Index numbers Flashcards
Index numbers in theory?
You have information about prices and quantities. How do you combine these to give an overall index of prices?
Problem: prices are in different units. £5 per main course meal; £1.30 per litre of petrol; £2 for one hours car parking. How do you combine these prices? Economy : Real output. You produce more cars, wheat, musicals and haircuts. How do you combine them?
Same units no problem: age, height or weight. Simply take average of sample or population. Average age of British (UK) person was 40 years in 2015: in 1974 it was 34 years.
Index approach: you look at each item, look at growth or price relative (how much at time t relative to base year). Combine the price relatives (some sort of average). Growth rates do not have units
You know quantities and prices?
You compare 2 periods and know n prices for each period t
base period t = 0
Laspeyres - use initial quantities to weight price pit (most used
Paasche - uses current period t quantities to weight prices pit
Lowe - use weights from past (before initial period)
What if we don’t know quantities?
Maybe expenditures
CPI as a Laspeyres like index?
The data - prices of different G/S at different times Pit
Transform data into price relatives (units don’t matter)
- price of goodi at time 0 is the reference point and we express the p at time t in terms of its value relative to reference
- price relative is unity R=1 if price is unchanged, less than 1 if p falls, greater if rises
- inflation data constructed by applying index formulae to p relatives not PL itself
- Fundamental equation – from PL to price relatives
CPI is Laspeyres-like, because the information on weights is old: as we shall see, the current weights are based on the previous year’s expenditure shares
- P relatives made w respect to Jan as base – bc expenditure shares are not Jan, it is not a true Laspeyres index, technically a Lowe index
CPI not really a big issue as we only care looking at prices (not bothered about quantity)
Laspeyres v Paasche?
- Laspeyres index is expenditure weighted S average of price relative R
- Irvin Fisher realised you could use expenditure weights
- Laspeyres, price index is a weighted arithmetic mean of price relatives
- Paasche price index is weighted harmonic mean of p relatives
Index number problem?
can we combine p and q data over time to obtain measure of relative changes in PL and level of output
Arithmetic V Geometic?
Arithmetic - Carli
Geometric - Jevons
Suppose we don’t have expenditure weights, can use simple averages
Iso-Jevons are rectangular hyperbolae.
Iso-Carli are straight lines.
Along 45 degree line, Jevons equals Carli.
As you move along Iso-Carli, Jevons falls.
RPI uses arithmetic, CPI geometric
Formula for Carli and Jevons?
Carli = (R1 + R2) / 2
Jevons = SQRT(R1.R2)
Summary?
- Index number enables you to combine objects measured in different units. ForInflation, use growth rates (price relatives) which do not depend on the units
- Classic measures: Laspeyres and Paasche Indices. Need both price and quantity information.
- But, can weight price relatives by expenditure. Expenditure is a common measure for all types of goods and services.
a. Laspeyres Index: expenditure weighted mean of price relatives.
b. Paasche Index: weighted harmonic mean of price relatives. - Unweighted means: Carli (arithmetic) and Jevons (geometric). Equal if all price relativesequal, arithmetic exceeds geometric when price relatives differ
Economists and price indices?
- economists look at price index as function of PL (not relatives) and as cost of attaining add utility
- start from primal problem, max u st budget constraint
- Cobb douglas gives geometric index in terms of PL
- so for cobb douglas preferences, geometric index of relatives gives p relative for index
Arithmetic corresponds to Leontief preferences
For Leontief preferences, relative of index not expressed in terms of relatives, but rather Dutot index (average each period and compare periods)
Cobb-douglas and Laspeyres are very different (log-linear v linear)
Axiomatic approach to index numbers?
- by statisticians
- Irvin Fisher
- specify some desirable properties of indices and see how many of these index numbers satisfy
- the best or ideal index put forward was the Fisher index, geometric mean of Paasche and Laspeyres
Carli fails
Jevons and Dutot pass
The fact carli fails is one of the reasons for why international bodies decided against Carli being used in CPI
Axiomatic approach - ‘obvious ones’?
Index should be positive, increasing in prices and equal to unity if all prices do not change (all relatives equal to unity)
Axiomatic approach - ‘less obvious’
- time reversal, if you reverse temporal order – switch time superscripts – then new index is inverse of the other
- to test, simply invert p relatives to see if index is inverted
Why does Carli fail - axiomatic?
- It is just like %, 10% increase from 100 is 110, 10% decrease from 110 is 99
- Carli takes gross % Ri from base period and averages
- if you take gross % from first period 1/Ri and average, you don’t get the inverse of the first index
- The axiomatic approach has issues – what axioms do we want to have, if there are more than one index meeting all axioms we like, how do we choose between
The formula effect?
- RPI and CPI measured differently, thus the weights of various items vary
- also deploy diff averaging techniques, the resulting difference is the formula effect
- if we compare any 2 months, Jevons always less than Carli
- Sequence of months and relative p defined w respect to first month Jan, cumulative inflation will be greater for Carli than Jevons
- but if we compare 2 months within sequence, Carli monthly may be less than Jevons
Month on month J can be greater than C because gap driven by dispersion of price relatives, if this goes down can outweigh formula effect
During 239 months 1997-2017 - 75 months in which RPIJ > RPI, so changes in relative p can dominate formula effect
