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Quarterly National Accounts Course
Joint Vienna Institute
August 5 - 16, 2013
L-11: Price and Volume Measures
Reproductions of this material or any parts of it should refer to the IMF Statistics Department as the source
Lecture Outline
I. Introduction
Scope and Main Issues
II. Main concepts and principles
A. Basic Relationship between Value, Quantity and Price
B. Laspeyres Volume Index and Paasche Price Index
C. The set of Laspeyres and Paasche Index Formulas
D. Basic Re ations ip etween Va ue, Vo ume an PriceIndices
III. GDP from the production approach
IV. GDP from the expenditure approach
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Why Do we Need Price and Volume
Measures?
To ro erl inter ret the chan es in nominal fi ureswhen relative prices and/or the general price level arechanging.
1. For goods and services this means that when thenominal value of goods and services transacted
roduced consumed im orted or ex orted chan es
how much is due to changes in quantity?
how much is due to changes in the prices of the goods orservices?
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To ro erl inter ret the chan es in nominal fi ures
Why Do we Need Price and Volume
Measures?
when relative prices and/or the general price level arechanging.
2. For nominal income this means that when suchincome increases or decreases
how much more or less of goods can be bought as a result ofthe change?
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Scope of Price and Volume Measures in the
System
r ce an vo ume measures are requ re
1. For goods and services
to factor the changes in their values intocomponents reflecting changes in their volumesand prices
2. And to measure the cash flows in real terms by deflating their values by price indices
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Scope of Price and Volume Measures in the
System
e scope o t e pr ce an vo umemeasures also includes:
3. Taxes and subsidies;
4. Trade margins;
. ,6. Compensation of employees;
7. Consumption of fixed capital;
8. Stocks of produced assets (inventories, fixed assets)
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Price and Volume Measures in the National
Accounts Main Issues
What is meant by price and volume measurement for
these items and their components
What is the relationship between the current price
value and the price and the volume measures for these
items?
How to aggregate them?
How to obtain price and volume measures in practice?
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Main SNA Recommendations for Price and
Volume Measures
a e o e ar
Ideal method:
Annually chained Fischer price and volume indices forGDP and its components.
Second best:
Annually chained Laspeyres volume indices andPaasche price indices; OR
Annually chained Paasche volume indices andLaspeyres price indices.
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Basic Relationship between Value, Quantity
and PriceQuantity: Unit for measuring amount of a good or service. Quantities
are additive onl at the sin le roduct level
Price:Value of one unit of a good or service (of same quality)
An average of the prices of different goods or services should notbe used to measure price changes over time
Value: Price multiplied by quantity. Additive across different products
At single product level:
At an aggregated level:
where:vit,pit, and qitare the value, price and quantity of item iin period t,and
Vit is the total value at current prices in period t
ititit qxpv
i ititi itit
qxpvV
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What is Meant by Value(s) at Constant Prices
It is the value(s) of a product(s) for the current period using its ownprice(s) from an earlier period.
Table 1 Car Production
Price Quantity Value Price Quantity Value Value
(000 $/un.) (No.) (000 $) (000 $/un.) (No.) (000 $) (year 0 $)
(1) (2) (3) = (1)*(2) (4) (5) (6)=(4)*(5) (7)=(1)*(5)
Model A 20 15 300 40 24 960 480
Model B 10 15 150 20 6 120 60
30 450 30 1,080 540
Year 0 Year 1
Values in column (6) are in current prices showing a 140 percentincrease over year 0 (index = 1080/450 = 240)
Values in column (7) are at constant prices of year 0, they reflectchanges in quantity and/or quality.Values at constant prices are an aggregated volume measure, expressed in moneyterms and additive
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Laspeyres Volume Index Formula
The change from the base year in constant prices or the ratio of
be expressed in index form as:
LQ0t = 540 x 100 / 450 = 120.0
This is also called Laspeyres (fixed-base) volume index (LQ0t).
Mathematically:
)1(0,0,
,0,
0
,0
0,0
,00
i ii
i tiittt qp
qp
VQ
LQ
Note: the two components of the index are ADDITIVE
The Laspeyres volume index can also be written as:
where wi,o is the base period weight, i.e. the items share in the total value inthe base period
)2(0,
,0,0 i
i
tiit q
qwLQ
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Relationship between Measures at Constant Prices and
Laspeyres Volume Index Formula
Accordin to this formulation 2 , L is derived asfollows using the data from the data in table 1:
Table 2
Price Quantity Value Weight Quantity
(000 $/un.) (No.) (000 $) (w0) (No.)
