Transcript index number

Price Indices: Part 1
MEASUREMENT ECONOMICS
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What is an index number?
The problem of how to construct an index
number is as much of economic theory as
of statistical technique. Frisch (1936)
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What is an index number?
• Definition 1:
An index number of prices shows the average percentage
change of prices from one point of time to another.
The percentage change in the price of a single product from
one time to another is found by dividing its price at time t by its
price in time 0.
Pt/P0: price relative of a commodity in relation to these two
points in time.
An index number of the prices of a collection of products is the
average of their price relatives.
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What is an index number?
• Definition 2:
An index number is limited to the measure of changes in a
magnitude between one situation to another. The two situations
compared are in no way restricted; they may be two time
periods (e.g. CPI), or two situations in a spatial sense (e.g. two
cities or two or more countries - PPPs), or two groups of
individuals (e.g. one and two-person pensioner families).
Since index numbers measure changes, they are expressed
with one selected situation as 100. This is called the “time”
reference base of the series of index numbers.
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What is an index number?
• Definition 2 (cont’d):
In an annual series for example, the reference base
is the year taken with the level of 100 for comparison.
In another year, the index number may be 126. This
shows an increase of 26% over these two years.
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Example (changing a time base)
2001
2002
2003
2004
Number of employees, 000’s
8741
8727
8432
8062
Series with 2001 as 100
100
99.8
96.5
92.2
The math…
Series with 2003 as 100
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100/96.5
x 100
103.7
99.8/96.5 96.5/96.5
x 100
x 100
103.5
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100
92.2/96.5
x 100
95.6
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The index number problem
How exactly should the microeconomic information involving
possibly millions of prices and quantities be aggregated into
a smaller number of price and quantity variables?
----The problem that arises is how to combine the relative
changes in the prices of various commodities into a single
index number that can meaningfully be interpreted as a
measure of the relative change in the general price level.
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The index number problem
An index number reduces all the distinct prices for the
class of goods in question to a single number.
1.Cuts through the noise thus helps in seeing the big picture.
2.Hides potentially important details, some prices may be
increasing but many others can be decreasing.
---Different index numbers, i.e., different forms of averages,
tend to lead to different results, some of them will register
positive changes and others negative, so that different
indices may be moving in different directions !!!!!
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Uses of index numbers
• Broad indexes
– It measures the economy’s price level.
– Changes in the Cost-of-living
– Help in measuring GDP
• Narrow indexes
– Tracks changes in the price of certain products
– Help in guiding investments
– Stumpage fees and other contracts.
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Some notable price indexes
•
•
•
•
•
CPI
IPPI
PCE deflator
GDP deflator
Productivity indexes
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Some basic index number formulas:
Historical overview
• 1738 Dutot compared prices for 24
items from the time of Louis XII and
Louis XIV using the following formula:
P P 
2
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1

n1 pn N
N
2

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1
p
n1 n N
N

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Some basic index number formulas:
Historical overview
• 1747 and 1780: Tabular format by the Colony of Massachusetts
to counter the effects of the depreciation of money. 5 bushels of
corn, sixty-eight Pounds and four-seventh parts of a Pound of
beef, 10 Pounds of sheep’s wool, and sixteen pounds of shoe
leather shall then cost, more or less than one hundred and thirty
pounds current money, at the then current prices of the said
articles.
P P   n1wpn
2
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1
N
2
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
N
1
n
wp
n1
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Some basic index number formulas:
Historical overview
• 1764: Carli in Italy looked at the change in the
prices for grain, wine, and oil between 1500
and 1750 to show the effect of the discovery
of America on the purchasing power of
money.

P P   n1 pn p
2
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1
N
2
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1
n
N
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Some basic index number formulas:
Historical overview
• 1822: Lowe index or fixed basket index.
P P   n1 pn qn
2
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1
N
2
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
N
1
n n
p
q
n1
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Some basic index number formulas:
Historical overview
• 1863: Jevons (father of index numbers) worked out
index numbers for for English prices back to 1782.
He was interested in showing the fall in the value of
gold as a result of the outpouring of gold mines
beginning in 1849.

P P  n1 pn p
2
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1
N
2
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1
n

1N
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Some basic index number formulas
• Laspeyres and Paasche price indexes in
response to the need for more precision with
regards to the basket.
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Laspeyres price index
1834 - 1913
P P   n1 pn q
2
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1
N
2
1
n
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
N
1 1
n n
p
q
n1
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Paasche price index
1851 - 1925
P P   n1 pn qn
2
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1
N
2
2
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
N
1
2
n n
p
q
n1
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Paasche quantity index
Q Q   n1 pn qn
2
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1
N
2
2
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
N
2
1
n
p
q
n
n1
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Laspeyres quantity index
Q Q   n1 p q
2
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1
N
1
2
n n
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
N
1 1
n n
p
q
n1
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Fisher indexes
1867 - 1947
P P  Paasche  Laspeyres
2
1
Q Q  Paasche  Laspeyres
2
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Marshall-Edgeworth price indexes

1 1
P P  n1 pn
qn  qn 2
2
2
1
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N
2


