Transcript Applied Quantitative Methods III. MBA course Montenegro

Applied Quantitative Methods
MBA course Montenegro
Peter Balogh
PhD
[email protected]
7. Index numbers
• It is often necessary to describe and interpret changes in
economic, business and social variables over time.
• Information on change may come from different types of data,
recorded in different ways.
• Index numbers can provide a simple summary of change by
aggregating the information available and making a comparison to
a starting figure of 100.
• A typical index then could take the form of 100, 105, 107, where
100 is the starting point, 105 shows the relative increase one year
later and 107 shows the relative increase two years on.
• Index numbers, therefore, are not concerned with absolute values
but rather the movement of values.
• The retail prices index (RPI) is one the better known indices and is
a general measure of how the prices of goods and services change,
rather than as an indicator of the absolute amounts we actually
spend each week.
7.1 The interpretation of index
numbers
• Indices provide a measure of change over
time, making reference to a base year value of
100.
7.1.1 Percentage changes
• An index is a scaling of numbers so that a start is
made from a base figure of 100.
• Suppose, for example, the price for bread over
the past 4 years was as shown in Table 7.1.
7.1.1 Percentage changes
• We would first need to decide which year should be
used for the base year, and then scale all figures
accordingly.
• If year 1 was chosen for the base year, we would
divide all the prices by 0.50 and multiply by 100, as
shown in Table 7.2.
7.1.1 Percentage changes
• The index numbers given for years 2-4 all measure
the change from the base year.
• The index number of 120 shows that there was a
20% increase from year 1 to year 2, and the index
number 160 shows that there was a 60% increase
from year 1 to year 3.
• To calculate a percentage increase, we first find the
difference between the two figures, divide by the
base figure and then multiply by 100.
7.1.1 Percentage changes
• The percentage increase from 100 to 188 is
• In the same way, the percentage increase in the
price of bread from £0.50 to £0.94 is
7.1.1 Percentage changes
• One important feature of index numbers is that by
starting from 100, the percentage increase from the
base year is found just by subtraction.
• However, the differences thereafter are referred to
as percentage points.
• It can be seen from Table 7.2 that there was a 28
percentage point increase from year 3 to year 4.
• The percentage increase, however, is
7.1.2 Changing the base year
• There are no hard and fast rules for the choice of a
base year and, as shown in Table 7.3, any year can
be made into the base year from a purely
mathematical point of view.
7.1.2 Changing the base year
• Each of the indices measures the same change over
time (to two decimal places).
• The percentage increase from year 1 to year 2 using
index 2, for example, is
• To change the base year (move the 100) requires only a
scaling of the index up or down.
• If we want index 1 to have year 2 as the base year
(construct index 2), we can use the equivalence
between 110 and 100 and multiply index 1 by this
scaling factor 100/110.
7.1.2 Changing the base year
• In practice, there are a number of important
considerations in the choice of a base year.
• As the index gets larger the same percentage change is
represented by a larger increase in percentage points.
• A change from 100 to 120 is the same as a change from
300 to 360 but the impression created can be very
different.
• If, for example, our index were used as a measure of
inflation, like the Retail Prices Index, we would not
want the index to move very far from 100.
• We would like the seen change (points) to be close to
the actual change (percentages).
7.1.2 Changing the base year
• An index number is typically a summary of what is
happening to a group of items (often referred to as
a basket of goods).
• From time to time we may review and change the
items to be included and this is often when the
index is again started at 100.
• Footnotes or other forms of referencing may
indicate these changes.
• Suppose a manufacturer constructed a productivity
index using as a measure of productivity the times
taken to make the most popular products.
7.1.2 Changing the base year
• As new products appear and established products
disappear, the manufacturer would need to
reconsider the basis of the index.
• The manufacturer would also need to consider the
compatibility of the indices produced as the new
products may be adding a different level of value
and involve different methods of production.
• An index can be unadjusted or adjusted.
• To show the general (underlying) trend in
unemployment, the index can be adjusted to allow
for predictable changes through the year, like the
number of school leavers.
7.1.2 Changing the base year
• A change in base year is shown in Table 7.4.