(1) (2) (3) = (1)*(2) (4)=(3)/
(3) (5) (6)=(5)/(2) (7)=(4)*(6)*100
Model A 20 15 300 66.7% 24 1.6 107
w0*QR
Year 0 Year 1
Quantity
relatives (QR)
Note Laspeyres (fixed-base) volume index is one of several volume index formulas Measures at constant prices are one of several alternative volume measures Others are, Paasche index formula, chain-linked indices and Fisher.
Model B 10 15 150 33.3% 6 0.4 13
30 450 30 120
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Paasche Price Index FormulaTo factor the change in the value of car production from year 0 toyear 1 arising from price changes. take the ratio of the value of out ut in current rices in ear 1 to the value of
output in year 1 measured in constant prices (prices of year 0) (and multiply itwith 100 to convert to an index form):
PP0t= 1080 x 100 / 540 = 200.0
Shows 100 percent increase or doubling in prices
The above ratio is also called the Paasche price index (PP0t).Algebraically:
or
where wi,t is the current period weight, i.e. the items share in the totalvalue in the current period
)1(,0,
,,
,00
i tii
i tt
t
tt qpQ
PP
)2(1,
0,,0 i
ti
itit p
pwPP
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According to the later formulation, PP0t is derived as follows from
the data in table 1:
Paasche Price Index Formula
Table 3
Year 0
Price Price Value Weight
(000 $/un.) (000 $/un.) (000 $) (w1)(1) (2) (3) (4)=(3)/
(3) (5)=(1)/(2) (6)=(4)*(5)*1 00
Model A 20 40 960 88.9% 0.5 0.44
Model B 10 20 120 11.1% 0.5 0.06
1,080 0.50
*
Price
relatives (PR)w1*PR
Year 1
The ratio of any aggregate in current prices to the aggregate inconstant prices yields an implicit Paasche price deflator
Price measures for the main national accounts aggregates are(always) derived implicitly
0
t= w 1 = . x .
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The change in the current price value of car production from year 0 toyear 1 in our example can be expressed algebraically as:
Relationship between Value, Volume and Price Indexes
Multiplying and dividing by ipi,0qi,t gives:
Value index = Laspeyres Volume index * Paasche Price index / 100
i iii titit qpqpV
V0,0,,,
0
)(*)( ,0,,,0,0,,0,0
i tiii titii iii tiit qpqpqpqpV
V
The volume and price effects of value change are multiplicative When VtandV0are known and PP0t is available the Laspeyres
volume index can be derived indirectly from above formula aprocess called price deflation
200
0
120
0
240
0
ttt
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Another set of volume and price indices may be obtained starting from thechange in the current price value of car production from year 0 to year 1:
Relationship between Value, Volume and Price Indexes
Multiplying and dividing by ipi,tqi,0 gives:
orValue index = Paasche Volume index * Laspeyres Price index / 100
i iii titit qpqpV
V0,0,,,
0
)(*)( 0,,,,0,0,,,0
i itii titii iii oitit qpqpqpqpV
V
PQ0tcan be obtained by inflating the base period values using the oftenavailable LP0t and then dividing the current price value by this amount.
100/100
120
0
200
0
240
0
ttt PQLPVV
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Reference, Base and Weight periods
Reference period (comparison period): The period in an indexnumber time series which is taken as = 100
Base period (pricing period)
Price base period: The period whose prices are used as denominators incalculating price relatives Pt/ P0(0is the price base period)
Quantity base period: The period whose quantities are used as denominators incalculating quantity relatives Qt/ Q0(0is the price base period)
Weight period: The period from which the weights are taken
Equal to the base period for a (fixed-base) Laspeyres index (wo) and to the currentperiod for a (fixed-base) Paasche index (wt)
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Fixed Base or Chain Index
When a fixed base Laspeyres is used over a long run of periods,the wei hts become ro ressivel out of date and irrelevant. Thisimplies that the base period has to be updated sooner or later andthe new index linked to the old. Chaining is simply the limitingcase in which the weights are updated each period.
