Lecture 1 Introduction and Historical Notes

Transcription

1 Lecture and Historical Notes s 5415 Index Number Theory Winter 2019

2 Outline Test Test 7

3 Required Readings Test Diewert, W.E. (1993) Early History of Price Index Research, in Diewert and Nakamura (eds), Essays in Index Number Theory, Volume 1, Amsterdam: North-Holland, Chapter 2. Boskin, Michael (1996) Prisoners of Faulty Statistics, Wall Street Journal, December 5, Kadlec, Daniel (1998) Measuring the New CPI, Time, March 9, 1998, Page 40. Surowiecki, James (2004) Hail to the Geek, The New Yorker, April 19, Many of the references can be found on the reading list.

4 Test Data and Information We live in the information age. Big data are everywhere on the Internet. Data are not information. Rain drops are data, but we need a proper instrument to measure the amount of rain falls. players need timely and accurate information to make decisions. 1 Households make decisions on spending and saving. 2 Business people form marketing, production, financial, and investment strategies. 3 Governments decide how much to tax and to spend, central banks set interest rates to stabilize the economy. 4 The U.N., the World Bank, and the IMF have their own mandates to maintain world peace, to promote human welfare, and to raise the standards of living in the developing world.

5 Test Inside the Ivory Towers Economists are social scientists. We observed patterns, propose theory and models, and test our knowledge with what we observe. Economists are also policy analysts. We need accurate measurement of the state of the economy to make policy recommendations. We also perform economic forecasting to help government and business to plan ahead. The accuracy of our forecast depends squarely on our models and information in forms of index numbers. Many economists think that measurement problems have been largely solved. On the contrary, most techniques are developed for the industrial economy. Many new problems arise in the service and digital economy we are now in. This course is about aggregating data into useful information.

6 Test What is an Index Number? An index number is a measure of change. The changes can be about prices, incomes, economic outputs, productivities, etc. Popular and important economic statistics that are index numbers: 1 Consumer price index (CPI) 2 Producer price index (PPI) 3 Gross domestic product (GDP) 4 Labour productivity 5 Multi-factor productivity (MFP) 6 Dow Jones index A bilateral index is a measure of change between two time periods or difference between two economies. indices are measures of changes over time or differences across many economies.

7 Standard of Living or GDP? Test

8 Labour Productivity Test

9 Which Price Index? Test There are many ways to calculate an index number. The methodology depends on the purpose, target, usage, data availability, and the theory behind the statistic in question. Different approaches often give different results. The choice has important policy implications.

10 Test

11 Test A Simple Example A bilateral index number is expressed in ratio, not in difference. Let s say we want to measure the price change of gala apple at Superstore in Thunder Bay, Ontario between December 15, 2016 and January 15, The observed prices are p 0 = $4.50 and p 1 = $5.22 per kg respectively. The price index is defined as P = p1 p 0 = = It means that the inflation rate of gala apple at Superstore during that period is 16%. The first time period (p 0 ) is called the base period and the second (p 1 ) is called the comparison period.

12 Test A Little More Complicated There is no index number problem in the above example because there is only one good. Suppose we gather price data from Superstore and Safeway for the price of one bottle of Pepsi in two time periods: Location p 0 i p 1 i Superstore Safeway Label Superstore 1 and Safeway 2, their price ratios are respectively p1 1 p1 0 = = 2.00, p 1 2 p 0 2 = = 0.50.

13 Test Three Elementary Price 1 The Carli index is the arithmetic mean of the price ratios: P C = M i=1 1 pi 1 M pi 0 = 1 2 ( ) = The Jevons index is the geometric mean of the price ratios: P J = M ( p 1 ) 1/M i = i=1 p 0 i ( ) 1/2 = The Dutot index is the ratio of the arithmetic means: P D = M i=1 1 M p1 i M i=1 1 M p0 i = =

14 Test es in Index Numbers We only measure the price change on a single homogeneous product in the last example. These are called elementary price indices. In general we need to aggregate the changes of different goods and services to get the overall inflation rate. For this purpose, price information alone is not enough. We cannot simply add the price of apple to the price of beef, or computers. Prices have to be weighted by corresponding quantities. The choice of the index number formula depends on the approaches we consider. The index number problem is effectively Karl Marx s problem of the commensurability of disparate goods and the conservation of value in exchange.

