Numbers
Consumer Price Index (CPI)
Definition
The CPI is the most widely known index number.
It measures the average change in prices over time in a fixed
market basket of goods and services.
The basket includes items such as:
Food
Clothing
Shelter
Fuel
Transportation fares
Charges for doctor’s services
Other daily goods and services
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Index Numbers
Uses of Consumer Price Index (CPI)
CPI is useful for:
Economic and social policy determination:
Assessing changes in the standard of living of consumers
Monitoring effectiveness of counter-inflation policies
Adjusting wages and salaries:
Usually adjusted upwards in line with price movements
Used as an important tool in wage negotiations
Ensures wage earners retain purchasing power
Determining cost of living:
Can be easily determined for a given place and sector of the
population
Calculation method:
If quantities are known, CPI can be calculated using the
Laspeyres approach
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Index Numbers
Example 5.1: CPI Calculation
Weighted Average of Price Relatives
P
(Pnij /P0ij × 100) wij
P
CPI(ij) =
wij
Pnij = price of i th item in group j (current period)
P0ij = price of i th item in group j (base period)
wij = weight of i th item in group j
CPIj = consumer price index for group j
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Index Numbers
Example 5.1: CPI Calculation
Weighted Average of Price Relatives
P
(Pnij /P0ij × 100) wij
P
CPI(ij) =
wij
Pnij = price of i th item in group j (current period)
P0ij = price of i th item in group j (base period)
wij = weight of i th item in group j
CPIj = consumer price index for group j
General CPI Formula
P
CPI =
CPI(j) × wj
P
wj
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Index Numbers
Table 7: Prices, Weights, and Relative Prices
Consider the table below showing the prices of two baskets of
goods (cereal and fuel and light), as well as their corresponding
weights for the years 1982 and 1992.
Groups
Weight
P0(1982)
Pn(1992)
Pn
× 100
P0
a) Cereal
Maize
Groundnut
Wheat
Total
60.5
10.2
8.5
79.2
0.45
0.15
0.75
1.25
2.30
1.85
278
1533
247
16819
15636.6
2099.5
34554.1
b) Fuel& Light
Kerosene
Petrol
Electricity
Total
30.5
25.2
40.8
96.5
0.60
0.82
0.25
1.00
1.23
0.56
167
150
224
5093.5
3780
9139.2
18012.7
Pn
× 100
P0
× wij
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Index Numbers
CPI for Each Basket
CPI Calculation
34554.1
= 436.29%
79.2
18012.7
CPI(b) =
= 186.6%
96.5
CPI(a) =
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Index Numbers
CPI for Each Basket
CPI Calculation
34554.1
= 436.29%
79.2
18012.7
CPI(b) =
= 186.6%
96.5
CPI(a) =
Interpretation
The CPI for cereals (group a) increased much more than for fuel
and light (group b) between 1982 and 1992.
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Index Numbers
General CPI
Overall CPI Calculation
436.29 × 79.2 + 186.6 × 96.5
General CPI =
= 229%
79.2 + 96.5
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Index Numbers
General CPI
Overall CPI Calculation
436.29 × 79.2 + 186.6 × 96.5
General CPI =
= 229%
79.2 + 96.5
Interpretation
On average, the price level of the combined basket increased to
229% of 1982 prices by 1992.
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Index Numbers
Inflation Rate
Definition
The inflation rate measures the percentage change in the
cost-of-living index over 12 months:
Inflation Rate =
CPI (current) − CPI (previous)
× 100
CPI (previous)
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Index Numbers
Inflation Rate
Definition
The inflation rate measures the percentage change in the
cost-of-living index over 12 months:
Inflation Rate =
CPI (current) − CPI (previous)
× 100
CPI (previous)
Practical Meaning
Increasing inflation → prices rise faster
Decreasing inflation → prices rise slower or fall
Helps policymakers, businesses, and consumers adjust wages,
budgets, and policy
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Index Numbers
Problems of Price Index
Common Issues in Constructing a Price Index
1
Selection of items for the basket
2
Selection of the base period
3
Choice of weights
4
Data collection
5
Availability and comparability of data
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