Post on 18-Jan-2016
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
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