7.1.2 Changing the base year
• We can use the equivalence of 150 in the 'old' index
with 100 in the 'new' index at year 5.
• We either scale down the 'old' index using a
multiplication factor of 100/150 as shown in Table
7.5 or scale up the 'new' index using 150/ 100 as
shown in Table 7.6
7.1.3 Nominal and real change
• Index numbers allow us to distinguish between nominal and
real values.
• Suppose your annual entertainment allowance had increased
from £500 to £510.
• This £10 increase is referred to as nominal value (and is given
in the original units of measurement, in this case £'s).
• However, you may be more concerned with the purchasing
power of the new £510 allowance and how this compares
with the £500 allowed in the previous year.
• Suppose that you are now told that the cost of entertainment
has increased by 5%. To maintain your purchasing power you
would need £525 (£500 plus the extra 5%, which is £25).
• We would now say that in real terms your purchasing power
has decreased. Indices can measure change in real or nominal
terms.
7.1.3 Nominal and real change
•
•
•
•
Case 2: the use and construction of indices
Managers are always likely to be concerned with
measures of change over time.
Case 2 data gives the number of business enquiries
received, the value of new business and an index of
inflation over a 3-year period.
It is relatively clear from the figures that the
Important business performance measurements of
the number of enquiries and the value of new
business are declining.
It is less clear that the rate of inflation is about 3%
and what impact the inflation rate is having on the
'real' value of new business.
7.1.3 Nominal and real change
•
•
•
•
•
•
Case 2: the use and construction of indices
Table 7.7 shows the construction of indices to illustrate the
changes using year 1, quarter 1 for the base.
The columns in Table 7.7 for the number of enquiries and the
value of new business both show the downward trend and a
quarterly variation.
The indices confirm this trend and show the greater variation
(in percentage terms) in the number of enquiries.
The spreadsheet shown as Table 7.8 can also be constructed
to show how the Veal' value of new business has declined.
It can be seen that if we are working with the purchasing
power of the £ in year 1, quarter 1 (real as opposed to
nominal pounds), the drop in the value of new business is
even greater.
The spreadsheet has also been used to show that the rate of
inflation and how that has been declining.
stakeholders, about the position of the company.
7.1.3 Nominal and real change
• The figures 'on the surface' look bad, but need to be interpreted
within their business context (business significance rather than
statistical significance).
• Trading conditions might have become particularly difficult and
the company may still have done better than other rivals (the
business could consider benchmarking against best practice
wherever that is to be found).
• The figures may reflect a change in company strategy where new
business of this kind has not been sought and existing business has
been consolidated.
• What is important is that the analysis is able to inform a debate
and highlight the realities that the company may face (which is
better understood through analysis and debate).
• Analysis should also inform policy formulation and change.
• It is important to explore the data for improved insight; you could,
for example, calculate the ratio of the value of new business to the
number of enquiries and consider what these figures mean.
7.2 The construction of index numbers
• Index numbers are perhaps best known for measuring the change of
price or prices over time.
• To illustrate the methods of calculation, we will use the information
given in Table 7.9.
• The price can be taken as the average amount paid in pence for a cup
and the quantity as the average number of cups drunk per person per
week.
7.2.1 The simple price index
• If we want to construct an index for the price of one
item only we first calculate that ratio of the 'new'
price to the base year price, the price relative, and
then multiply by 100.
• In terms of a notation
• where P0 is the base year price and Pn is the 'new'
price.
• A simple index for the price of tea, taking year 0 as
the base year, can be calculated as in Table 7.10.
• The doubling of the price of tea from 32p to 64p over
the 3-year period gives a 100% increase in the index,
from 100 to 200.
• The increase from 48p to 64p is a 50 percentage point
increase (the index increases from 150 to 200) or a
percentage increase of 334%.
• In reality we are likely to drink more than just tea.
When constructing an index of beverage prices, for
example, we may wish to include coffee and chocolate
drinks.
7.2.2 The simple aggregate price index
• To include all items, we could sum the prices year
by year and construct an index from this sum. If the
sum of the prices in the base year is P0 and the
sum of the prices in year n is
Pn then the
simple aggregate price index is


• The calculations are shown in Table 7.11.
