PRICE INDEX AGGREGATION: PLUTOCRATIC WEIGHTS, DEMOCRATIC WEIGHTS, AND VALUE JUDGMENTS

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1 Revised June 10, 2003 PRICE INDEX AGGREGATION: PLUTOCRATIC WEIGHTS, DEMOCRATIC WEIGHTS, AND VALUE JUDGMENTS Franklin M. Fiser Jane Berkowitz Carlton and Dennis William Carlton Professor of Economics Massacusetts Institute of Tecnology * ffiser@mit.edu Abstract Te question of ow to aggregate individual cost-of-living indexes is not a simple one. Te recent National Researc Council Panel s report (Scultze et al. (2002). At Wat Price? Conceptualizing and Measuring Cost-of-Living and Price Indexes. Wasington: National Academy Press) discusses two metods. Te first, te plutocratic metod uses expenditure weigts in te aggregation; te second, te democratic metod weigts eac ouseold equally. Tis paper argues tat te coice of weigts depends on value judgments about income distribution. It is sown tat, save in very special circumstances, neiter te plutocratic nor te democratic metod is satisfactory. 1. Introduction Te teory of te cost-of-living index is well developed so far as concerns a single ouseold. Wen it comes to aggregation over ouseolds, owever, tere is far less agreement as to ow to proceed. Yet suc aggregation is inevitable. Wat ten is to be done? Te recent report of te National Researc Council s Panel on Conceptual Measurement and Oter Statistical Issues in Developing Cost-of-Living Indexes (Scultze, et al., 2002) 1 examines tis issue (among many oters) and offers two alternative aggregation metods neiter of wic seems fully satisfactory. 2 In tis paper, I evaluate tose metods by stating te aggregation problem in a form in wic value judgments about income distribution can be made explicit. I sow tat wile te so-called plutocratic metod, as te panel notes, will usually be undesirable, te so- * I am grateful to E. R. Berndt for comments but retain responsibility for error. 1

2 called democratic metod, wic te panel appears to prefer, is difficult or impossible to rationalize under any set of circumstances and reasonable value judgments about income distribution. Wile I ave unfortunately been unable to offer any practical recommendation, I ope tat setting out te explicit teory will eventually prove elpful. I begin by defining terms. Tere are H ouseolds, indexed by = 1,,H and n goods. Te t ouseold s consumption vector is denoted by x, and te vector of total consumption over all ouseolds is denoted by X. Subscripts denote consumption of a particular good. Te reference (or base) period is denoted by t = 0 and te comparison (or current) period by t = 1. Te income of te t ouseold in te tt period is denoted by y t, and total income of all ouseolds by Y t. It is assumed tat all ouseolds face te same prices. 3 For convenience, te price vector in te comparison period is denoted simply by p, and tat in te reference period by p 0. Again, te price of a particular good is denoted by a subscript. Te utility of te t ouseold (in te base period) is given by u 0, and, at tat utility level, its expenditure function in te comparison period is denoted by E (p, u 0 ). Of course, E (p 0, u 0 ) = y 0, so tat s cost -of-living index can be written as: (1.1) I E (p, u 0 )/ E (p 0, u 0 ) = E (p, u 0 )/ y 0, = 1,,H. Te Laspeyres index corresponding to tis is: 1 All references are to tis unless oterwise stated. 2 See pp and Capter 8, especially pp

3 (1.2) L px 0 / p 0 x 0 = px 0 / y 0, were te obvious inner products are involved. Te aggregation problem is tat of combining te indexes given in (1.1) and (1.2) into single aggregate indexes for te entire group of H ouseolds. As discussed by te panel (pp ), following Pollak (1980, 1981), one way to proceed is to define wat e calls te social cost -of-living index and ask for te minimum income tat would be required in te comparison period to give eac ouseold te same utility it acieved in te reference period. Tat income, summed across ouseolds, is ten divided by te total income required to acieve tose utilities in te reference period. Tis yields an index: (1.3) J E (p, u 0 )/ Y 0 = {E (p, u )/ y 0 }{ { y 0 /Y 0 } = I { { y 0 /Y 0 }, an average of te individual indexes weigted by te respective expenditures of te ouseolds. Te corresponding Laspeyres index is ten: (1.4) L L { y 0 /Y 0 } = px/p 0 X, te standard form of te consumer price index. 3 Te problem is quite ard enoug wit tis assumption. 3

