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Index Numbers
Facultad de Comercio, Turismo y Ciencias Sociales Jovellanos
Index Numbers
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Introduction
Simple indexes
Composite indexes
Index Numbers
Intro. SimpleComp.
What’s an index number
I (^) Example:
Total number of travelers and overnight stays. January to August 2000 1999 x 00 − x 99 x 00 /x 99 No. travelers 40,612,613 40,016,747 595,866 1. Overnight stays 160,320,642 158,511,595 1,809,407 1.
I (^) An index number is a number indicating change in magnitude (wrt time or space), as of price, wage, employment, or production shifts, relative to the magnitude at a specified point.
Intro. SimpleComp.
Simple index and rate of change
Given two values of a quantity, in time, either in the space, we define: I (^) Simple index (´ındice simple):
I 0 t =
xt x 0
(It is usually expressed as a percentage, x xt 0 100%) I (^) Rate of change -ROC- (tasa de variaci´on):
ROC 0 t = xt − x 0 x 0
(It is usually expressed as a percentage, xt^ − x 0 x 0 100%)
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Base and current period
I (^) Base period or reference period (per´ıodo base o de referencia): A specific period of time which is used in the comparison of data, in order to have a frame of reference. Nomenclature: t 0 or 0. I (^) Comparison period or Current period (per´ıodo actual o corriente): The period for which the index is computed. Nomenclature: t. Example: Workforce (annual Spanish average, thousands people)
Workforce 1988 (base p.) 1997 (current p.) I 8897 · 100% ROC Workers 11,773 12,765 108.42% 8.42% Unemployed 2,848 3,356 117.84% 17.84% Total 14,621 16,121 110.26% 10.26%
Source: I.N.E. Index Numbers
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Simple spatial indices vs simple time indices: example
Sydney Melbourne Brisbane Adelaide Perth Food 100.0 99.9 97.9 95.2 99. Alcohol and tobacco 100.0 98.9 98.7 100.7 101. Clothing and footwear 100.0 98.1 103.8 99.5 99. Household furnishings, supplies and services 100.0 95.5 94.3 95.4 96. Health 100.0 108.9 98.7 103.2 100. Transportation 100.0 93.8 91.8 91.6 93. Communication 100.0 99.9 99.9 100.1 100. Recreation 100.0 97.1 97.1 96.8 95. Education 100.0 99.6 96.7 104.7 98.
Index Numbers
Intro. SimpleComp.
Properties of simple indices
I (^) Identity.- I (^) tt = 1 I (^) Circularity.- I (^) tt 01 · I (^) tt 12 · I (^) tt 20 = 1 I (^) Inversion.- I (^) tt 12 =
I (^) tt 21 According to the above properties, we obtain:
I (^) tt 02 =
I (^) tt 20
= I (^) tt 01 · I (^) tt 12
and therefore we can change a base period as follows:
I (^) tt 12 =
I (^) tt 02 I (^) tt 01
Intro. SimpleComp.
Chain indices
I (^) Chain indices are obtained by linking indices for consecutive periods. I (^) The index of any given period is related to the value of its immediately preceding period. (Distinct from the fixed-base index, where the value of every period in a time series is directly related to the same value of one fixed base period.) I (^) Simple index numbers satisfy the following property:
I (^) tt 0 n = I (^) tt 01 · I (^) tt 12 ·... · I (^) ttnn− 1
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Price indices
I (^) Laspeyres price index: LP =
∑ ∑^ i^ pit^ qi^0 i pi^0 qi^0
, where pit (resp. pi 0 ): price of item i at time t(resp. at time 0); qi 0 : quantity of item i at time 0. I (^) Alternative equivalent formula (weighted mean):
LP =
∑ i (^) ∑wi^ ·^ (Ii^ )t 0 i wi
,
where wi = pi 0 · qi 0 , and (Ii )t 0 = (^) pp 0 itt. I (^) A similar formula is used to calculate Consumer Price Indices (IPC in Spain, RPI in UK, HICP in Europe, etc). CPI are: I (^) published monthly I (^) used as basis for measuring price inflation I (^) it considers products in the “shopping cart” I (^) qi 0 are updated periodically I (^) Alternative methods to compute composite indices: Paasche, etc. Index Numbers
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Example of calculation of Laspeyres price index
item pi 0 pit qi 0 qit wi = pi 0 · qi 0 (Ii )t 0 = (^) ppiti 0 1 0.9 1.0 11 10 0. 9 · 11 = 9. 9 10 ..^09 = 1. 11
2 0.8 0.9 15 16 0. 8 · 15 = 12 00 ..^98 = 1. 13
3 1.0 0.7 10 12 1. 0 · 10 = 10 01 ..^70 = 0. 7
LP =
Index Numbers
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Applications of CPI: Inflation and deflation
I (^) Inflation.- from time 0 (base) to time t.- (CPI 0 t − 1) · 100%. Example: If CPI 9293 = 1.10, then prices increased (inflated) by 10% from 92 to 93. I (^) Deflation.- When we want to compare salaries, pensions, rental expenses in different periods, we need to “deflate” them. (Covert them into constant units.) I (^) Example: I (^) A company has invested $23000 Eur in R&D in 2005. I (^) In 1995, it had invested $21000 Eur in R&D. I (^) Prices have increased by 1.28% from 1995 to 2005. I (^) Does the company really increased its investments on R&D? I (^) An investment of $23000 in 2005 would be equivalent to an investment of $17968.75 in 1995. If we use “constant dollars”, their investment in R&D has decreased. I (^) Deflation of price x in period t wrt base period: (^) CPIx t
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Properties of Laspeyres price indices
I (^) In general, price indices calculated by means of Laspeyres formula do not satisfy circularity and inversion. I (^) It can be easily checked that they hold when qit = qi 0 , ∀ i. I (^) In practice, qit is not very far from qi 0. I (^) Therefore, we assume that CPI satisfy the same properties as simple indices. I (^) Examples: I (^) CPI (^) tt 12 = CPI^
t 2 t 0 CPI (^) tt 01 I (^) Deflation: year t 2 constant (currency) unit = year t 1 constant unit · CPI (^) tt 01 CPI (^) tt 02