15 Test The Often called the tabular standard or the Lowe index (after Joseph Lowe, 1823). There are N goods and services. Prices and quantities are well-defined. Two time periods: 0 and 1. Price vectors: p 0 = (p 0 1, p 0 2,..., p 0 N ), p 1 = (p 1 1, p 1 2,..., p 1 N ). The prices are weighted by a constant basket of goods and services, represented by a quantity vector: x = (x 1, x 2,..., x N ).

16 Test The Cost-of-Goods Index The fixed basket (Lowe) price index is defined as P FB (p 0, p 1, x) = p1 x N p 0 x = i=1 p1 i x i N i=1 p0 i x. (1) i The denominator in definition (1), p 0 x, is the cost (or expenditure) to purchase the fixed basket in period 0, while the numerator is that in period 1. For this reason a fixed basket index is often called a cost-of-goods index. Where do we get the representative basket x? William Fleetwood (1460): 5 quarter of wheat, 4 hogsheads of beer, and 6 yards of cloth. Legislature of Massachusetts (1780): 5 bushels of corn, 68 and 4/7 pounds of beef, 10 pounds of sheep s wool, and 16 pounds of sole leather.

17 Test The Constant Basket Laspeyres (1871) proposes the use of the observed quantity vector x 0 in the base period as the fixed basket. Equation (1) becomes the Laspeyres price index: P L (p 0, p 1, x 0, x 1 ) = p1 x 0 N p 0 x 0 = i=1 p1 i x i 0 N i=1 p0 i x 0 i. (2) Paasche (1874) instead recommends the comparison period quantity vector x 1. The Paasche price index is defined as P P (p 0, p 1, x 0, x 1 ) = p1 x 1 p 0 x 1 = N i=1 p1 i x 1 i N i=1 p0 i x 1 i. (3) There is no reasons to pick either x 0 or x 1.

18 Test Some Kinds of Averages Other economists recommend taking the average of periods 0 and 1: Irving Fisher (1922): P F (p 0, p 1, x 0, x 1 ) = (P L P P ) 1/2 = C.M. Walsh (1901): P W (p 0, p 1, x 0, x 1 ) = ( p1 x 0 p 1 x 1 ) 1/2 p 0 x 0 p 0 x 1. N ( i=1 x 0 i xi 1 ) 1/2 p 1 i N ( ) i=1 x 0 i xi 1 1/2. p 0 i F.Y. Edgeworth (1887) and Alfred Marshall (1887): P EM (p 0, p 1, x 0, x 1 ) = N i=1 1 2 (x 0 i N i=1 1 2 (x 0 i + xi 1)p1 i. + xi 1)p0 i

19 Test The Canadian CPI The expenditure share of good i in period t is defined as s t i = pi tx i t N j=1 pt j x. j t (Exercise:) The Laspeyres price index in equation (2) can be written as N ( p 1 ) P L = i si 0. (4) i=1 Instead of using formula (4), the Canadian CPI uses expenditure shares s i from the Survey of Household Spending and updates the numbers every two years. p 0 i The CPI is published every month. Therefore the p 1 i /p0 i ratios are from consecutive months.