1 1
2
p
q

q
n1 2 n n
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N
1
n

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Value aggregates
V   n1 p q V   n1 pn qn
N
1
1 1
n n
V V   n1 pn qn
2
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1
N
N
2
2
2
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
N
2
1 1
n n
p
q
n1
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2
Example: Simple index
June
•
•
•
•
•
Granny Smith:
Red Delicious:
Fuji:
Gala:
Russets:
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$0.72/each
$0.75/each
$0.50/each
$0.75/each
$0.90/each
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Example: Simple index (cont’d)
July
•
•
•
•
•
Granny Smith:
Red Delicious:
Fuji:
Gala:
Russets:
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$0.80/each
$0.85/each
$0.55/each
$0.77/each
$0.90/each
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Example: Simple index (cont’d)
0.72 + 0.75 + 0.50 + 0.75 + 0.90
=
0.72
=
0.774
5
0.80 + 0.85 + 0.55 + 0.77 + 0.90
5
Dutot = 0.774/0.72 = 0.075 or 7.5%
Carli = 3.87/3.62 = 0.069 or 6.9%
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Basic index number theory
•
Choosing an index number formula: two
approaches
1. “AXIOMATIC” approach: the theoretical
underpinnings of index numbers are built upon
certain postulates (or axioms). Irving Fisher, 1922;
Eichhorn and Voeller (EV), 1983
2. “ECONOMIC THEORETIC” approach: seeks to
define the price or volume indices with reference to
underlying utility or production functions.
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Axiomatic approach
• The axiomatic approach to the selection of an
appropriate index formulation specifies a number of
desirable properties an index formulation should
possess.
• These properties are imbedded in axioms.
• Potential indexes are then evaluated against the
specified properties and the index that passes the
most tests is the preferred one.
• EV define an index as a function of the observed
prices and quantities which satisfies four basic
axioms.
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Before we start, some simple notation
Pij: price index number for year j compared with year i.
Qij: quantity index number for year j compared with year i.
Vij: value index number for year j compared with year i.
Pji: price index number for year i compared with year j.
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Axiomatic (or test) approach
Four basic axioms:
1. Monotonicity: a price index is increased
whenever any of the prices in the current period
are increased or any of the prices in the base
period are lowered.
2. Proportionality: when all prices in the current
period are uniformly greater or lower than those
in the base period by some fixed proportion, the
index should equal that proportion.
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Axiomatic (or test) approach (cont’d)
3. Price dimensionality: the same
proportional change in the unit of currency
in both periods (e.g., from dollars to
euros) does not change the index.
4. Commensurability: a change in the unit
of quantity for any commodity in both
periods (e.g., from pounds to kilos) does
not change the index.
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Axiomatic (or test) approach (cont’d)
•
•
These axioms are described as “basic properties
which are desirable for every price (or quantity)
index”.
Indexes that have these properties automatically
satisfy various tests of the type which Irving Fisher
proposed.
•
•
•
•
Identity test
Weak proportionality test
Mean value test
Here are some others:
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Fisher’s proposed tests
1. Time-reversal: requires that Pij x Pji = 1
2. Factor-reversal: requires that Pij x Qij = Vij
3. Determinateness: requires that Pij and Qij shall be
positive, finite and determinate regardless of the
price and quantity value assumed by an individual
commodity.
4. Commensurability: requires that Pij and Qij be
independent of the scale of measurement of the
prices and quantities of the individual commodities.
5. Proportionality: requires that Pij and Qij are linear
homogeneous functions of the observed prices and
quantities, respectively.
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Eichhorn and Voeller tests
•
•
The number of indices satisfying the four basic
properties are too broad.
Eichhorn and Voeller (EV) suggest adding the
following conditions:
– Product test: The product of a price and a quantity
index should equal the expenditure ratio where the
price and quantity indices do not necessarily have to
have the same form, but must satisfy the previous
four basic axioms of a price index.
– Time-reversal test
– Factor-reversal test
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Eichhorn and Voeller tests
•
More on the product test
–
–
–
–
Weak version of Fisher’s “factor reversal” test
It is extremely important for economic analysis whenever
time-series data are available in current values.
The product test requires that when the change in the
current values is divided by a price index, we should come
out with a recognisable and acceptable quantity index,
even if it has a different form or properties from the price
index.
E.G.: Laspeyres Quantity Index with Paasche Price Index
satisfy the product test but not the factor-reversal test.
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Eichhorn and Voeller tests
•
•
•
•
•
By adding the product test to the four basic axioms,
the number of acceptable indexes is till quite (too)
broad.
But if Fisher’s circular test (P1,2 x P2,3 = P1,3) is
added, then the set of possible indices is “empty”.
Example of an “inconsistency theorem” or “nonexistence theorem”.
Circular test = circularity test = transitivity.
Circularity means that a direct comparison
between periods 1 and 3, should give the same
result as and indirect comparison via period 2.
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Eichhorn and Voeller tests
•
•
•
It can and will be shown that the and index
cannot satisfy the proportionality test and at
the same time satisfy the transitivity
condition.
Which of the conditions should be relaxed in
order to be able to define an index?
More on this later.
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Economic theoretic approach
• Remember that with the axiomatic
approach, prices and quantities were treated
as separate independent variables.
– Two price vectors and two quantity vectors
• With the economic approach, the quantities
are a function of the prices.
– Two price vectors plus a functional relationship
connecting the prices to the quantities in periods 1
and 2 respectively.
– Parameters of the function are unknown hence
economic theoretic indices cannot be estimated.
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Economic theoretic approach
• Two main functions relating Qs to Ps:
– Utility functions
– Production functions
– Or the generic: “aggregator function”
• The classic example of an economic theoretic
index is the Cost-of-living-index (COLI).
• COLI: “the ratio of the minimum expenditures
required to attain a particular indifference
curve (level of welfare) under two price
regimes”.
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END
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