Fixed base index (Laspeyres)compares between periods 0 and t.
with 1, then 1 with 2,..,then t-1 with t.
Any index number formula can be used for the links.
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ChainingA chain index between periods 0 and t is path dependant- it
depends not just on the prices and quantities in 0 andt, but alsoon the rices and uantities in all the intervenin laces.
If the path is fairly smooth then the additional price and quantityinformation will tend to lead to a better measure of the overallchange. It will tend to reduce spread between Laspeyres andPaasche.
quantity information may increase the index number spread and
lead to drift, e.g. if the prices in 0 are the same as those in t, achain Laspeyres index may exceed 100 if the path is not smooth.
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Change of Reference Period or Re-Referencing
Example of re-referencing:
Table 6
2000 2005 2010 2011
Index
(reference period 2000=100)
Growth rate
(percent)
New Index 83.3 91.7 100 108.3
(reference period 2010=100) (100/120) (110/120) (120/120) (130/120)
Growth rate
100 110 120 130
10.0 9.1 8.3
Note:
Growth rate remains the same
Re-referencing shifts focus to new reference year
Values of the other periods are now compared with the value in this year
(percent)
. . .
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Change of Base Year. Effect on growth rates.An example:Table 7
2000 2005 2010 2011 2000-05 2005-10 2010-11
Values in current prices
Growth rate (percent)
Price 5 10 20 22 100.0 100.0 10.0
Quantity 4 5 6 7 25.0 20.0 16.7
Value 20 50 120 154 150.0 140.0 28.3
Mutton
Price 15 10 5 4 -33.3 -50.0 -20.0
Quantity 11 10 8 7 -9.1 -20.0 -12.5
Value 165 100 40 28 -39.4 -60.0 -30.0
TOTAL
Value 185 150 160 182 -18.9 6.7 13.8
Values in constant prices of 2 000
Wool 20 25 30 35 25.0 20.0 16.7
Mutton 165 150 120 105 -9.1 -20.0 -12.5
TOTAL 185 175 150 140 -5.4 -14.3 -6.7
Values in constant prices of 2 005
Wool 50 60 70 20.0 16.7 Mutton 100 80 70 - 20 .0 - 12 .5
TOTAL 150 140 140 -6.7 0.0
Values in constant prices of 2 010
Wool 120 140 16.7
Mutton 40 35 -12.5
TOTAL 160 175 9.4IMF Statistics Department JVI/QNA/L11 : 21
Why Change of Base Year
Because of structural changes in:
Production
Consumption
Relative prices, and because of
appearance of new products/ disappearance of old and qualityimprovements so that goods and services are no longer comparablebetween periods that may be too far apart.
For example: Norways GDP growth rate in 1989 was 0.6 percent in 1984rices and 5 ercent in 1988 rices because
Price of crude oil jumped during 1984-1988 (Norway is an oil producer andexporter)
Oil production increased significantly (by 25 percent) during 1988-1989
Consequences:
How to derive continuous time series of index numbers from series ofindex numbers with fixed bases?
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Volume Measures and Measures in Real Terms
I. Changes in the values of flows or stocks at current prices, whichhave quantity and price dimensions, can be decomposed into:
changes in their prices; and
changes in their volumes
Measured at constant prices (prices which prevailed in the past), thevalues of such flows and stocks are said to be in volumeterms.