4 Because of te expenditure weigts, tese indexes are called plutocratic. Tey give more weigt to te indexes of te ric tan to te indexes of te poor. Te panel finds tis unappealing for some purposes 4 (and I agree), and wonders weter unweigted averages called democratic indexes would not be better. Te democratic index is defined as: (1.5) D (1/H) {E (p, u )/ y 0 } wit te corresponding Laspeyres index: (1.6) F (1/H) L. 2. Formally Modeling te Problem It is clear tat te issue ere necessarily involves value judgments, especially value judgments about income distributions. It follows tat te analysis sould be framed in a way in wic suc value judgments can be made explicit. I do so in te following way. 5 Let W(u 1,, u H ) W(u) be a social welfare function defined over te utilities of te ouseolds. Parallel to te definition of te expenditure function in te teory of te individual ouseold, we can define a social welfare expenditure function (WEF). In its unrestricted form, tat function is defined as: 4 But not for all. Te plutocratic index is consistent wit accounting identities in te national accounts. 5 Te formal set-up is similar to tat in Fiser (2003), altoug te problem is different. 4

5 (2.1) E* = E*(p, W) Min px, subject to W(u) = W. In oter words, E* is te minimum total money income needed by te ouseold sector to attain social welfare W wen prices are p, provided tat suc total income can be allocated by te social planner wose welfare function is W( ). But, of course (and, peraps, fortunately), social planners or, more generally, people presenting teir views as to te relative desirability of different allocations typically do not ave suc powers, and tis will increase te cost of obtaining welfare level, W. 6 Tus, wile E* measures te total cost of acieving a given welfare level if te distribution of income can be altered by te planner, we need to examine te (greater or equal) total cost of acieving tat level wit te income distribution eld constant. Tis is a second-best solution from te planner s viewpoint. Formally, let s denote s sare of total income in te fixed income distribution. Define te restricted welfare expenditure function by: (2.2) E = E(p, W) Min px, subject to W(u) = W and y / Y = s ( = 1,, H). It is evident tat E E*, since te latter is calculated as a minimum witout restrictions. Tere is anoter way to look at te restricted problem (2.2), and tat way brings us back to te price-index aggregation problem. Let te fixed income distribution be tat of te reference period. Suppose tat a single, aggregate price index is to be publised and tat suc aggregate is to be used to adjust eac ouseold s income by te same 6 Pollak (1981, pp ) considers tis but in a different way tan is done in te present paper. 5

6 percentage. Ten te restricted problem is seen to be tat of minimizing te total amount of expenditure required under suc conditions to acieve a given level of welfare. Of course, in bot te restricted and unrestricted cases, te minimization problem is to be solved at comparison period prices and te result divided by Y 0 (te solution at reference period prices) to obtain te desired index. 7 Now consider te Pollak social cost-of-living index given in (1.3). Te denominator is, of course, Y 0, wile te numerator is te total expenditure required to give eac ouseold te same utility at prices p 1 tat it acieved in te reference period at prices p 0. 8 Evidently, tis is one way of acieving te reference level of social welfare as in te unconstrained problem. It will generally not be te efficient way, owever, for, isomorpic to te reasons tat a Laspeyres index overstates te corresponding cost-ofliving index, te plutocratic index, J, fails to allow for substitution among individuals in te social welfare function and terefore overstates te unrestricted first-best solution. In oter words, J E*/Y 0. Furter, te plutocratic Laspeyres, L, will certainly overstate J, so tat: (2.3) L J E*/Y 0. 7 A reader as commented tat e sees no reason to tink tat individual product prices would be te same in te current and optimal income distribution worlds. I agree. Tat issue, owever, is irrelevant to te discussion in tis paper wic takes te current prices as given and constructs indices comparing tem to tose of te reference period. 8 Tis is a loose way of stating wat is involved. Te prase te same utility as no operational meaning, and one sould say making te ouseold indifferent between facing te comparison period prices wit te calculated income and facing te reference period prices wit reference period income. See Fiser and Sell (1972, pp. 2-4). In te present context, tere is no need for tat precision, but I would not like it to be tougt tat my views on te subject ave canged. 6

