INEXACT INDEX NUMBERS AND ECONOMIC
THE GDP IMPLICIT PRICE DEFLATOR
Several countries have recently switched – or are about to do so – to chained price and
quantity indexes in the framework for their national accounts. Thus, the United States and Canada have adopted the chained Fisher indexes, whereas the United Kingdom, Switzerland, Australia and New Zealand have opted for chained Laspeyres indexes for real GDP, and chained Paasche for the implicit price deflator. Nonetheless, the vast majority of countries, including most OECD members, have not yet embraced chaining. In these countries the GDP implicit price deflator is still computed as a direct (or fixed-base) Paasche price index. Time series data on the deflator are obtained by taking runs (or sequences) of these direct indexes. Changes in the price level over consecutive periods are measured by the change in the direct Paasche index, a use for which it is ill suited.
Indeed, using the economic approach to index numbers in the context of supply theory, we show that runs of direct Paasche indexes fail an economic monotonicity test if the number of 2periods exceeds two. That is, the price index can register a drop between consecutive periods even though none of the disaggregate prices has fallen, and some have actually increased.
The purpose of this paper is thus to draw attention to some of the undesirable properties of the direct Paasche price index, and, more generally, to the problems that the absence of chaining can raise. Examples based on a constant elasticity of substitution or constant elasticity of transformation (CES or CET) aggregator function are provided. A similar result holds for runs of direct Fisher indexes. This provides a powerful argument in favor of chaining. It should also serve as a warning against the use of unit values as elementary price indexes at the most disaggregate level.
Our analysis is based on the economic approach to index numbers. Much of this literature focuses on exact index numbers, to use the terminology introduced by Diewert (1976). That is,
1 When this paper was written, the author was the chief economist, Swiss National Bank. He is now with the University of Geneva and can be reached at Ulrich.Kohli@unige.ch. He is grateful to W. Erwin Diewert, Andreas Fischer, Kevin J. Fox, Alice Nakamura, and Ludwig von Auer for their comments on earlier drafts of this paper, but they are not responsible for any errors or omissions. 2 This problem also arises in the context of demand theory; see Kohli (1986). However, Paasche index numbers are probably most prevalent in supply theory, since they are widely used in the national accounts.
Citation: Ulrich Kohli (2010), “Inexact Index Numbers and Economic Monotonicity Violations:
The GDP Implicit Price Deflator,” chapter 14, pp. 317-328 in
W.E. Diewert, B.M. Balk, D. Fixler, K.J. Fox and A.O. Nakamura (2010),
PRICE AND PRODUCTIVITY MEASUREMENT: Volume 6 -- Index Number Theory. Trafford Press.
Also available as a free e-publication at www.vancouvervolumes.com and www.indexmeasures.com.
? Alice Nakamura, 2010. Permission to link to, or copy or reprint, these materials is granted without restriction, including for use in commercial textbooks, with due credit to the authors and editors.
knowing the precise form of the aggregator function (e.g. the production possibilities frontier), one seeks to find an index number formula that is exact for it. Alternatively, starting from an arbitrary index number formula, one looks for the aggregator function for which this index 3number would be exact. For instance, it turns out that the Paasche price index is exact as long as the transformation function is either linear or Leontief.
The strategy followed in this paper is somewhat different. We investigate the properties of a given, commonly used index number formula when we know in advance that this index is not exact for the underlying aggregator function, which itself is assumed to be fairly general and well behaved. Specifically, we will look at the direct Paasche price index (the Laspeyres and the Fisher indexes will be briefly examined as well), while assuming that the production possibilities frontier is strictly concave.
The case of the CET aggregator function can be thought of as an example of a well-
behaved production possibilities frontier. Clearly, one could argue that if the aggregator function is CET, then it would be a simple matter to use a CET price index, which would then be exact.
The point, though, is that analysts and commentators often have no choice in this matter, statistical agencies typically supplying Paasche, Laspeyres and Fisher indexes only. In any case, the selection of the CET is only meant as an illustration. Any other functional form that allows the production possibilities frontier to be strictly concave would yield similar results.
2. Runs of Direct Paasche Price Indexes
P The direct Paasche price index () makes a direct comparison between the cost of a Pt,0
basket of goods in the current period (period t) and the cost of the same basket at base period
(period 0) prices. Period t quantities are used for this comparison. Formally, the direct Paasche price index can be defined as follows:
Pi(1) P, ，t,0pq?i,i,t0
where and denote the price and the quantity of good i at time t. pqi,ti,t
A run of direct Paasche price indexes is then given by the following sequence:
iiii(2) 1, , , …, , , pqpqpqpq??010i,i,i,i,t?i,0i,2?i,0i,t？1
or, in more compact form:
3 Much emphasis has been devoted to superlative indexes; i.e., index numbers that are exact for flexible functional forms. A functional form is flexible if it can provide a second-order approximation to an arbitrary aggregator function. The terms “flexible,” “superlative,” and “aggregator function” were coined by Diewert (1974, 1976).