7.2.2 The simple aggregate price index
• This particular index ignores the amounts
consumed of tea, coffee and chocolate drinks.
• In particular, the construction of this index ignores
both consumption patterns and the units to which
price refers.
• If, for example, we were given the price of tea for a
pot rather than a cup, the index values would differ.
7.2.3 The average price relatives index
• To overcome the problem of units, we could
consider price ratios of individual commodities
instead of their absolute prices and treat all price
movements as equally important.
• In many cases, the goods we wish to include will be
measured in very different units.
• Breakfast cereal could be in price per packet,
potatoes price per kilo and milk price per pint
bottle.
7.2.3 The average price relatives index
• As an alternative to the simple aggregate price
index we can use the average price relatives index:
• where k is the number of goods.
• Here the price relative,  Pn /  P0 for a stated
commodity will have the same value whatever the
units.
• The calculations are shown in Table 7.12
7.2.3 The average price relatives index
• Comparing Tables 7.12 and 7.11 we can see that the
average price relatives index, in this case, shows larger
increases than the simple aggregate price index.
• To explain this difference we could consider just one of
the items: tea.
• The value of tea is low in comparison to other drinks so
it has a smaller impact on the totals in Table 7.11.
• In contrast, the changes in the price of tea are larger
than any of the other drinks and this makes a greater
impact on the totals in Table 7.12.
• To construct a price index for all goods and sections of
the community we need to take account of the
quantities bought.
7.2.3 The average price relatives index
• It is not just a matter of comparing what is spent year
by year on drinks, food, transport or housing.
• If prices and quantities are both allowed to vary, an
index for the amount spent could be constructed but
not an index for prices.
• If we want a price index we need to control quantities.
• In practice, we consider a typical basket of goods in
which the quantity of goods of each kind is fixed and
we find how the cost of that basket has changed over
time.
• To construct an index for the price of beverages we
need the quantity information for a selected year as
given in Table 7.9.
7.2.4 The Laspeyre index
• This index uses the quantities bought in the base
year to define the typical basket.
• It is referred to as a base-weighted index and
compares the cost of this basket of goods over time.
• This index is calculated as
PQ


PQ
n
0
0
0
100
• where P0Q0 is the cost of the base year basket of
PnQ0 is the cost of
goods in the base year and
the base year basket of goods in any year
(thereafter) n.

7.2.4 The Laspeyre index
• It can be seen from Table 7.13 that we only require
the quantities from the chosen base year (Q0 in this
case).
• The index implicitly assumes that whatever the
price changes, the quantities purchased will remain
the same.
• In terms of economic theory, no substitution is
allowed to take place. Even if goods become
relatively more expensive it assumes that the same
quantities are bought. As a result, this index tends
to overstate inflation.
7.2.5 The Paasche index
• This index uses the quantities bought in the current
year for the typical basket.
• This current year weighting compares what a basket of
goods bought now (in the current year) would cost,
with cost of the same basket of goods in the base year.
Pn Qn
• This index is calculated as


PQ
0
100
n
• where PnQn is the cost of the basket of goods
bought in the year n at year n prices and
P0Qn is
the cost of the year n basket of goods at base year
prices.
• The calculations are shown in Table 7.14.

7.2.5 The Paasche index
• As the basket of goods is allowed to change year by
year, the Paasche index is not strictly a price index and
as such, has a number of disadvantages.
• Firstly, the effects of substitution would mean that
greater importance is placed on goods that are
relatively cheaper now than they were in the base year.
• As a consequence, the Paasche index tends to
understate inflation.
• Secondly, the comparison between years is difficult
because the index reflects both changes in price and
the basket of goods.
• Finally, the index requires information on the current
quantities and this may be difficult or expensive to
obtain.
7.2.6 Other indices
• The Laspeyre and Paasche methods of index
construction can also be used to measure quantity
movements with prices as the weights.
• Laspeyre quantity index using base year prices as
weights:
• Paasche quantity index using current year prices as
weights:
7.2.6 Other indices
• To measure the change in value the following 'value'
index can be used:
• These calculations are shown, along with the
Laspeyre and Paasche index, in Table 7.15.