7 But, of course, tis means tat te plutocratic indexes (including te standard consumer price index) overstate an index wic is itself lower tan te index of interest te restricted index, E/Y 0. Tis does not mean tat te standard indexes are likely to be rigt, owever. To examine tat question, we must look at te restricted problem in more detail. It will be convenient to do tis in terms of Laspeyres indexes. Formally, a Laspeyres consumer price index is a linear approximation to te cost-of-living index taken at te reference year s purcases. We sall compare te weigt given to a particular price (p i), in te plutocratic Laspeyres index, L (te actual consumer price index), wit tat wic would be given in a Laspeyres index approximating E/Y 0. We do tat by evaluating te derivatives of E at te original prices. 9 Te LaGrangian for te restricted minimization problem is: (2.4) Ë = px - ì {W(u) W} - δ { px - s px }, were ì and te ä are LaGrange multipliers. One (relatively minor) point before proceeding. 10 Te constraints in (2.2) are not independent, since if every ouseold but one as its current-period income sare equal to its reference-period income sare, so will te remaining ouseold. Tis means tat we sould omit tat constraint for one of te ouseold, say te Ht. To avoid complicating te notation in (2.4) and later equations, I simply define ä H 0. It may elp to fix ideas to take tat ouseold as one wit te median income in te referenceperiod income distribution, and I do so in te following discussion. 9 I am indebted to Eduardo Ley for calling to my attention te need to make tis explicit. 7

8 Note tat, by te Envelope Teorem: (2.5) ä = E/ s px ( = 1,, H). Hence, if, according to te value judgments of te social planner, as too ig a sare of income in te fixed distribution relative to te median, ten ä > 0, since a lower sare for would reduce te cost of acieving te given welfare level. Similarly, ä < 0 corresponds to too low a sare for relative to te median. If all ä = 0, ten te given income distribution is te same as would be imposed by te social planner in te firstbest solution, and E = E* wen bot are minimized. Again using te Envelope Teorem, (2.6) E/ p i = Ë/ p i = X i - δ { x s Xi so tat te weigt given to p i in a Laspeyres index would be: i } (i = 1,,n), (2.7) w i (1/Y 0 )( X i - 1 δ { xi s Xi }), wereas te weigt given to p i in te plutocratic index is simply (1/Y 0 )( X i ). Tese will be te same for all i under any of te following conditions 11 : Case 1: x s X 1 = 0, all i. In tis case, eac ouseold consumes eac good in i1 i te same proportion to its income as does every oter ouseold. If tis olds globally, ten te ouseold s all ave te same omotetic indifference map, and teir beavior 10 Troug an oversigt, tis paragrap was omitted from Fiser (2003). 11 And, except for cases of measure zero, only ten. 8

9 can be represented as tat of an aggregate ouseold. 12 In tis case, te plutocratic and democratic indexes are te same. Indeed, any weigted average of individual cost-ofliving indexes would give te same result. Case 2: ä = 0, = 1,, H. Here tere are two possibilities. a. We are indifferent to income distribution questions; b. We regard te existing income distribution as optimal. Note tat, in all tree of tese cases, te restricted minimum problem as te same solution as te unrestricted one; tat is, E = E*. Note furter, tat, in Case 2b, te P in Pluto cratic Index migt just as well stand for Panglossian or for te Popian as in Alexander Pope, since te view taken can be described eiter as All s for te best in tis best of all possible worlds, or Watever is, is rigt, at least wen it come s to income distribution. Tese are not attractive views. Given tis, te reader may now tink tat I sall come down in favor of te democratic index, but tat is not te case. For te democratic index, D, defined in (1.5) to coincide wit E/Y 0, it must be te case tat te two corresponding Laspeyres indices also coincide. Tis requires tat, for all i, te weigt given to p i in F, defined in (1.6) equal w i, defined in (2.7). Tis means: (2.8) (1/H) {x i / y 0 } = (1/Y 0 )( X i - δ { x 1 i s X i }), i = 1,, n. conditions: Except for cases of measure zero, tis can only appen under one of te following 12 If te condition olds locally, ten aggregation is possible in some neigborood. 9