PPPP, , …, , . (3) 1, PPPP1,02,0t,0？10t,
It is common practice to use elements of this sequence to make comparisons between arbitrary pairs of periods. For instance, if one wanted to compare period t with period t-1, one would
calculate defined as follows: ?t,t？1
pqpq?i,ti,t?i,i,t0？1PPt,0ii(4) . ?，；t,t？1PpqpqP??i,i,ti,ti,t0？1？1t,？10ii
It is important to note that, although rests on the comparison between two Paasche ?t,t？14indexes, it is not itself a Paasche index, unless period t-1 happens to be the base period.
Although its properties are little known, is routinely used in economic analysis. ?t,t？1
Consecutive changes in the GDP price deflator, in particular, are often used as a broad measure of inflation. If turns out to be greater than one, one might be inclined to conclude that, on ?t,t？1
average, prices have gone up between period t-1 and period t. As we shall see below, this
conclusion could be diametrically wrong.
3. Profit Maximization and the Economic Approach to Index Numbers
The economic approach to index numbers assumes that the observed quantities () are qi,t
not random, but rather that they are the outcome of economic decisions. In particular, they reflect prices, technology, and optimizing behavior.
Assume for simplicity that national production involves just one input (e.g., an aggregate of labor and capital), the quantity of which we denote by , and two outputs (i = 1, 2). We xt
assume constant returns to scale, nondecreasing marginal rates of transformation, and profit maximization. Let the country’s production possibilities frontier be given by the following transformation (or factor requirements) function:
(5) . x；h(q,q)t1,t2,t
Constant returns to scale imply that is linearly homogeneous, and the assumption of h(?)
nondecreasing marginal rates of transformation means that is convex. Profit maximization h(?)
implies that the marginal rate of transformation is equal to the output price ratio:
h(q,q)p11,t2,t1,t(6) , ；h(q,q)p21,t2,t2,t
4 This is also pointed out by Afriat (1977). It is interesting to note that can be viewed as the value index ?t,t？1
divided by a Lowe quantity index, and as such it could be termed an implicit Lowe price index; see Kohli (2004b). Moreover, as pointed out to me by Bert Balk, it can be seen from (4) that ?violates the Proportionality and t,t？1
Identity tests, which bodes ill for its other properties.
, i = 1, 2. It is possible to solve (5) and (6) for : where h(?)，：h(?)：qqii,ti,t
(7) , q；q(p,p,x)；y(p,p)xi,ti1,t2,tti1,t2,tt
where can be interpreted as unit output (output per unit of input). The linear y(?)，q(?)xiit
homogeneity of with respect to results from the linear homogeneity of . Moreover, q(?)xh(?)it
the assumption of optimization and the convexity of (concavity of the production h(?)
possibilities frontier) imply that – and for that matter – is homogeneous of degree q(?)y(?)ii
zero in prices and nondecreasing in its own price:
：q(?)：q(?)q：(?)iii(8) , , i = 1, 2. p；p；0?01,t2,tp：：p：pi,t1,t2,t
The economic approach to index numbers then amounts to introducing expressions such as (7) into index number formulas such as (1). In the two-good case, the direct Paasche price index becomes:
pq(p,p,x)；pq(p,p,x)1,t11,t2,tt2,t21,t2,ttPP；t,0pq(p,p,x)；pq(p,p,x)1,011,t2,tt2,021,t2,tt(9) . py(p,p)；py(p,p)1,t11,t2,t2,t21,t2,t；py(p,p)；py(p,p)1,011,t2,t2,021,t2,t
It is customary to normalize base period (period 0) prices to unity (). Thus, p；p；11,02,0
expression (9) can be rewritten as:
py(p,p)；py(p,p)1,t11,t2,t2,t21,t2,tPP(10) P；P(p,p)； . 1,t2,tt,0t,0y(p,p)；y(p,p)11,t2,t21,t2,t
4. Economic Monotonicity Test
It seems reasonable to expect a well-behaved measure of the general price level to be 5monotonically increasing – or at least nondecreasing – in its arguments. That is, if one
disaggregate price were to rise, while all other prices are held constant, one would like to see the aggregate price index increase, or at least not fall, after having allowed for the endogenous
adjustment in quantities.
