Chapter 7 THE ECONOMIC THEORY OF INDEX NUMBERS: A SURVEY* W.E. Diewert 1. Introduction The literature on index numbers is so vast that we can cover only a small fraction of it in this chapter. Frisch [1936] distinguishes three approaches to index number theory: (i) 'statistical' approaches, (ii) the test approach, and (iii) the functional approach, which Wold [1953; 135] calls the preference field approach and Samuelson and Swamy [1974; 573] call the economic theory of index numbers. We shall mainly cover the essentials of the third approach. In the following two sections, we define the different economic index number concepts that have been suggested in the literature and develop various numerical bounds. Then in Section 4, we briefly survey some of the other approaches to index number theory. In Section 5, we relate various functional forms for utility or production functions to various index number formulae. In Section 6, we develop the link between 'flexible' functional forms and 'superlative' index number formulae. The final section offers a few historical notes and some comments on some related topics such as the measurement of consumer surplus and the Divisia index. Essays in Index Number Theory, Volume I W.E. Diewert and A.O. Nakamura (Editors) ©1993 Elsevier Science Publishers B.V. All rights reserved. *First published in Essays in the Theory and Measurement of Consumer Behaviour in Honour of Sir Richard Stone, edited by A. Deaton, London: Cambridge University Press, 1981, pp. 163-208. The financial support of the Canada Council is gratefully acknowledged as are the helpful comments of R.C. Allen, C. Blackorby, and A. Deaton, who are not responsible for the remaining shortcomings of this chapter. It is a pleasure to dedicate this chapter to Professor Stone, since the author first learned of the existence of the index number problem as a graduate student at Berkeley by reading some of Professor Stone's work. 178 Essays in Index Number Theory 7. The Economic Theory 179 2. Price Indexes and the Koniis Cost of Living Index We assume that a consumer is maximizing a utility function F(x) subject to the expenditure constraint pTx = ^2,f=1PiXi < y where x = (x\,... ,x^)T > On is a nonnegative vector of commodity rentals, p = (pi,... ,pn)t ^ Ojv is a positive vector of commodity prices1 and y > 0 is expenditure on the N commodities. We could also assume that a producer is maximizing a production function F(x) subject to the expenditure constraint pTx < y where x > On is now an input vector, p On is an input price vector and y > 0 is expenditure on the inputs. In order to cover both the consumer and producer theory applications, we shall call the utility or production function F an aggregator function in what follows. The consumer's (or producer's) aggregator maximization problem can be decomposed into two stages: in the first stage, the consumer (or producer) attempts to minimize the cost of achieving a given utility (or output) level, and, in the second stage, he chooses the maximal utility (or output) level that is just consistent with his budget constraint. The solution to the first stage problem defines the consumer's (or producer's) cost function C: (1) C(u,p) = min{pTx : F(x) > u, x > On} x The cost function C turns out to play a pivotal role in the economic approach to index number theory. Throughout much of this chapter, we shall assume that the aggregator function F satisfies the following conditions I: F is a real valued function of N variables defined over the nonnegative orthant 51 = {x : x > On} which has the three properties of (i) continuity, (ii) increasingness2 and (iii) quasiconcavity.3 Let U be the range of F. From I(i) and (ii), it can be seen that U = {u : u < u < ou} where u = F(0n) < ou. Note that the least upper bound ou could be a finite number or +oo. In the context of production theory, typically u — 0 and ou — +oo, but, for consumer theory applications, there is no reason to restrict the range of the utility function F in this manner. Notation: x > On means each component of the column vector x is nonnegative, x On means each component is positive, x > On means x > On but x ^ On where On is &n N dimensional vector of zeros, and xT denotes the transpose of x. 2If x" > x' > 0N, then F{x") > F(x'). 