10 Case 1: x i1 s X i1 = 0, all i. In tis case, eac ouseold consumes eac good in te same proportion to its income as does every oter ouseold. We ave already seen tat tis is te case of perfect aggregation over ouseolds, and tat, if it olds, te plutocratic and democratic indexes are te same. Case 2: For every = 1,, H, ä = 0, and y 0 = Y 0 / H. In tis case, te income distribution is egalitarian, and (as we sould expect, given te democratic nature of te index to be used), suc egalitarian distribution is also considered to be te optimal one. Note tat, in tis case also, te democratic and plutocratic indexes coincide. Te interesting feature of Case 2 is tat te democratic index is optimal only if te income distribution is already egalitarian and considered optimal. Te reason for tis is fairly easy to see. Assume tat te aggregate price index is to be used to provide te same adjustment for price canges to eac ouseold s income. Te use of te democratic index provides tat equal weigt is to be given to eac ouseold. But tis provides equal treatment for eac ouseold only going forward equal treatment as regards canges from te reference to te comparison period. Te Case 2 result sows tat te value judgment as to income distribution tat tis relies on is tat te optimal distribution involves equal treatment for eac ouseold. But, if te income distribution is not already egalitarian, and if particular price canges ave different (proportional) effects on te costs of living of different ouseolds (te absence of Case 1), ten different ouseolds sould be weigted differently, so as to correct for te lack of optimality of te income distribution. An increase in te price of goods eavily bougt by te poor sould get a iger weigt tan an increase in te price of goods eavily bougt by te 10

11 ric, even if te number of ouseolds in eac group is te same. Tis would alter te distribution of real income in te direction of equality wile leaving te distribution of money income uncanged. Hence, te same value judgment on income distribution tat leads to te use of te democratic index also implies tat suc use is suboptimal unless te income distribution is already egalitarian. 3. Summary and Conclusions We ave sown tat, wen embedded in a context permitting value judgments on income distribution to be made explicit, neiter te plutocratic nor te democratic index are really satisfactory apart from te obvious case in wic aggregation of preferences is possible a case in wic te two indexes coincide and also coincide wit any oter metod of averaging individual indexes. Te plutocratic index is optimal only if we eiter do not care about income distribution at all or else believe tat te existing income distribution is optimal. Te democratic index is optimal only if we already ave an egalitarian income distribution and believe tat distribution to be optimal formally a more restrictive set of circumstances tan tat required for te plutocratic index (even toug tey would coincide in tat case). Having said tis, I wis I ad a truly constructive suggestion to offer, but I do not. I sympatize wit te view tat te democratic index is to be preferred to te plutocratic one, but note tat it only provides equal treatment going forward from a given position tat is likely to be unequal. Since te use of aggregation metods to affect te 11

12 distribution of real income will doubtless be controversial, peraps te democratic index is te best we can do. Of course, all tis suggests tat te use of a single aggregate index for large, disparate groups of ouseolds is likely to be unsatisfactory. Te obvious conclusion is tat different indexes sould be constructed for different groups werever possible. 12

13 References Fiser, F.M. (2003). A Metric for Asse ssing te Goodness of Income Distributions and te Effect of Price Canges. Journal of Economic Teory 109, Fiser, F.M. and K. Sell (1972). Te Economic Teory of Price Indices. New York:Academic Press. Fiser, F.M. and Z. Grilices (1995). Aggregate Price Indices, New Goods, and Generics. Quarterly Journal of Economics 109, Pollak, R.A. (1980). "Group Cost-of-Living Indexes". American Economic Review 70, (1981). "Te Social Cost-of-Living Index". Journal of Public Economics 15, Scultze, C. et al. (2002). At Wat Price? Conceptualizing and Measuring Cost-of- Living and Price Indexes. Wasington: National Academy Press. 13