Index numbers

Value-, price-, quantity indices

Measuring changes in time

Indices: relative change in price, quantity and value of products or services in a given period

Marking: p=unit price; q=quantity; v= value (p*q) Value is highly important in

examining economic phenomena.Periods• Object for comparison: current period (p1,q1,v1)• Base for comparison: base period (p0,q0,v0)

Simple indicesA simple index shows the relative change in price, quantity or value of a given product in the current period compared to the base period.

•Simple value index

•Simple price index

•Simple quantity index

0

1

p

p i p

0

1

q

q iq

0

1v v

v i

pq ii i v

Aggregate values

• Aggregation• Fixed price aggregate:

– At base price: p0q1--------- p0q0

– At current price: p1q1 --------- p1q0

N

ii

N

iii pq

11

v

Aggregate indices

An aggregate index shows the relative change in price, quantity or value of a given basket of goods (product group) in the current period compared to the base period.•Value index: Iv•Price index: Ip•Quantity index: Iq

Value index

0

1v v

v I

v

1

1v v

v

i

I

Product group’s value change at current price

Average value change of the products of a

product group

Mean of simple value indices

Weigh: value

00

11v

pq

pqI

0

v0v v

v

iI

A value index expresses the relative change of a phenomenon measured in value.

Problem of value index

• E.g.: Revenue growth is real performance increase?

• „Heterogeneous”: contains the price change and the quantity change as well

• The inflation, the change of prices occurs in value data. If we want to filter out of that, we have to count the relative change of prices:

• This is the price index

Solution: calculation of price- and quantity indices

• Price index– Average change of price level for a product group

(more products)

• Quantity index– Average change of quantities for a product group

(more products)

Price index

• There are a lot of cases when price changes should be calculated for different products.

• Therefore we consider the average price changes of each product.

• The weighted mean of the products’ price changes (simple indices) should be calculated. The weights are values. Which period’s values?– Base period weights– Current period weights

Price indices

v

v

00

000

1

0

00

pq

pqpp

iI p

p

v

v

111

0

11

1

11

pqp

p

pq

i

I

p

p

The average price change for the products of a product group

Base period weights: Laspeyres' price index

Current period weights: Paasche's price index

00

100 pq

pqI p

01

111 pq

pqI p

Value change due to price changes

Price indices

10pp

Fp II I Geometric mean of the two indices:

Fishers’ price index

•Average value change due to price changes

Special price index: Consumer Price Index (CPI)

• Based on 960 representative items of 141 product groups • 35-150 prices/item, together 70000 prices/month• Simple price index of representative items: ratio of average

prices• Consumer Price Index based on simple price indices:

Laspeyres’ price index• Weights: represents the ratio of product- and services groups

in the consumption of households. (Source: consumption data of national accounts, household statistics surveys).

• Revision of weights happens on a yearly basis, based on the household consumption two years before the current year.

Index relationships

10 * qpv III

01 * qpv III

Fq

Fpv III *

Meaning: decomposition of the value index (value change ) for two components:•Value change due to the price changes •Value change dute to the quantity changes

Quantity indices

• If we divide the value index by the price index, we can filter out the price change resulting the quantity index.

• The quantity index reflects the change in quantities.• Rearranging the index relationships: the quantity index

expressing the changes in quantities is equal to the ratio of the value index and price index

Fp

vFq

Fq

Fpv

p

vqqpv

p

vqqpv

I

IIIII

I

IIIII

I

IIIII

*

*

*

1001

0110

Base period weights: Laspeyres' quantity index

Current period weights: Paasche's quantity index

Quantity indices

00

000

1

0

00

v

v

qp

qpqq

iI q

q

The average quantity change for the products of a product group

00

010

pq

pq Iq

10

111

pq

pq Iq

• Value change of a product group at fixed price• Value change due to quantity changes

v

v

0

111

11

1

11

qq

qp

qp

i

I

q

q

Current period weights: Paasche's quantity index

Base period weights: Laspeyres' quantity index

Quantity indices

•Value change of a product group at fixed price•Average value change due to quantity changes

10qq

Fq II I Geometric mean of the two indices:

Fishers’ quantity index