3For every u G range F, the upper level set L(u) = {x : F(x) > u} is a convex set. A set S is convex iff x' G S, x" G S, 0 < X < 1 implies Xx' + (1 - X)x" G S: i.e. the line segment joining any two points belonging to S also belongs to S. Define the set of positive prices p = {p : p 3> On}- It can be shown that (see Diewert [1978c]) if F satisfies conditions I, then the cost function C defined by (1) satisfies the following conditions II: (i) C(u,p) is a real valued function of N +1 variables defined over UxP and is jointly continuous in (u,p) over this domain. (ii) C(u,p) — 0 for every peP. (iii) C{u,p) is increasing in u for every peP; i.e., if p G P, u', u" G U, with u' < u", then C(u',p) < C(u",p). (iv) C(ou,p) — +oo for every peP; i.e., if p G P, un G U, lim„ un — u, then lim„ C{un ,p) — +oo. (v) C{u,p) is (positively) linearly homogenous in p for every u E U; i.e., u elf, A>0, peP implies C(u, Xp) — XC(u,p). (vi) C{u, p) is concave in p for every u £ (7; i.e., if p' 3> On, p" 3> On, 0 < \ XC{u,p') + {\-X)C{u,p"). (vii) C{u,p) is increasing in p for u >u and u G t/. (viii) C is such that the function F*(x) = m&xu{u : pTx > C(u,p) for every p£P, u G U} is continuous for x > On- For some of the theorems to be presented in this chapter, we can weaken the regularity conditions on the aggregator function F to just continuity from above.4 Under this weakened hypothesis on F, the cost function C defined by (1) will still satisfy many of the properties in conditions II above.5 Finally, some of the theorems below make use of the following (stronger) regularity conditions on the aggregator function: we say that F is a neoclassical aggregator function if it is defined over the positive orthant {x : x On} and is (i) positive, i.e. F(x) > 0 for x On, (ii) (positively) linearly homogeneous, and (iii) concave over {x : x On}- Under these conditions (let us call them conditions III) F can be extended to the nonnegative orthant 51, and the extended F will be nonnegative, linearly homogeneous, concave, increasing and continuous over 51 (see Diewert [1978c]). Moreover, if F is neoclassical, then _F's cost function C factors into (2) C{u,p)=uC{l,p) = uc{p) 4F is continuous from above over x > On iff for every u G range F, L(u) = {x : F(x) > u} is a closed set. 5Specifically, Diewert [1978c] shows that C will satisfy the following conditions II": (i) C(u,p) is a real valued function of N + 1 variables defined over UxP and is continuous in p for fixed u and continuous from below in u for fixed p (the set U is now the convex hull of the range of F), (ii) C(u,p) > 0 for every u G U and peP, (iii) C{u,p) is nondecreasing in u for fixed p, (iv) C{u,p) is nondecreasing in p for fixed u, and properties (v) and (vi) are the same as (v) and (vi) of conditions II. 180 Essays in Index Number Theory 7. The Economic Theory 181 for u > 0 and p On where c{p) = C(l,p) is F's unit cost function. It can be shown that c satisfies the same regularity conditions as F; i.e. c is also a neoclassical function. Also, if we are given a neoclassical unit cost function c, then the underlying aggregator function F can be defined for x On by F(x) = max{« : C(u,p) < pTx for every p > On} u — max{u : uc(p) < pTx for every p > On, pTx — 1} u (3) — min{l/c(p) : p > On, pT x — 1} p (4) = 1/max{c(p) : pTx — 1, p > Oat}. p Now that we have disposed of the mathematical preliminaries, we can define the Koniis [1924] cost of living index® Pk'- for p° 3> Oat, p1 3> Oat and x > On (5) PKlp^p^x) = C[F{x)1p1]/C[F{x)1p\ Thus Pk depends on three sets of variables: (i) p°, a vector of period 0 or base period prices, (ii) p1, a vector of period 1 or current period prices,7 and (iii) x, a reference vector of quantities.8 In the consumer context, Pk can be interpreted as follows. Pick a reference indifference surface indexed by the quantity vector x > On- Then Pk(p°, p1 ,x) is the minimum cost of achieving the standard of living indexed by x when the consumer faces period 1 prices p1 relative to the minimum cost of achieving the same standard of living when the consumer faces period 0 prices p°. Thus Pk can be interpreted as a level of prices in period 1 relative to a level of prices in period 0. If the number of goods is only one (i.e. N — 1), then it is easy to see that Pk(pi,p\,xi) — p\/p\ for all xi > 0. Note that the mathematical properties of Pk with respect to p° , p1 and x are determined by the mathematical properties of F and C given by conditions I and II above. In particular, for A > 0, p° On, p1 On and x On, we 6Or cost of production index in the producer context. 7In the theory of international comparisons, p° and p1 can be interpreted as price vectors that a given consumer (whose utility level is indexed by the quantity vector x) faces in countries 0 and 1. 