• Having constructed a spreadsheet you can experiment by
making changes to price or quantity information and observing
the overall effect.
• (This spreadsheet (sp715.XLS) is available on the website).
7.3 The weighting of index numbers
• Weights can be considered as a measure of
importance.
• The Laspeyre index and the Paasche index both
refer to a typical basket of goods.
• The prices are weighted by the quantities in these
baskets.
• In measuring a diverse range of items, it is often
more convenient to use amount spent as a weight
rather than a quantity.
• If we consider travel, for example, it could be more
meaningful to define expenditure on public
transport than the number of journeys.
7.3 The weighting of index numbers
• In the same way, we would enquire about the
expenditure on meals bought and consumed
outside the home rather than the number of meals
and their price.
• Expenditure on public transport, meals outside the
home and other items are additive since money
units are homogeneous; the number of journeys,
number of meals and number of shirts are not.
7.3 The weighting of index numbers
• In constructing a base-weighted index we can use
• where Pn/P0 are the price relatives (see Section 7.2.1)
and w are the weights.
• Each weight is the amount spent on the item in the
base year.
• Consider again our example from Section 7.2 (Table
7.16).
• It is no coincidence that this base-weighted index is
identical to the Laspeyre index of Table 7.13.
• The identity is proven below:
7.3 The weighting of index numbers
• The identity is proven below:
7.3 The weighting of index numbers
• The weights only need to represent the relative
order of magnitude and in practice are scaled to
sum to 1000.
• (If we were to multiply each of the weights in Table
7.16 by 1000/748, the value of the index would not
change but the sum of weights would add to 1000.)
• The items included in the Retail Prices Index are
assigned weights in this way.
7.4 The Retail Prices Index (RPI)
• The general Retail Prices Index (RPI) is the main index used to
measure of inflation in the UK.
• It measures the average change on a monthly basis of the
prices of goods and services purchased by most households.
• It is the measure of inflation reported in the media, debated
by politicians and used to revise benefits and pensions.
• Increases in wages are often justified in terms of the RPI, with
recent or anticipated changes often forming the basis of a
wage claim.
• In many cases, savings and pensions are index-linked; they
increase in line with the index.
• All forms of economic planning take some account of
inflation, and economists will use both real and nominal
values in their analysis.
7.4 The Retail Prices Index (RPI)
• The RPI covers a range of goods and services bought
by a typical household.
• It is useful to think of the RPI as representing the
changing cost of a large 'basket of goods and
services' reflecting the full range of things that
people buy including leisure goods, fuel, food and
footwear.
• In many ways, the RPI is not a 'cost of living' index
as it does not attempt to provide a measure of the
cost of staying alive.
• A 'cost of living' index would imply some definition
or knowledge of what were essential purchases.
• Who could make such a judgement?
7.4 The Retail Prices Index (RPI)
• The index reflects what people choose to buy; for
example, some people buy cigarettes and alcohol,
so these are included in the index.
• Coverage includes housing and travel but excludes
items like savings, investments, charges for credit,
betting and cash gifts.
• The expenditure of certain higher income
households and of pensioner households mainly
dependent on state benefit is excluded.
7.4 The Retail Prices Index (RPI)
• The 'basket of goods' is kept fixed for a year at a time,
so that only changes in prices are recorded that year.
• The basket is reviewed each year to keep it as up to
date as possible.
• Changes made in 'year 2000' basket include broccoli,
prepacked salad and takeaway/delivered pizza in the
'food and catering category', and the introduction of PC
printers in the 'leisure goods' category.
• It has also been decided that the RPI should begin to
reflect Internet prices, and some books and toys
typically bought on the Internet have been included.
7.4 The Retail Prices Index (RPI)
• The prices of more than 600 separate goods and
services are collected each month.
• The movements in these prices are taken as
representative of all price movement in the goods and
services covered by the index.
• There are six price indicators for beef, for example
(January 2000), which are combined together to
estimate the overall change in beef price.
• The base period is January of each year and current
prices are compared to this base period.
• The RPI for any month is calculated by weighting
(averaged) price relatives.
• Essentially, the RPI is a Las-peyres base-weighted index.
