Chain indices of the cost of living and the path-dependence problem: an empirical solution Nicholas Oulton Centre for Economic Performance, London School of Economics Presented to the 2008 World Congress on National Accounts and Economic Performance Measures for Nations, Washington, D.C., May, 2008

SUMMARY (1) The true cost-of-living index, or Konüs price index, is the Gold Standard. It can in theory be estimated if we knew the parameters of the consumer’s expenditure function. This has been done but only at a very aggregate level, eg 5 commodities (food, shelter, clothing, etc, as in Banks et al. (1997)). But real world price indices use hundreds of commodities. So the econometric approach seems impossible at this level --- too many parameters and not enough data. Nowadays real world price indices are usually chain indices. Chaining reduces substitution bias but may increase “path- dependence” bias (failure of the circularity test). Path-dependence would not be a problem if all income elasticities were equal to one. But this is very implausible (consider Engel’s Law). 2

SUMMARY (2) Chain indices can be thought of as discrete approximations to Divisia indices. Divisia and Konüs price indices are closely related: the Divisia uses actual budget shares, the Konüs uses compensated shares (ie hypothetical shares where utility is held constant) [Balk, 2005]. This paper proposes a new method of estimating the Konüs price index econometrically, for an arbitrarily large number of commodities. I derive an equation relating actual and compensated shares for the Quadratic Almost Ideal Demand System (QAIDS). This equation shows that to derive the compensated shares from the actual ones we need to know only 2 parameters per commodity. I apply this method to estimate Konüs price indices for 70 products covering the UK’s Retail Prices Index. 3

Divisia indices are generally path-dependent (and chain indices may fail the circularity test): 4 0T 1 Path A Path B p1,p2p1,p2 time

Path-independence theorem A Divisia price index is path-independent if and only if the utility function is homothetic, ie all income elasticities are equal to one [Hulten, 1973; Balk, 2005]. A chain index number passes the circularity test if and only if the utility function is homothetic, ie all income elasticities are equal to one [Samuelson and Swamy, 1974]. But this restriction is decisively rejected by empirical studies of demand, eg Engel’s Law! 5

The true cost-of-living (Konüs) index Definition The ratio of the minimum cost of buying the reference utility level at the prices of time t to the minimum cost of buying the same utility level at the prices of time 0: 6

Why bother with index numbers at all? Why not just estimate the consumer’s expenditure function using some flexible functional form? Answer: in a flexible functional form the number of parameters is approximately proportional to N 2 /2 where N is the number of products. The UK’s Retail Prices Index has 650 “items”, so we would need more than 325 years of data on each product(!). 7

Dilemma Any chain index number is likely to be flawed by path-dependence. But (apparently) we can’t estimate the true cost-of-living index number due to lack of data. Actually, we can! 8

Konüs versus Divisia 9

The Quadratic Almost Ideal Demand System (QAIDS) 10 (Deaton and Muellbauer, 1980; Banks, Blundell and Lewbel, 1997)

Compensated and actual shares in the QAIDS 11 [see equation (10) of Oulton (2008)]

Put this together with the earlier result: 12

Two feasible ways to estimate the betas and lambdas 1.Cross section approach. Estimate the betas and lambdas by fitting the QAIDS to microdata on household expenditures in a given year, assuming prices are the same for all households. [ie, regress budget shares on nominal expenditure and nominal expenditure squared]. 2.Aggregate time series approach. Replace the N prices by a smaller number of principal components. Fit the QAIDS to aggregate time series data. This approach uses just the data used by statistical agencies to construct conventional price indices, namely prices and budget shares. 13

The data 70 products, covering nearly all the UK’s Retail Prices Index, ; ie N = 70, T = budget shares and prices. Source: ONS and IFS (Blow et al., 2004). This is much the largest number of products for which the QAIDS has been fitted: Banks et al. (1997) used 5 groups, Neary (2004) used

Methodology Reduce the 69 relative prices to 6 principal components Form initial estimate of QAIDS price index P Estimate the system equation-by-equation using OLS (also IV) Form new estimate of P Iterate till convergence achieved Calculate Konüs price indices, one for each base year for utility, using estimated compensated shares 15

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Other applications of the method Cost functions: estimate the extent to which economies of scale are input-biased International comparisons of living standards 17

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