8The index Pk can also be written as Pk(p°,px,u) = C(m,p1)/C(m,p°) where u is the reference output or utility level. Written in this form, the symmetry of the Koniis price index Pk with the Malmquist quantity index to be introduced later becomes apparent. However, our present notation for Pk is more convenient when we set the reference consumption vector x equal to the observed consumption vector xr in period r. have Pk(p°,Xp°,x) — A and Pk(p°,px, x) — 1/Pk(px,p°, x). Thus if period 1 prices are proportional to period 0 prices, then Pk is equal to the common factor of proportionality for any reference quantity vector x. However, if prices are not proportional, then in general Pk depends on the reference vector x, except when preferences are homothetic as is shown in the following result. Theorem 1. (Malmquist [1953; 215], Pollak [1971a; 31], Samuelson and Swamy [1974; 569-570]): Let the aggregator function F satisfy conditions I. Then Pk(p° jP1 , x) is independent of x if and only if F is homothetic.9 Proof: If F is homothetic, then, by definition, there exists a continuous, monotonically increasing function of one variable G, with G(u) — 0 such that G[F(x)] = f(x) is a neoclassical aggregator function (i.e. / satisfies conditions III above). Under these conditions, F's cost function decomposes as follows: for u > 0, p ^> On, C(u,p) = min{pTx : F(x) > u} x = min{pTx : G[F(x)} > G(u)} x (6) = G{u)c{p) where c is the unit cost function which corresponds to the neoclassical aggregator function /. Thus for p° 3> Oat, p1 3> Oat and x > On, we have PK(p0,p\x) = C[F(x),p1]/C[F(x),p0} = GiFixMp^/GiFixMp0) (7) = cip^/cip0) which is independent of x. Conversely, if Pk is independent of x, then we must have the factorization (7) ; i.e. we must have for every x On, p ^ Oat (8) C(F(x),p) = G[F(x)]c(p) for some functions G and c, whose regularity properties must be such that C satisfies conditions II. It can be verified that the regularity conditions on C and the decomposition (8) imply that the functions c and G(F) both satisfy conditions III,10 so that, in particular, G[F(x)] is (positively) linearly homogeneous in x. Thus F is homothetic.qed 9It seems clear that earlier researchers such as Frisch [1936; 25] also knew this result, but they had some difficulty in stating it precisely, since the concept of homotheticity was not invented until 1953 (by Shephard [1953] and Malmquist [1953]). 10Linear homogeneity of G(F) follows from the following identity which can be derived in a manner analogous to (4): G[F(x)] — 1/maxp{c(p) : p > On, pTx — 1} for every x ^ On- 182 Essays in Index Number Theory 7. The Economic Theory 183 Thus in the case of a homothetic aggregator function, the Koniis cost of living index Pk(p°, p1, x) is independent of the reference quantity vector x and is equal to a ratio of unit cost functions, c(p1)/c(p°). If we knew the consumer's preferences (or the producer's production function), then we could construct the cost function C(u,p) and the Koniis price index Pk- However, usually we do not know F or C and thus it is useful to develop nonparametric bounds on Pk', i.e. bounds that do not depend on the functional form for the aggregator function F (or its cost function dual C). Theorem 2. (Lerner [1935-36], Joseph [1935-36; 149], Samuelson [1947; 159], Pollak [1971a; 12]): If the aggregator function F is continuous from above, then, for every p° = (pi, ■ ■ ■ ,p°N)T > Oat, p1 = (p{,... ,pN)T > Oat and x > On where F(x) > F(0N), (9) minfoVrf : i = 1, .. ., iV} < PK(p0,p\x) < maxfaVrf : 1......V |: i.e. Pk lies between the smallest and the largest price ratio. Proof: Let p° > Ojv, p1 > Oat, x > On where F(x) > F(0n) and let x° > On and x1 > 0 solve the following cost minimization problems: (10) C[F(x),p°] = min{p0Tx : F(x) > F(x)} = p0Tx° > 0 x (11) ClFix)^1] = min{p1Tx : F(x) > F(x)} = p1Tx1 > 0. x Then ClFix)^1] = min{p1Tx : F(x) > F(x)} x > mlii{p1Tx : p0Tx > p0Tx°, x > 0N} x since {x : F(x) > F(x)} C {x : p0Tx > p0Tx°, x > 0N} (12) = min{rf(P°T^0M°) = i = 1, ■ ■ ■ , N} i since the solution to the linear programming problem iiimx{p1Tx : p0Tx > p0Tx°, x > On} can be taken to be a corner solution. Similarly, C[F(x),p°] > min{PlVT*7Pi) :i = l,...,N} or (13) l/C[F(x),p0} < m&xipl/pVp^x1 : i = l,...,N}. i Since PrtG^p^x) = C^^^^/C^^),^0], (10) and (12) imply the lower limit of (9) while (11) and (13) imply the upper limit.qed The geometric idea behind the above algebraic proof is that the sets {x : p0Tx > p0Tx°, x > On} and {x : p1Tx1 > p1Txx, x > On} form outer approximations to the true utility (or production) possibility set {x : F(x) > F(x)}. Moreover, it can be seen that the bounds on Pk given by (9) are the best possible,11 i.e., if F(x) = p0Tx, then Pk will attain the lower bound while, if F(x) = p1Tx, then Pk will attain the upper bound in (9). It is natural to assume that we can observe the consumer's (or producer's) quantity choices, x° > On and x1 > On, made during periods 0 and 1 in addition to the prices which prevailed during those periods, p° On and p1 3> Oat. In the remainder of this section, we shall also assume that the consumer (or producer) is engaging in cost