20 Annual Consumer Price Index, 2008 to 2018 Test Source(s): Statistics Canada Table

21 CPI Major Components Test Source(s): Statistics Canada Table

22 Provincial CPI Test Source(s): Statistics Canada Table

23 Test The The idea originated from William Jevons in the mid 19th century. The price ratios of goods and services tend to follow a general trend, with random fluctuations for individual items. Depending on the probability distribution of the random term, a particular index formula can be justified. If the random term is normally distributed, then the Carli index should be used. For a log-normal random term, the Jevons index should be used. Using the expenditure shares as the probability distribution of a dollar spent, Henri Theil (1967) justifies the use of the Törnqvist price index: N ( p 1 ) 1 2 (s0 i +s1 i ) P T = i. i=1 p 0 i

24 Test Core Inflation Central banks want to eliminate the noise from monthly CPI and reveal the so-called core inflation of the economy. The information is important for conducting monetary policy. Many central banks have the explicit objective of inflation targeting. Edgeworth (1923) advocates the use of the median price ratio as an estimate of the inflation trend. Stephen Cecchetti (1997) recommends a trimmed method to eliminate data on both tails of the distribution. Also, monthly price indices are too volatile for the purpose of the central banks. Quarterly data perform much better. There is no unifying theory behind the concept of core-inflation. A variety of methods are used in practice.

25 The Test The concept of economic indices should be based on economic theory and evidence. For example, when price of a consumer good goes up, we expect quantity sold goes down. Therefore prices and quantities are not independent variables in the index formulae. The economic approach uses neoclassical theory to derive the characteristics and properties of index numbers. In the process, we assume that consumers maximize utility, producers minimize production costs and maximize profits. On the consumer side, the resulting price index is called a cost-of-living index.

26 Test Neoclassical Theory The consumer is assumed to have a rational preference relation on a consumption set X R N +. The preference relation is represented by a utility function U : X R that is continuous, increasing, and strictly quasi-concave. The expenditure is defined as E(p, u) = min {p x : U(x) u}. x X In the context of production theory, a firm produces a single output y from inputs x R N + with a production function y = f (x). The cost function is C(p, y) = min x {p x : f (x) y}.

27 Cost-of-Living Index Test Suppose that the utility level u represent a certain standard of living. A consumer in period 0 faces market prices p 0. The minimum cost to achieve the standard of living at level u is E(p 0, u). Similarly, the cost to achieve the same standard of living in period 1 with market prices p 1 is E(p 1, u). The Konüs cost of living index from period 0 to period 1 is defined as P K (p 0, p 1, u) = E(p1, u) E(p 0, u). (5)

28 Test Konüs Since the utility function U is ordinal and u is unobservable, the Konüs index is often written as P K (p 0, p 1, x) = E(p1, U(x)) E(p 0, U(x)), where x is a reference basket. The natural choice for the reference basket is the observed base period basket x 0, which gives the Laspeyres-Konüs index, P K (p 0, p 1, x 0 ). Or the comparison basket x 1, which give the Paasche-Konüs index, P K (p 0, p 1, x 1 ). Using the assumptions in neoclassical theory and expenditure minimization, we can obtain some bounds for the Konüs index.

29 Theory of Bounds Test Konüs (1924) shows that there exists a convex combination of basket x = λx 0 + (1 λ)x 1, with 0 λ 1, such that the cost of living index is bounded between the Laspeyres and the Paasche indices. That is, either or P P (p 0, p 1, x 0, x 1 ) P K (p 0, p 1, x ) P L (p 0, p 1, x 0, x 1 ), P L (p 0, p 1, x 0, x 1 ) P K (p 0, p 1, x ) P P (p 0, p 1, x 0, x 1 ).

30 Exact Index Numbers Test Suppose that E(p, u) is the expenditure function derived from a utility function U. An index formula (or function) P(p 0, p 1, x 0, x 1 ) is said to be exact for the utility function U if P(p 0, p 1, x 0, x 1 ) = E(p1, u) E(p 0, u), for some reference utility level u. Therefore, if we assume the utility function of the consumer is of a particular function form, it is possible to find a cost of living index to match that function.