To investigate the slope properties of the direct Paasche price index, it suffices to differentiate (10) with respect to a disaggregate price, say the first one:
5 See Kohli (1986) for an examination of the monotonicity properties of index numbers in the context of demand theory.
where and are the price elasticities of ?，：lny：lnp?0?，：lny：lnp，011,t1,t1,t21,t2,t1,t
output supply with respect to at time t; and ??, where )，h(?)h(?)h(?)h(?)，0p12,t12121,t
2, is the elasticity of transformation between the two outputs. Note h(?)，：h(?)(：q：q)121,t2,t
sthat and , with being the revenue share of output i at time ?；)s?；？)si,t21,t12,t1,t11,t12,t2,t
67t, so that . ?？?；？)?011,t21,t12,t
P It is apparent from the last line of (11) that can be negative if the ratio ：P：p1,tt,0
is sufficiently large, i.e. if is small enough relative to , unless happens pppp)2,t1,t1,t2,t12,t8to be nil. If the production possibilities frontier is strictly concave, on the other hand, and if production is diversified, is strictly negative. In that case, which can be viewed as the )12,t
normal case, the Paasche price index (10) fails to be globally monotonically increasing in its components. That is, it may register a fall as increases and is held constant. pp1,t2,t
Since the price ratio is unity in the base period, economic monotonicity violations can only occur at a point away from the base period; that is, if one compares two situations that do not encompass the base period.
6 See Kohli (1991), for instance. 7??0?；？?，0 Note that it follows from (8) that and . 11,t21,t22,t8 This would be the case if the factor requirements function were linear or Leontief, in which case the Paasche price index would be exact and thus necessarily well behaved.
5. Numerical Illustration
A simple numerical example might help to show how economic monotonicity violations
might occur. Assume three periods: periods 0, 1, and 2. Table 1 shows the given values of p1
and for the three periods, together with the optimizing values of and , assuming that qpq221
922the factor requirements function is given by , a quarter circle in the first quadrant. 2；q；qtt1,2,
Economic monotonicity violations: Numerical illustration
Pt ppqq1,t2,t1,t2,t Pt,0
0 1.0000 1.0000 1.0000 1.0000 1.0000
1 0.2000 1.0000 0.2774 1.3868 0.8667
2 0.4000 1.0000 0.5252 1.3131 0.8286
It is then straightforward to compute the direct Paasche price index. It is shown in the last column, and it can be seen that between periods 0 and 1, as falls from 1 to 0.2 while pp12
Premains unchanged, P drops from 1 to 0.8667. Between periods 1 and 2, recovers and p1
Pincreases from 0.2 to 0.4, while is still being held constant, but keeps dropping, from Pp2
0.8667 to 0.8286. That is, the GDP price deflator registers a fall, thus suggesting deflation (at a rate of 4.4%), even though one disaggregate price has doubled and the other one has remained unchanged.
6. The CET Aggregator Function
The behavior of the direct Paasche price index can be further investigated with the help of some simulations. Let the country's factor requirements function (5) have the following CET 10form:
??1/?(12) , . ?~1x；(q；q)ttt1,2,
922q；p2(p；p) The supply of output i is then given by ; see expression (13) below. i,ti,t1,t2,t10 The name CET stands for constant elasticity of transformation. The elasticity of transformation implied by (12)
can be calculated as )；1(1？?). 12
Under profit maximization, the output supply functions are:
?1/(？1)p,it；, i = 1,2 . (13) qxi,tt1/?/(？1)/(？1)??????；pp1,2,tt11The GDP function can then be written as:
(1)/?？?/(1)/(1)？？????(14) , ??((p,p,x)；p；px；r(p,p)x1,t2,ttt1,t2,tt1,t2,t
is the unit revenue function. The following CET price index will then be exact in the sense of Diewert (1976):
??(？1)/????/(？1)/(？1)??p；p(,)rpp1,2,1,2,tttt(16) . (,)Ppp；；,01,2,ttt(?？1)/?(,)rpp21,02,0
Given that , it follows from (13) that . The direct Laspeyres q；qp；p；11,02,01,02,0
price index can thus be written as:
pq；pq11,t1,02,t2,0L(17) . P(p,p)，；(p；p)1,t2,t1,t2,tt,0q；q21,02,0
It is obvious from (17) that the direct Laspeyres price index is monotonically increasing in prices.
Next, making use of (13), we can derive the direct Paasche price index. It is as follows:
????/(？1)/(？1)p；ppq；pqtttttt1,1,2,2,1,2,P(18) Ppp，； . (,)ttt,01,2,1/(?？1)1/(?？1)q；qttp；p1,2,tt1,2,
As indicated by (11), and given that is strictly negative in the CET case, the direct Paasche )12
price index (18) is not globally monotonically increasing in prices.