minimizing behavior during the two periods. Thus we assume: (14) p0Tx° = C[F(x°),p0}; p^x1 ^CiFix1)^1}; p^p1 » 0^; x°,x1>0N. Given the above assumptions, we now have two natural choices for the quantity vector x which occurs in the definition of the Koniis cost of living index Pk(p°"tP1 ,x): x° or x1. The Laspeyres-Koniis cost of living index is defined as Pk{p°,px, x°) and the Paasche-Koniis cost of living index is defined as Pk(p°,p1, x1)-12 It turns out that the Laspeyres-Koniis index Pk(p0,px, x°) is related to the Laspeyres price index Pl(p°, p1, x°, x1) = p1Tx°/p0Tx° while the Paasche-Koniis index Pk(lAp1, a^1) is related to the Paasche price index Pp(p°,p1,x°,x1) = p1T x1 / p0T x1. Theorem 3. (Koniis [1924; 17-19]): Suppose F is continuous from above and (14) holds. Then (15) PK(p°,p\x°) < p1Tx°/p0Tx° = PL and (16) PKip^p1^1) > p1Tx1/p0Tx1 = Pp. Proof: PK(jP,p\x°) = C^l.p'l/ClFfx0),;] = C[F(x^^^/p^x0 using (14) = min{p1Tx : F(x) > F(x°)}/p0Tx° x Fix1)} > p1Tx1 /p0Tx1. qed nThis point is made by Pollak [1971a; 28]. 12 The terminology is due to Wold [1953; 136]. 184 Essays in Index Number Theory 7. The Economic Theory 185 Corollary 3.1. (Pollak [1971a; 17]): (17) minl^/rf -i = 1, ■ ■ ■, N} < PK(p°,p\x0) < p1Tx°/p0Tx0 = PL. i Corollary 3.2. (Pollak [1971a; 18]): (18) PP = p1Tx1/pQTx1 < PxijP^1,!1) < maxfe'/ft0 : i = 1,... ,N}. i Corollary 3.3. (Frisch [1936; 25]): If in addition F is homothetic, then for x > Oat, (19) PP = pXTxxjpQTxx < PK(p0,p\x) < p1Tx°/p0Tx° = PL. The first two corollaries follow from Theorems 2 and 3, while the third corollary follows from Theorems 1 and 2. Note that PL ee p^/p^x0 = , (pI/p°)(pW/p°tx°) Z-'2 — 1 = , (Pi/Pi)3°i ^ ™*{pi/p° ■ i = l,2,-..,N} since a share weighted average of the price ratios p\/pf will always be equal to or less than the maximum price ratio. Thus the bounds given by (17) will generally be sharper than the Joseph-Pollak bounds given by (9). Similarly, PP ee p"V/ «^1 ^ 1 (p}/Pi)(Pix}/p0tx1) Z-'2 — 1 >min{PlVp? :i = l,2,...,iV}, i so that the bounds (18) are generally sharper than the bounds (9). The geometric idea behind the proof of Theorem 3 is that the sets {x : x — x0} and {x : x — x1} form inner approximations to the true utility (or production) possibility sets {x : F(x) > F(x0)} and {x : F(x) > F(xx)} respectively. Moreover, it can be seen that the bounds on Pk given by (15) and (16) are attainable if F is a Leontief aggregator function (so that the corresponding cost function is linear in prices).13 13Pollak [1971a; 20] makes this well known point. F is a Leontief aggregator function if F(x\, X2, ■ ■ ■, xjv) ee mirij{xj/ai : i — 1,2, .. ., N} where aT = (ai, a2, ■ ■ ■, aw) > Oat. In this case C(u,p) — upTa. Theorem 4. (Koniis [1924; 20-21]): Let F satisfy conditions I and suppose (14) holds. Then there exists a X* such that 0 < A* < 1 and Pk[p°',px,X*X± + (1 — X*)x°] lies between Pl and Pp ; i.e. either (20) PL ee p1Tx°/p0Tx° < PK[p°,p\X*x1 + (1 - X*)x°] Kp^x^p^x^Pp or (21) PP < PK[p°,p\X*x1 + (1 - X*)x°] < PL. Proof: Define h(X) = PK(p°,p1,Xx1 + (1 - X)x°) ee C^Ax1 + (1 + X)x°),p1]/C[F(Xx1 + (1 — X)x°),p0]. Since both F and C are continuous with respect to their arguments, h is continuous over the closed interval [0,1]. Note that h(0) = PftrQ^p1^0) and h(l) = PK{p°,p1 ,x1). There are 4! = 24 possible inequalities between the four numbers Pl, Pp, h(0) and h(l). However, from Theorem 3, we have the restrictions h(0) < Pl and Pp < h(l). These restrictions imply that there are only six possible inequalities between the four numbers: (1) h(0) < PL < PP < h(l), (2) h(0) < PP < PL < h(l), (3) h(0) < PP < h(l) < PL, (4) PP < h(0) < PL < h(l), (5) PP < h(l) < h(0) < PL and (6) Pp < h(0) < h(l) < Pl- Since h(X) is continuous over (0,1) and thus assumes all intermediate values between h(0) and h(l), it can be seen that we can choose A between 0 and 1 so that Pl < h(X*) < Pp for case (1) or so that Pp < h(X*) < PL for cases (2) to (6), which establishes (20) or (21).qed It should be noted that A* can be chosen so that (20) or (21) is satisfied and in addition ^[A*^1 + (1 — X*)x°] lies between F(x°) and i^x1). Thus the Paasche and Laspeyres indexes provide bounds for the Koniis cost of living index for some reference indifference surface which lies between the period 0 and period 1 indifference surfaces. The above theorems provide bounds for the Koniis price index Pk(p°p1, x) under various hypotheses. We cannot improve upon these bounds unless we are willing to make specific assumptions about the functional form for the aggregator function F, a strategy we will pursue in Sections 5 and 6. 