# Chapter 4 Test for index numbers Timereversal test

Chapter 4 Test for index numbers Time-reversal test Dr. A. PHILIP AROKIADOSS Assistant Professor Department of Statistics St. Joseph’s College (Autonomous) Tiruchirappalli-620 002.

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)

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/P 0: 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.

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.

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.

Example (changing a time base) 2001 2002 2003 2004 Number of employees, 000’s 8741 8727 8432 8062 Series with 2001 as 100 99. 8 96. 5 92. 2 The math… Series with 2003 as 100/96. 5 x 100 103. 7 99. 8/96. 5/96. 5 x 100 103. 5 100 92. 2/96. 5 x 100 95. 6

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. ----

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 !!!!!

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.

Some notable price indexes • • • CPI IPPI PCE deflator GDP deflator Productivity indexes

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:

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 then current prices of the said articles.

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.

Some basic index number formulas: Historical overview • 1822: Lowe index or fixed basket index.

Some basic index number formulas: Historical overview • 1863: Jevons (father of index numbers) worked out index numbers 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.

Some basic index number formulas • Laspeyres and Paasche price indexes in response to the need for more precision with regards to the basket.

Laspeyres price index 1834 - 1913

Paasche price index 1851 - 1925

Paasche quantity index

Laspeyres quantity index

Fisher indexes 1867 - 1947

Marshall-Edgeworth price indexes

Value aggregates

Example: Simple index June • • • Granny Smith: Red Delicious: Fuji: Gala: Russets: $0. 72/each $0. 75/each $0. 50/each $0. 75/each $0. 90/each

Example: Simple index (cont’d) July • • • Granny Smith: Red Delicious: Fuji: Gala: Russets: $0. 80/each $0. 85/each $0. 55/each $0. 77/each $0. 90/each

Example: Simple index (cont’d) 0. 72 + 0. 75 + 0. 50 + 0. 75 + 0. 90 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% = 0. 72 = 0. 774

Basic index number theory • Choosing an index number formula: two approaches 1. “AXIOMATIC” approach: 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.

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.

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.

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.

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.

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:

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.

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

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.

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 (P 1, 2 x P 2, 3 = P 1, 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.

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.

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.

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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