II. Flows or stocks, which do not have quantity and pricedimensions, are measured in real terms (at constant purchasing
power) by
deflating by taking into account the change in the prices of somerelevant basket of goods and services or assets, or the change in thegeneral price level.IMF Statistics Department JVI/QNA/L11 : 23
How are Price and Volume Measures
obtained for GDP?Through price and volume measures for its components:
From the production approach:
for value added by industry PLUS
for taxes less subsidies on products
From the expenditure approach:
for government final consumption expenditure PLUS
or ouse o s ina consumption expen iture PLUSfor NPISHs final consumption expenditure PLUS
for capital formation (and changes in invent.) PLUS
for exports MINUS
for imports
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Price and Volume Measures
for Gross Value AddedI. Double deflation double extrapolation
Gross value added is derived as the difference between the output andintermediate consumption at constant prices both obtained separately
II. Single extrapolation
Gross value added is extrapolated using output data
Gross value added is extrapolated using employment data
III. S ng e e at on
Gross value added is deflated using output deflator
Gross value added is deflated using the wage index Gross value added is deflated using a general measure of inflation like CPI
(does not result in a volume measure but one of real income)
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Double Deflation
Since gross value added in current prices is derived indirectly as the residualitem in the production account
It has no observable flows of oods and services as counter art and
No quantity and price components. Hence,
It may not be revalued in the prices of a base year
Gross value added at constant prices is obtained as
the difference between the output and intermediate inputs at constant prices - aprocess called double deflation
Double deflation means output and intermediate consumption in currentprices
are deflated using an appropriate (Paasche type) price index, or extrapolated from base year values by an appropriate Laspeyres-type volume index
Double deflation is sensitive to error especially
when input data are not accurate, and/or
value added is a small proportion of output
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Double Deflation
Algebraically, value added at constant prices of the base year may be written as
tt qPQP 00where P0QtandP0qt are revaluations of outputs and intermediate inputs inthe prices of the base year 0
The deflation of current price gross value added data using the followingPaasche-type price index
yields the same results as double deflation
ttttttt qPQPqPQPPPVA 000
The Laspeyres-type volume index for gross value added is given by
Note: When chain indices are compiled for output, intermediate consumption and valueadded, they are not additively consistent (Look at Handout for Double Deflation)
ttttt qPQPqPQPLQVA 00000
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Single Extrapolation/Deflation
As approximation of double deflation, the volume index for gross outputmay be used to extrapolate the volume of gross value added when
input data are not accurate
The underlying assumption is that
input-output coefficients are fixed so that price measures for intermediateconsumption are derived implicitly
Alternatively, the Paasche price deflator for output may be used to deflate
the value added data in current prices
The underlying assumption is that
output and input prices do not diverge significantly
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Illustration of Single Indicator MethodsPrimary data
Output at current Intermediate Value added current
prices consump on curren prices
(1) (2) (3)=(1)-(2)
2000 3,200 2,400 800
2001 2,940 2,100 840
2001 3,680 2,700 980
Date
Output at constant
2000 prices
Paasche price deflator
for outputOutput volume index
(4) (5)=(1)/(4)*100 (6)=(4) / 3200 * 100
2000 3,200 100.0 100.0
2001 3,000 98.0 93.8
2002 3,100 118.7 96.9
Date
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Illustration of Single Indicator Methods
Single extrapolation
Laspeyres volume Value added constant
index output 2000 prices
(6) (7)=800*(6) / 100
2000 800 100.0 800*1.000 = 800.0
2001 ...... 93.8 800*0.938 = 750.0
2002 ...... 96.9 800*0.969 = 775.0
Single deflation
Date
Paasche price deflatorfor output
Value added currentprices
Value added constant2000 prices
(5) (13)=(1)-(2) (14)=(13)/(5) * 100
2000 100.0 800.0 800/1.000 = 800.0
2001 98.0 840.0 840/0.980 = 857.1
2002 118.7 980.0 980/1.187 = 825.5
Date
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Intermediate Inputs as an Indicator
Normally, the volume index for output is preferred to one based on inputs,which has greater bias because
num er an var e y o ou pu s s sma er an e num er o n erme a egoods and services consumed in the production process