3. The Koniis, Allen and Malmquist Quantity Indexes In the case of only one commodity, a quantity index could be defined as x\/x\, the ratio of the quantity in period 1 to the quantity in period 0. This ratio is also equal to the ratio of expenditures in the two periods, p\x\/p1x1, divided by the price index p\/p\. This suggests that a reasonable notion of a quantity 186 Essays in Index Number Theory 7. The Economic Theory 187 index in the general N commodity case could be the expenditure ratio deflated by the Koniis cost of living index. Thus we define the Konus-Pollak [1971a; 64] implicit quantity index for p° 3> Oat, p1 3> Oat, x° > Oat, x1 > Oat and x > Oat as and the Paasche-Konüs implicit quantity index as (22) (23) ?*r(p°,p\ x°, x\ x) = p^x'/p^x^Kip^p^x) _ ClFix1)^1} /ClFix)^1} C[F(x< ),P \ I °),p°}/ C[F(x),p»] where (23) follows if the consumer or producer is engaging in cost minimizing behavior during the two periods; i.e. (23) follows if (14) is true. Note that Qk depends on the period 0 prices and quantities, p° and x°, the period 1 prices and quantities, p1 and x1, and the reference indifference surface indexed by the quantity vector x. The following result shows that Qk gives the correct answer (at least ordinally) if the reference quantity vector x is chosen appropriately. Theorem 5. Suppose F satisfies conditions I and (14) holds, (i) If F(x1) > F(x°), then for every x > 0N such that Fix1) > F(x) > F(x°), Qk{p°,PX, a;0, a;1, a;) > 1. (ii) If F(xx) — F(x°), then, for every x > Oat such that F(x) = ^(a;1) = F(x°), Qxip0^1, a;0,a;1,a;) = 1. (Hi) If Fix1) < F(x°), then for every x > 0N such that F(xx) < F(x) < F(x°), Qxip0^1, a;0,a;1,a;) < 1. Proof of (i): ?K(pt\p\z0,x1,:z) CjFjx1)^1] C[F(x),p°] . C[F{x),pi] C[F(x°),p°] wm&W > 1 since Fix1) > F(x) implies C^a;1),p1} > C[.F(a;), p1] and F(x) > F(x°) implies C[F(x),p°] > C[F(x°),p°] with at least one of the inequalities holding strictly, using property (iii) on the cost function C. Parts (ii) and (iii) follow in an analogous manner.qed It can be verified that if F(xx) > _F(a;0) > F(x), then, if F is not homo-thetic, it is not necessarily the case that Qk(p0,px, x°, x1, x) > 1- However, if we choose x to be a;0 or x1, then the resulting Qk will have the desirable properties outlined in Theorem 5. Thus define the Laspeyres-Koniis implicit quantity index as Qk(p°, p\ x°, x\ x°) = p^x'/p^x^Kip0,?1^0) = cwix^^ycmx0)^] ■ (cwix^^ycwix0)^]) using (5) and (14) (24) =C[F(x%P*yC[F(x°),P*\ IK (p0,P1,x0,x\x1)=p1tx1/p0tx0Pk(p°,p\x1) (25) = C[F(x1),p»]/C[F(x0),p°i where (25) follows using definition (5) for Pk and the assumptions (14) of cost minimizing behavior. It turns out that the quantity indexes defined by (24) and (25) are special cases of another class of quantity indexes. For x° > 0^, x1 > 0^ and p^- On, define the Allen [1949; 199] quantity index as (26) )A(x°,x\p) = C[JF(a:1),p]/C[JF(a:0),p]. Note that Qk(p°, p1, a;0, a;1, a;) — Qa(x°, x, p°)Qa(x, x1 ^p1) and that the Laspeyres-Allen quantity index Qa^jX1^0) equals the Paasche-Koniis implicit quantity index Qk(p°^p1, x°, x1, x1) while the Paasche-Allen quantity index Qa(x°, x1,p1) equals Qk(p0,px, x°, x1, x°), assuming that (14) holds. Theorem 6. Suppose F satisfies conditions I. (i) If F(x1) > F(x°) > u, then Qa{x0,xx,p) > 1 for every p ^> Oat- (ii) If F(xx) — F(x°) > u, then Qa(x°,xx,p) — 1 for every p > Oat. (Hi) If u < F(x1) < F(x°), then Qa(^°,^1,p) < 1 for every p ^> Oat- The proof of the above lemma follows directly from definition (26) and property (iii) for the cost function C(u,p): increasingness in u.14 It turns out that Allen quantity indexes do not satisfy bounds analogous to those given by Theorem 2 for the Koniis price indexes. However, there is a counterpart to Theorem 3. Theorem 7. (Samuelson [1947; 162], Allen [1949; 199]): If the aggregator function F is continuous from above and (14) holds, then (27) QA(x°,x1,p°) p1T x1/p1T x° = QP(p° ,px, x°, x1); i.e. the Laspeyres-Allen quantity index is hounded from above by the Laspeyres quantity index Ql and the Paasche-Allen quantity index is bounded below by the Paasche quantity index Qp. Proof: Qa^0,!1,?0) = C[F(x^^^/p^x0 using (26) and (14) = min{p0Tx : F(x) > F^j/p^x0 x Kp^x'/p^x0 14 We also utilize property (ii) for C : C(u, p) — 0 for every p 3> 0 n- 188 Essays in Index Number Theory since x1 is feasible for the minimization problem. Similarly, Qa^V1,/) =p1Tx1/mm{p1Tx : F(x) > F(x0)} x >p1TxVp1Tx° since x° is feasible for the minimization problem and p1Tx° > O.qed Theorem 8. If F is homothetic (so that there exists a continuous, monotoni-cally increasing function of one variable such that G[F(x)] is neoclassical) and (14) holds, then for every x ^ Oat and p ^ Oat 7. The Economic Theory 189 (29) Proof: )k(j>°\pX ,x°\x1 ,x) = QA(x°,x1,p) = G[F(x1)]/G[F(x»)]. , n nix CVFix1)^1] /ClFtx)^1] . . . (p°,p, x°, x1, x) = \ ) ('"l /t4^H^ usin§ (23) C[F(x°),p0]/ C[F(x),p°] G[F(x1)}^1) /GiFix)}^1) -/->/ ( G[F(x°)}c(p0)/ G[F(x)]c(p°) by homotheticity of F = G[F(x1)]/G[F(x0)] = GlFix^cW/GlFix0)]^) = C[F(x1),p]/C[F(x°),p] by homotheticity again = QA(x°,x1,p). qed Corollary 8.1. (Samuelson and Swamy [1974; 570]): If Qa(x°,xx,p) is independent of p and F satisfies conditions I, then F must he homothetic. Proof: If QA(x°, xx,p) is independent of p, then C[F(x1),p]/C[F(x1),p] is independent of p for all x° ^> Oat and x1 ^> Oat. Thus we must have C[F(x),p] — G(F(x)]c(p) for some functions G and c which implies that F is homothetic.qed Corollary 8.2. If F is neoclassical (so that G(u) = u) and (14) holds, then for every x On, and every p Oat' (30) (p°,p\x°,x\x) = QA(x°,x\p) = Fix^/Fix0). Corollary 8.3. If F is homothetic and (14) holds, then for every x > Oat and p^> Oat •' p1Tx1/p1Tx° < QK(p0,p1,x0,x1,x)=QA(x0,x\p) Proof: From (28), Qp°] ^(p^x'/p^x^/Pl^Qp. Similarly, if (21) holds then QP < QK[x°, x1 ,p°,px, X*xx + (1 - X*)x°] < Ql-QED Theorem 10. Let F be continuous from above and suppose (14) holds. Then there exists a A* such that 0 < A* < 1 and Qa[x°, x1, X*p1 + (1 — X*)p°] lies between Ql and Qp. Proof: Define h(X) = QA[x°, x1, Xp1 + (1 - X)p°] = C[F(x1)^1 + (1 -X)p°]/C[F(x°), Xp1 + (1 — X)p0]. Since F is continuous from above, C(u,p) is continuous in p and thus h(X) is continuous for 0 < A < 1. Note that h(0) = QA(x°,xx,p) and h(l) = QA(x°, x1^1). From Theorem 7, h(0) < QL and Qp < h(l). Now repeat the proof of Theorem 9 with Ql and Qp replacing Pl and Pp.qed Thus the Paasche and Laspeyres quantity indexes (which are observable) bound both the implicit Koniis quantity index Qk and the Allen quantity index Qa, provided that we choose appropriate reference vectors between x° and x1 or p° and p1 respectively. However, it is also necessary to assume cost minimizing behavior on the part of the consumer or producer during the two periods in order to derive the above bounds. 190 Essays in Index Number Theory 7. The Economic Theory 191 Recall that the Koniis price index Pk had the desirable property that Pk(p°, Xp°,x) = \PK(p°,p°,x) for all A > 0, p° > Oat, and x > Ojv; i.e. if the current period prices were proportional to the base period prices, then the price index equalled this common factor of proportionality A. It would be desirable if an analogous homogeneity property held for the quantity indexes. Unfortunately, it is not always the case that Qk(x°, Xx°,p°^p1, x) — A or that Qa(x°, Xx°,p) — A. However, the following quantity index does have this desirable homogeneity property. For x > Oat, x° > Oat, x1 > Oat, define the Malmquist [1953; 232] quantity index as (32) QmOeV1,^) = D[F (x),^1]/^ (x),x°] where D[u, x] = maxfc{/j : F(x/k) > u, k > 0} is the deflation function15 which corresponds to the aggregator function F. Thus D[F(x), x1] is the biggest number which will just deflate the period 1 quantity vector x1 onto the boundary of the utility (or production) possibility set [x : F(x) > F(x),x > Oat] indexed by the quantity vector x while D[F(x),x°] is the biggest number which will just deflate the period 0 quantity vector x° onto the utility possibility set indexed by x, and Qm is the ratio of these two deflation factors. Note that the assumption of cost minimizing behavior is not required in order to define the Malmquist quantity index Qm- Theorem 11. (Malmquist [1953; 231], Pollak [1971a; 62]): If F satisfies conditions I, then (i) A > 0, x° 3> Oat, x 3> Oat implies Qm(x°, Xx°,x) — A and (ii) x° 3> Oat, x1 3> Oat, x2 3> Oat, x 3> Oat implies Qm(x°,x1,x)Qm(x1,x2,x) — Qm (x® , x2, x). Proof: (i) If F is merely continuous from above and increasing, then D[F(x),x] is well defined for all x 3> Oat and x 3> Oat- Moreover, D has the 15If F satisfies conditions I, then it can be shown (e.g., see Diewert, [1978c]), that the deflation function D satisfies conditions IV: (i) D{u, x) is a real valued function of N +1 variables defined over Int U x Int 51 — {u : u < u < ou} x {x : x Oat} and is continuous over this domain, (ii) D(u, x) — +oo for every x G Int 51; i.e., un G Int U, limun — u, x G Int 51 implies lim„ D(un,x) — +oo, (iii) D(u, x) is decreasing in u for every x G Int 51; i.e., if x G Int 51, u', u" G Int U with u' < u", then D(u', x) > D(u", x), (iv) D(ou, x) — 0 for every x G Int 51; i.e. u" G Int U, limu" — ou, x G Int 51 implies Yivnn D(un,x) — 0, (v) D(u,x) is (positively) linearly homogeneous in x for every u G Int U; i.e., u G Int U, A > 0, x G Int 51 implies D(u,Xx) — XD(u,x), (vi) D(u,x) is concave in x for every u G Int U, (vii) D(u,x) is increasing in x for every w G Int U; i.e., m G Int U, x', x" G Int 51 implies D(u,x' + x") > D(u,x'), and (viii) D is such that the function F(x) = {u : u G Int