• For a more detailed description of the RPI refer to
http//www.statistics.gov.uk or http//www.ons.gov.uk.
• CPI and RPI
• CPI and RPI measure change in the prices charged for goods
and services bought for consumption in the UK. Prices are
recorded monthly for a typical selection of products (referred
to as the ‘basket of goods’) , using a large sample of shops
and other outlets.
• Each month price collectors record about 120,000 prices for
over 650 goods and services. These prices are ‘weighted’ to
ensure they reflect the relative importance of the items in the
average shopping basket. The ‘basket’ is updated on an
annual basis to ensure that the indices reflect UK consumer
spending patterns.
• After the price data have been processed, the Office for
National Statistics (ONS) calculates an overall average price
change, which forms the basis of the monthly CPI and RPI.
These data are published each month on the National
Statistics website in the form of a First Release, which
contains relevant tables, texts and charts.
7.4 The Retail Prices Index (RPI)
• Weights are used to allow for the relative importance
of the various categories of goods and services.
• The weights are derived mainly from the Family
Expenditure Survey (see Figure 7.3).
• The Family Expenditure Survey is based on a set sample
size of about 11 000 households each year (11 400 in
1998/99 – see Figure 11.3) and uses addresses from
the postcode address file.
• Selected households are asked to keep records of what
they spend over a 2-week period and are also asked to
give details of their major purchases over a longer
period.
• The response rate given in Figure 7.3 is 59%, with a
more recent response rate of 63% being reported.
• Analysis is based on about 7000 households.
• The weights used in the RPI for 1990, 1995
and 2000 are shown in Table 7.17.
7.4 The Retail Prices Index (RPI)
• As far as the RPI is concerned, using the weights given in
Table 7.17, food accounted for 15.8% of the typical basket in
1990, 13.9% in 1995 and 11.8% in 2000.
• You could also note the recent decrease in the relative
expenditure on alcoholic drink (from 7.7% in 1990 to 6.5% in
2000) and the recent increase on the relative expenditure on
motoring (from 13.1% in 1990 to 14.6% in 2000).
• The weightings therefore provide a useful guide to changing
patterns of expenditure.
• The weighting can be used to demonstrate the effect of a
price change in one category on the overall RPI.
• If, for example, the 'price' of fuel and light increased by 10%
in 2000, the overall impact would be 0.32% (under A%), as
the category accounts for 3.2% of expenditure (a weight of 32
out of 1000).
• To show this using weights, the weight for fuel and light
would increase from 32 to 35.2 (a 10% increase).
7.4 The Retail Prices Index (RPI)
• The sum of the weight would become 1003.2 and the percentage
increase would be:
• In practice, calculations can be more complex because a number
of changes take place at the same time.
• The typical basket of goods and services indicated by the weights
shown in Table 7.17 will only reflect completely the expenditure of
a proportion of households.
• Some families will spend more on some items and less on others,
particularly on an annual basis.
• The RPI, like all aggregated statistics, will have an averaging-out
effect and will reasonably describe most families most of the time.
7.5 Conclusions
• Index numbers play an important role in describing the economy,
managing the economy and measuring the performance of
business.
• Percentage increases from the base year can be seen at a glance,
and that the numbers provide a manageable and understandable
sequence.
• As we have seen, we are able to aggregate a wide range of
different items into a single index series, which will enhance our
comprehension of an overall situation, for example, the level of
inflation in an economy.
• In the presentation of accounting information allowance needs to
be made for inflation.
• Historic cost accounting (with no allowance for inflation) only
works well in periods of stable prices.
• In current cost accounting (CCA), adjustments are made in
proportion to relevant indices.
• The Office for National Statistics publishes price index numbers for
current cost accounting.
7.5 Conclusions
• Index numbers can be misleading if care is not taken.
• When an index is rebased it is important to compare
the last value of the previous series to the starting
value of the new series and make any necessary
adjustments.
• When items are excluded, or new items included in an
index, there may be drastic movements in the series,
which do not reflect major changes in prices or
quantities, but merely the changed composition of the
index.
• Crime statistics, in particular, are statistically (and
politically) very sensitive to changes in definition and
reporting.