31 Test Example of an Exact Index Suppose that utility is of the perfect complement type { x1 U(x 1,..., x N ) = min, x 2,..., x } N. (6) i α 1 α 2 α N The expenditure function is the linear function E(p, u) = u N α i p i. It can be shown that the Konüs cost of living index corresponding to U is either the Paasche price index or the Laspeyres price index. In other words, by using the Paasche or the Laspeyres price indices, we implicitly assume that the consumer s preference relation is represented by the utility function in equation (6). i=1

32 Test 1 Cobb-Douglas utility function: U(x) = N i=1 x α i i, α i > 0, More Examples N α i = 1. i=1 The exact index is the geometric price index P G (p 0, p 1, x 0, x 1 ) = N ( p 1 ) si i = i=1 p 0 i N ( p 1 ) αi i. i=1 p 0 i 2 Homogeneous quadratic utility function: U(x) = (x Ax) 1/2, A = A T. The exact index is the Fisher price index.

33 Superlative Index Numbers Test In economic theory we assume particular utility functional forms for analytical convenience. In reality we do not know the true functional form. A flexible functional form is capable of approximating an arbitrary linearly homogeneous utility function to the second order. Examples are the homogeneous quadratic function, translog function, normalized quadratic function, etc. Erwin Diewert (1976) defines a superlative price index as an index formula that is exact for a flexible utility function or its dual expenditure function. The Fisher, Walsh, Edgeworth-Marshall, and the Törnqvist price indices are examples of superlative indices.

34 Test Quantity output measures such as GDP are quantity indices. Using the same framework as for price indices, we can define a bilateral quantity index as a function of the price and quantity vectors, Q(p 0, p 1, x 0, x 1 ). In the economic approach to index number, an example is the Allen (1949) quantity index: Q A (x 0, x 1, p) = E(p, U(x 1 )) E(p, U(x 0 )), where p is a reference price vector. Under the neoclassical assumptions, the concepts of bounds, exact index, and superlative index also apply to quantity index numbers.

35 Test The In the (1926) approach, time is modelled as a continuous variable. (See Balk (2005) for a survey.) Prices p i (t) and quantities x i (t) of good i are functions of time. Total expenditure (or value) is therefore v(t) = p(t) x(t) = N i=1 p i(t)x i (t). The rate of change of value at time t is v (t) = N x i p i(t) + i=1 N p i x i (t). i=1 Dividing the above equation by v(t) to get the growth rate of value: v (t) N v(t) = i=1 x i p i (t) v(t) + N i=1 p i x i (t) v(t). (7)

36 Test Aggregate Price and Quantity Consider the identify v(t) = P(t)Q(t), where P(t) and Q(t) are aggregate price and quantity levels at time t. The growth rate of value is then 1 v (t) v(t) = P (t) P(t) + Q (t) Q(t). (8) Comparing equations (7) and (8), we obtain the following two differential equations: P (t) N P(t) = x i p i (t) p(t) x i (t) ; i=1 Q (t) N Q(t) = p i x i (t) p(t) x i (t). (9) i=1 1 Exercise in macroeconomics: Let γ x, γ y, and γ z be the growth rates of X t, Y t, and Z t respectively. Show that if Z t = X ty t, then γ z = γ x + γ y.

37 Test Price and Quantity With a little mathematics from equations (9), the price and quantity indices from period 0 to period 1 are log P Div (0, 1) = log Q Div (0, 1) = N i=1 N i=1 s i (t) d log p i(t) dt s i (t) d log x i(t) dt dt, In practice, the continuous functions p(t) and x(t) must be approximated by the discrete data. Unfortunately, many index formulae can be used to approximate the indices. These include the Paasche, Laspeyres, and the Törnqvist indices. Therefore the approach is a useful analytic tool but not very practical in calculating actual index numbers. dt

38 Test The Test In the test approach, we consider a set of mathematical properties, called axioms or tests, that we want the index number formulae to satisfy. Given a set of tests, if only one formula satisfies all the tests, then call the set of tests a characterization of the index. Sometimes the tests are in conflict with each other, so no index number formula can satisfy all the tests. The tests are not always independent. A subset of the tests may imply another test. Various tests have been proposed by Jevons (1884), Laspeyres (1871), Westergaard (1890), Pierson (1896), Walsh (1901), Fisher (1911),.... Fisher (1922) is the classic reference on the test approach.