U, D(u, x) — 1} defined for x 3> Oat has a continuous extension to x > Oat- following homogeneity property (recall property (v) of conditions IV on D): for A > 0, D[F{x),Xx] = XD[F{x),x\. Thus QM(x°,Xx°,x) = D[F(x),Xx0}/ D[F(x),x°] = AD[F(x),x°]/D[F(x),x°] = A. (ii) follows directly from definition (32).qed Property (ii) in the above theorem is a desirable transitivity property of Qm- Qk, Qa, Pa and Pk all possess the analogous transitivity property (or circularity property as it is sometimes called in the index number literature). Theorem 12. If F satisfies conditions I, x° > Oat, x1 > Oat, x > Oat and F(x) is between F(x°) and F(x1), then the Malmquist quantity index Qm(x°', x1 ,x) will correctly indicate whether the aggregate has remained constant, increased or decreased from period 0 to period 1. Proof: (i) Suppose F(x°) = F(x) = F(xx). Then QM(x°, x1, x) = F>[F(x), x1]/F'[F(x), x°] = 1/1 = 1. (ii) Suppose F(x°) < F(x) < Fix1) with F(x°) < Fix1). Then QM(x°, x1, x) = k1 /k° where F(x1//j1) = F(x) < Fix1) which implies k1 > 1 and F(x°/k°) — F(x) > F(x°) which implies 0 < k° < 1. Since at least one of the inequalities F(x) < F(xx) and F(x) < F(x°) is strict; at least one of the inequalities k1 > 1 and k° < 1 must also be strict. Thus Qm(x°,xx,x) — kx/k° > 1. The remaining case is similar.qed If F is nonhomothetic, then the restriction that the reference indifference surface indexed by F(x) lie between the indifference surfaces indexed by F(x°) and F(xx) is necessary in order to prove Theorem 12; e.g. if F(x°) < F(xx) < F(x), then it need not be the case that Qm{x°,xx,x) > 1. The following result shows that the Malmquist quantity index satisfies the analogue to the Joseph-Pollak bounds for the Koniis price index. Theorem 13. If F satisfies conditions I and x° > Oat, x1 > Oat, x > Oat, then (33) _ min{x^ jx\ : i — 1,. .., N} < Qm(x°, x1 ,x) < maxji- jx\ : i — 1, .. ., N}. i i Proof: If F satisfies conditions I, then the deflation function D satisfies conditions IV. Thus D(u,x) satisfies the same mathematical regularity properties with respect to x as C(u,p) satisfies with respect to p. Since C[F(aT),p1]/C[F(aT),p°] = Pk(p°,px,x) satisfies the inequalities in (9), D[F(x), x1]/D(F(x), x°] = Qm{x°tX1 ,x) will satisfy the analogous inequalities (33).16qed 16More explicitly, C[F(x),p] is the support function for the set L[F(x)] = {x : pTx > C[F(x),p] for every p 3> Oat} and the sets {x : p0Tx > p0Tx°, x > Oat} and {x : p1Tx > p1Txx,x > Oat} form outer approximations to this set where x° G dpC[F(x),p°] and x1 G dpClFffljp1]. dpC(u,p°) denotes the set of 192 Essays in Index Number Theory 7. The Economic Theory 193 In general, the Malmquist quantity index will depend on the reference indifference surface indexed by x. As usual, two natural choices for x are x° or x1, the observed quantity choices during period 0 or 1. Thus the Laspeyres-Malmquist quantity index is defined as Qm^x0,!1,!0) = DiFix^^^/DiFix0)^0] = D[F (a;0), a;1] since D[F(x°),x°] — 1 if F is continuous from above and increasing, and the Paasche-Malmquist quantity index is defined as QmOW1) = DlFix^^^/DlFix1)^0} = l/DjFfa:1),!0] since D[F{x1), x1] — 1 if F is continuous from above and increasing. Theorem 14. (Malmquist [1953; 231]): Suppose F satisfies conditions I and (14) holds. Then (34) (35) Proof: )M(x°, x1, x°) < p0Tx1/p0Tx° = QL and )M(x°, x\ x1) > p1Tx1/p1Tx° = QP. 1m(x°,x\x°) = DiFix0)^1} = max{k : F(xx/k) > F(x°), k > 0} = k1 where Fix1/k1) = F(x°). Now p0Tx0 = C{F(x°),p0} = mm{p0Tx : F(x) > F(x0)} x Kp^x'/k1 since x1 /k1 is feasible for the cost minimization problem. Thus k1 — Qm(x°, x1, x°) D[F(x), x] for every x 3> Oat} and the sets {p : pTx° > p0Tx°,p > On} and {p : pTx1 > p1Txx,p > Oat} form outer approximations to this set where p° e dxD[F(x),x°] and p1 e dxD[F(x), x1]. Theorem 15. Suppose F satisfies conditions I and (14) holds. Then there exists a X* such that 0 < A* < 1 and Qm(x°, x1, X*x1 + (1 — X*)x°) lies between Ql and Qp. Proof: Define h(X) = QM[x°, x1, Xx1 + (1 - X)x°] = ^[^[Aa;1 + (1 - A)^0],^1 /D FlXx1 + (1 - X)x°],x° . Since ^[Aa;1 + (1 - X)x°] is continuous with respect to A and D(u,x) is continuous with respect to u (recall property (i) of conditions IV on D, h(X) is continuous for A between 0 and 1. Moreover, h(0) = QM(x°,a;1,a;0) and h(l) = QM{x°,x1 ,x1). From Theorem 14, h(0) < Ql and Qp < h(l). Now repeat the proof of Theorem 10.qed It should be noted that A* can be chosen so that 0 < A* < 1 and Qm[%°, x1, X*x1 + (1—X*)x°] lies between Ql and Qp, and in addition, F[X*xx + (1 — X*)x°] lies between F(x°) and i^x1). Thus the Paasche and Laspeyres quantity indexes provide bounds for the Malmquist quantity index for some reference indifference surface which lies between the period 0 and period 