Chapter 6 Index numbers

Index Numbers Index numbers allow relative comparisons over time It measures the percentage change in the value of some economic commodity over time Index numbers are reported relative to a Base Period Index Base period index = 100 by definition

Notes ‘Economic commodity’: used to describe anything measurable which has some economic relevance. ‘Economic commodity’ can be: price, quantity, wage, productivity…. Index numbers must always be related to some time period (i.e. base time period and another time period)

Examples If the index number values are:

Index Relatives Index relative or simple index number is an index number which measures the change in a single distinct commodity

Index Relatives Formula: where I y = index number of commodity ‘y’ y t = value of commodity ‘y’ at time t y 0 = value of commodity ‘y’ in the base period

Index Numbers: Example Company orders from 1995 to 2003: YearNumber of Orders Index (base year = 2000)

Index Numbers: Interpretation

Price relatives Formula Where: p t : price at time t p 0 : price in the base period The price index for 2003, based on 2000(as 100), is 120

Quantity relatives Formula Where: q t : quantity at time t q 0 : quantity in the base period The quantity index for 2003, based on 2000 (as 100), is 150

Example Calculate price index relatives and quantity index relatives ItemsUnit Price (USD)Quantity (p 0 )(p t )(q 0 )(q t ) Akg3,04, Bm5,06, Cl2,02,

Time series of relatives Describe how the values of an index relative change overtime. There are two distinct ways in which relatives can be calculated - Fixed base relatives: each relative is calculated based on the same fixed time point - Chain base relatives: each relative is calculated with respect to the immediately preceding time point

Fixed base relatives Year Sales (1000 USD) Fixed base relatives (2000=100) Fixed base relative (2002=100)

Chain base relatives Year Sales (1000 USD) Chain base relative

Changing the base of fixed base relative Used when the old base time point is no longer relevant (usually it is too far in the past or out of date) Step 1: Choose the required new base time point and identify the corresponding relative. Step 2: Divide each relative in the set by the value of the relative identified above and multiply the result by 100.

Changing the base of fixed base relative Year Old Index (1980=100) New Index (2000=100)

Time series deflation A technique used to obtain index relatives that measure the changes in the real value of some commodity with respect to a given indicator Year Average daily wage (USD) CPI What is the real wage index for 2005?

Time series deflation: procedure Step 1: Choose a base for the index of real values of the series Step 2: For each time point, find the ratio of the current value to the base value

Time series deflation: procedure Step 3: Multiply by the ratio of the base indicator to the current indicator Step 4: Multiply the result by 100

Real value Index (RVI) Formula: Where: The value at time t of a given time series The value at base time point of a given time series The indicator at time t The indicator at base time point

Time series deflation: example Year Average daily wage (USD) CPI Real wage index

Composite index numbers A composite index number is an index number obtained by combining the information from a set of economic commodities. It measure the percentage changes of a group of items (not one item)

Weighting of components Usually, a composite index can be calculated by weighting each component. A weighting factor is an indicator of the importance of each component in calculating the composite index The need for weights (read in the textbook)

Types of composite index number Weighted average of relatives Weighting the index relative calculated for each component Weighted aggregates Multiplying each component value by its corresponding weight and adding these products to form an aggregate

Weighted average of relatives Formula: Where: w: weighting factor of each component I: index relative of each component

Weighted average of relatives ItemsPriceStandard quantity A B C

Weighted average of relatives ItemsPriceStandard quantity A B C Totals 28 Price relative

Weighted average of relatives Weighted average of price index relatives

Weighted aggregates Formula where Value of commodity at time t Value of commodity at base time point Weight factor

Weighted aggregate: Example ItemsPriceStandard quantity A B C

Weighted aggregates ItemsPriceStandard quantity A B C Totals

Weighted aggregates Weighted aggregate of price index:

Weighted Aggregate Price Indexes Paasche index Laspeyres index A Paasche price index uses current time period quantities as weights A Laspeyres price index uses base time period quantities as weights

Weighted Aggregate Quantity Indexes Paasche index Laspeyres index A Paasche quantity index uses current time period prices as weights A Laspeyres quantity index uses base time period prices as weights

Example ItemsUnit Price (USD)Quantity (p 0 )(p t )(q 0 )(q t ) Akg3,04, Bm5,06, Cl2,02, Calculate the Pasache and Laspeyres price and quantity index?