The direct Fisher index, finally, can be obtained as:
?/(?？1)?/(?？1)(19) . p；p1tt1,2,；(p；p)1,t2,t1/(?？1)1/(?？1)2p；ptt1,2,
FNote that when ?；2, ; i.e., the direct Fisher index is exact for P(p,p)；P(p,p)1,t2,tt,01,t2,tt,0
aggregator function (12). This is because in that case the CET factor requirements function
11 See Kohli (1978, 1991) and Woodland (1982) for details.
becomes a special case of a quadratic mean of order 2 (the square rooted quadratic function), and,
as shown by Diewert (1976), the Fisher index is exact for that functional form. In this case, the
direct Fisher price index is necessarily monotonically increasing in prices. However, for other
this may no longer be true, since it might well be that the adverse behavior of the values of ?
Paasche component dominates that of its Laspeyres counterpart.
Our results can easily be illustrated with the help of some simulations. We show in Figure
P1 as a function of for alternative values of , after having set . It is P(p,p)p?p；1,01221t
Papparent that is not monotonically increasing in for low values of . That is, as P(p,p)pp,01211t
increases, the GDP price deflator will actually fall, even though is held constant by pp21
Direct Paasche price index for alternative values of ρ
? ； 1.25
1.4? ； 1.50
P? ； 2.001.2P
? ； 4.001.0
? ； 8.00
FPL Next, in Figure 2, we show , , and as functions of P(p,p)P(p,p)P(p,p),012,01212ttt,0
PFL for . It can be seen that, although as P(p,p)?P(p,p)?P(p,p)?；4p1121212t,0t,0t,0expected, there are ranges for where p1PFLP, and where, moreover, and ：P(p,p)/：p：：P(p,p)/：p：：P(p,p)/：pP(p,p),0121,0121,0121,012tttt
F are decreasing in . P(p,p)p,0121t
Direct Paasche, Fisher and Laspeyres indexes for ρ = 4 1.6
PPFinally, it can be seen from Figures 1 and 2 that for , if and only if P(p,1)?P(1,1)p~0t,01t,01
. The same is true for the direct Fisher index. That is, relative to the base period, both p?11
indexes are monotonically increasing in prices. This provides a strong argument in favor of
7. Unit Values
Economic monotonicity violations are likely to plague unit values even more so than they do direct Paasche indexes. Yet unit values are routinely used at the elementary level to aggregate goods seemingly belonging to a same category. The average price of an apparently homogeneous product that is nevertheless sold at different times, different places, and under different conditions is often calculated by simply dividing total revenues by the number of items sold. The use of unit values is particularly prevalent in the area of foreign trade, where differences in quality and specification are not always taken into account when calculating the average price of exported cars, cameras, or watches. Thus, import and export price deflators are often computed on the basis of unit values.
Unit values can be thought of as a special (and rather perverse) case of direct Paasche price indexes, for it is defined relative to an arbitrary (and often imaginary) base period. The unit value index can be defined as follows:
pqpq；1,t1,t2,t2,tU . (20) Ppp(,)，t1,t2,tqq；1,t2,t
UPComparing (20) with (18), it can be seen that is equivalent to if we interpret as being PPtt,0
defined relative to a base period for which . p；p1,02,0
Naturally, when we aggregate different types of goods, such as luxury automobiles and compact cars, there may well never have been such a time. Since direct indexes only make sense relative to the base period, one must conclude that they are meaningless if that base period has never existed. Economic monotonicity violations are likely to be particularly important in the case of unit values since relative prices will tend to differ greatly from unity. As an example, high performance automobiles can easily cost 20 times a much as compact cars, top-of-the-line cameras can cost 50 times more than more basic ones, luxury watches can be worth 1,000 times more than low-end ones, and so on.
It is well known that, in the context of supply theory, the Paasche price index tends to lie above the true price level, and the Laspeyres index underneath it. This is often understood to mean that the Paasche price index overstates price increases, and that the Laspeyres index underestimates them. This is clearly does not need to be true if the reference period is not the base period, i.e. the period for which the data are normalized. Note also that relative to the base period, the direct Paasche price index will understate price falls, whereas the Laspeyres index will exaggerate them. At a time when several countries have been flirting with deflation, this might be something to keep in mind.
The fact that the direct Paasche GDP deflator is not monotonically increasing in prices makes it a poor indicator of inflation, since it might point at a price increase when prices are