commodity composition of inputs is more variable over time
Volume index for inputs may be used as a single indicator for value added inexceptional cases:
for example, in construction and capital goods industries where the output
for example, in construction, ship-building, etc. where production period isvery long compared with accounting year
Even in these cases, input data are usually employed in combination withemployment indicators
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Employment as an Indicator
Volume index for inputs of labor services is an alternative to the volumeindex for intermediate inputs
Indices easier to compile are quantity indices of time spent at work by labor,weighted by hourly wages paid to different kinds of workers, and thus takeinto account
changes in the total number of hours worked
changes in the composition of labor force
In practice, it is common to find numbers employed (employment) as
in icators o c anges in rea va ue a e in government services (since services are labor-intensive and output is
difficult to measure - non-market services are without prices and physicalcharacteristics are difficult to quantify)
in financial, business, entertainment services (because of practicaldifficulties of quantifying their output)
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Illustration of Employment plus other Inputs as IndicatorsPart 1: Construction of compound price and volume indices of inputs
Intermediate consumption and compensation of employees at constant prices
Intermediate CompensationTotal input at
Dateconsump on
constant 2000
rices
Hours workedages per our
2000
o empoyees a
constant 2000
rices
constant 2000
prices
oume n ex
total input
(1) (2) (3) (4)=(2)*6 (5)=(1)+(4) (6)=(5)/3060*100
2000 2,400.0 110.0 6.0 660.0 3,060.0 100.0
2001 2,282.6 102.0 6.2 612.0 2,894.6 94.6
2002 2,647.1 107.8 6.5 646.8 3,293.9 107.6
Total input at current prices - compound price deflator
Date
Intermediate
consumption
current prices
Compensation
of employees at
current prices
Total input at
current prices
Price deflator
total input
(7) (8)=(2)*(3) (9)=(7)+(8) (10=(9)/(5) *100
2000 2,400.0 660.0 3,060.0 100.0
2001 2,100.0 632.4 2,732.4 94.4
2002 2,700.0 700.7 3,400.7 103.2
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Illustration of Employment plus other Inputs as indicatorsPart 2: Output and value added at constant prices
Input based deflation
Output at Implicit price
Dateu pu a
current pricesconstant 2000
prices
a ue a e a
constant prices
a ue a e a
current pricesdeflator value
added
(11) (12)=(11)/(10)*100 (13)=(12)-(1) (14)=(11)-(7) (14)/(13)*100
2000 3,200.0 3,200.0 800.0 800.0 100.0
2001 2,940.0 3,114.5 831.9 840.0 101.0
2002 3,680.0 3,564.4 917.3 980.0 106.8
Input based volume extrapolation
Note: Using the dataon wages per hourto deflate valueadded directly givesa different result:840/(6.2/6) = 812.9980/(6.5/6) = 904.6
DateVolume index
total input
Output at
constant 2000
prices
Value added at
constant prices
Implicit price
deflator value
added
(6) (15)=(6)*3200/100 (16)=(15)-(1) (14)/(16)
2000 100.0 3,200.0 800.0 100.0
2001 94.6 3,027.0 744.4 112.8
2002 107.6 3,444.6 797.5 122.9
Note: Using the data onemployment to extrapolatevalue added directly givesdifferent result:800 * (102/110) = 741.8800 * (107.5/110) = 781.8
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Illustration of Double Deflation MethodDouble Deflation Example
2003
Current prices Constant (2000) pricesPrice indexes
GO IC GVA PPI ICI GO IC GVA
(000 $) (000 $) (000 $) (2000=100) (2000=100) (000 $) (000 $) (000 $)
(1) (2) (3)=(1)-(2) (4) (5) (6)=(1)/(4)*100 (7)/(5)*100 (8)=(6)-(7)
Mining 7,300.0 3,800.0 3,500.0 210.0 215.0 3,476.2 1,767.4 1,708.7
Manufacturing 12,800.0 6,300.0 6,500.0 185.0 206.0 6,918.9 3,058.3 3,860.7
Total 20,100.0 10,100.0 10,000.0 ---- ---- 10,395.1 4,825.7 5,569.4
2000
Current
prices
2003
GVAGVA volume
index
GVA implicit
deflator
(000 $) (2000=100) (2000=100)
(9)(10)=(8)/(9)*10
0
(11)=(3)/(8)*10
0
Mining 1,735.0 98.5 204.8
Manufacturing 3,680.0 104.9 168.4
Total 5,415.0 102.9 179.6
GO: Gross OutputIC: Intermediate ConsumptionGVA: Gross Value AddedPPI: Producer Price IndexICI: Intermediate Consumption price Index
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GDP by Expenditure Categories
GDP at constant prices is derived as the sum of expenditurecomponents at constant prices
Expenditure components of GDP are aggregates of transactions thatcan be compiled by observing and recording actual transactions
Value of these transactions can be factored into their own prices andquantities
Better measures of price and volume for GDP conceptually may be
obtained from the expenditure approach (CPI for HC, export and importprices)
Commonly, the deflation of current values is used to derive the dataat constant prices for most of expenditure items, although extrapolationby volume index is also used
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