39 Test Examples of Tests or Axioms Consider a bilateral price index P(p 0, p 1, x 0, x 1 ) from period 0 to period 1. 1 Identity test: If p 0 = p 1, then P(p 0, p 1, x 0, x 1 ) = 1. 2 Proportionality test: For all α > 0, 3 Circularity test: P(p 0, αp 1, x 0, x 1 ) = αp(p 0, p 1, x 0, x 1 ). P(p 0, p 1, x 0, x 1 )P(p 1, p 2, x 1, x 2 ) = P(p 0, p 2, x 0, x 2 ). 4 Time reversal test: P(p 0, p 1, x 0, x 1 ) = 1 P(p 1, p 0, x 1, x 0 ).

40 Test The Product Test The product identity, called the product test in the test approach, is the fundamental problem in index number theory. We often observe the total values of transaction in a market or economy in two periods: v 0 = p 0 x 0, and v 1 = p 1 x 1. Our objective is to decompose the value ratio v 1 /v 0 into a price change component (P(p 0, p 1, x 0, x 1 )) and a quantity change component (Q(p 0, p 1, x 0, x 1 )). The product test requires that P(p 0, p 1, x 0, x 1 )Q(p 0, p 1, x 0, x 1 ) = v 1 v 0. (10)

41 Test Real GDP Equation (10) means that once we have calculated a price index P(p 0, p 1, x 0, x 1 ), the quantity index is implicitly defined as Q(p 0, p 1, x 0, x 1 ) = 1 P(p 0, p 1, x 0, x 1 ) ( v 1 v 0 ). (11) In the context of the national accounts, v 0 and v 1 are the nominal (or market value) GDP in the two periods. Equation (11) can be written as Q(p 0, p 1, x 0, x 1 )v 0 = v 1 P(p 0, p 1, x 0, x 1 ). The left hand side is period 1 s real GDP in period 0 dollar. The price index P is called the GDP deflator.

42 Real GDP in Canada Test

43 Quantity Theory of Money Test Consider the identity PY = VM, where P is the general price level, Y (or Q) is the real output, V is velocity of money, and M is money supply. Assuming that V is constant and Y has a constant growth rate, then in the long run P is directly proportional to M. Milton Friedman claims that inflation is always and everywhere a monetary phenomenon. Empirical studies of the theory rely on accurate decomposition of nominal output into the price and quantity components. See Wen (2006) for a discussion.

44 Test So far we have looked at approaches in bilateral indices. data are often times series or cross-sectional aggregates. These are multilateral indices. For time series data, there are generally two methods: 1 Fixed base index The price index in period t is a bilateral index compared to a fixed base period 0. Therefore the time series data, starting from period 0, are 1, P(p 0, p 1, x 0, x 1 ), P(p 0, p 2, x 0, x 2 ), P(p 0, p 3, x 0, x 3 ),... 2 Chained index Bilateral indices of consecutive periods are calculated. The price index in period t is the product of all the preceding indices: 1, P(p 0, p 1, x 0, x 1 ), P(p 0, p 1, x 0, x 1 )P(p 1, p 2, x 1, x 2 ), P(p 0, p 1, x 0, x 1 )P(p 1, p 2, x 1, x 2 )P(p 2, p 3, x 2, x 3 ),...

45 Spatial Test Spatial indices are cross-sectional comparison. Examples are cost of living between cities or countries, GDP comparison between countries, etc. Since there is no natural ordering of locations, the fixed base or chain principles do not apply. A further complication is the use of different currencies in different countries. The exchange rate fluctuations present an additional challenge. A popular class of methods to compare international output is the use of purchasing power parities, which calculates common price indices for all countries. Other methods such as the Gini-EKS system and the own share system are based on bilateral quantity indices.

46 A Spatial Index Test