1 indifference surfaces. The following theorem relates the Paasche and Laspeyres Malmquist quantity indexes to the Paasche and Laspeyres implicit Koniis and Allen quantity indexes. Theorem 16. (Malmquist [1953; 233]): Suppose F satisfies conditions I and (14) holds. Then (36) Qm(x0,x1,x0) febW,!1,!1) = Qa(x°,x\p°). Proof: )m(^W°) = -D^OAx1] = k1 say where F^/k1) = F(x°). Also )A{x\x\p1)=p1Tx1/C[F{x°),p1 ig (26) and (14) = Qk(p°,p\x0,x\x0) using (23) = p1Tx1/min{p1Tx : F(x) > F(x0)} x < p1Tx1/p1T(x1/k1) since x1 /k1 is feasible but not necessarily optimal — k1 194 Essays in Index Number Theory 7. The Economic Theory 195 which establishes (36). (37) follows in a similar manner.qed It is obvious that an implicit Malmquist price index Pm can be defined as the expenditure ratio for the two periods deflated by Qm'- i-e. define (38) Pm (p°, p1 ,x°,x1,x) = p1tx°/p0tx°Qm (x° ,x\x). However, the resulting price index does not have the desirable homogeneity property Pm(p°, Xp°, x°, x1, x) — A. Thus Pm has properties analogous to the implicit Koniis quantity index Qk, except that the role of prices and quantities is reversed. Now that we have studied price and quantity indexes separately, it is time to observe that it is essential to study them together. For empirical work, it is highly desirable that the product of the price index P and the quantity index Q equal the actual expenditure ratio for the two periods under consideration, p1Jx1 /p0Jx°. If P and Q satisfy this property, then we say that P and Q satisfy the weak factor reversal test17 or the product test.1® We have seen that the Koniis price index Pk is a desirable price index and that the Malmquist quantity index Qm is a desirable quantity index since they each have a desirable homogeneity property. The following result shows that there exists at least one reference indifference surface such that Pk and Qm satisfy the product test. Theorem 17. (Malmquist [1953; 234]): Suppose the aggregator function F satisfies conditions I and (14) holds. Then there exists a A* such that 0 < A* < 1 and (39) PK[p°,p\ X*x1 + (l-X*)x°}QM[x0,x1,X*x1 + (l-X*)x0} = p1Tx1/p0Tx°. Proof: For 0 < A < 1, define the continuous function h(X) = PkIp0^1, Xx1 + (1 - X)x°}Qm[x°, x1, Xx1 + (1 - X)x0}. Thus h(0) = Pk(p0,p1,x0)Qm(x0,x1,x°) = C[F(x0)y]/C[F(x0),p0]] iDiFix^^x^/DiFix0)^0} by (5) and (32) ClFix0)^1} ClFix1)^1} < C[F(x°),p0} CiFix0)^1 sing (36) and (26) 17The concept is associated with Irving Fisher [1922]. 18This terminology is due to Frisch [1930]. < p1Tx1/p0Tx° using (14) cif^1),?1]/^1),?0]] [ciFix^yyciFix0),?0] CiFix1)^1} DiFix1)^1} . using (37), (26) and (32) CiFix1)^0] DiFix1)^0] — Pk(p°jP1 , ^1)Qm(^°, x1, x1) using (5) and (32) = h{l). Since h(X) is continuous over [0,1] and since h(0) < p1Jx1 /p0Jx° < h(l), there exists 0 < A* < 1 such that h(X*) — p^x1 /p0Tx° and thus (39) is satisfied. Moreover, since h(X) = (C^Ax1 + (1 - X)x°],p1] /C^Xx1 + (1 -X)x°],p°]) (D [FlXx1 + (1 -X)x°], x1]/D [FlXx1 + (1 -X)x°],x0]), we can choose A* so that ^[A*^1 + (1 - X*)x°] lies between F(x°) and F^.qed Thus the reference indifference surface indexed by X*x1 + (1 — X*)x° which occurs in the above theorem lies between the surfaces indexed by x° and x1, the quantity vectors observed during periods 0 and 1. The final result in this section shows that all three quantity indexes that we have considered coincide (and are independent of reference price or quantity vectors) if the aggregator function is homothetic. Theorem 18. (Pollak [1971a; 65]): If F is homothetic (so that there exists a continuous, monotonically increasing function of one variable such that G[F(x)] is neoclassical) and (14) holds, then for every x ^ Oat and p ^ Oat (40) ^mO^0,^1,^) = Qk(p°',px ,x° ,xx ,x) = Qa^^x1^) ^GlFix^/GlFix0)}. Proof: maxi/j : Fix1 Ik) > F(x)}/maxi/j : Fix0Ik) > Fix)} k>0 k>0 QmO^V1,^) = D[F(x),x1]/D[F(x),x° = maxi/j : Fix1 Ik) > F( k>0 _ maxfcjfc : G[F(x1 /k)] > G[F(x)],k > 0} ~ maxfc[/j : G[F(x°/k)} > G[F(x)},k> 0} = k1/kQ say where G[F(x1/A:1)] = G[F(x)] and G[F(x°/k0)} = G[F(x)}. Since G[F(x)} is linearly homogeneous in x, the last two equations imply k1 — G[F(x1)]/G[F(x)] and k° — G[F(x°)]/G[F(x)] which in turn implies k1 /k° — Qm(x°\xx,x) — G[F(x1)]/G[F(x0)]. The other two equalities in (40) now follow from (29) and (30).qed 196 Essays in Index Number Theory 7. The Economic Theory 197 Corollary 18.1. Qp < Om^V1,1) — Qk(p°\p*,x°,xx,x) — Qa(x°,xx,p) < QL. Proof: Follows from (40) and (31).qed Corollary 18.2. If Qm(x°, x1, x) is independent of x > Oat for all x° > Oat and x1 3> Oat and F satisfies conditions I, then F must be homothetic. Proof: If Qm(x°, x1, x) is independent of x, then D[F(x), x1]/D[F(x),x°] is independent of x for all x° ^> Oat and x1 ^> Oat- Thus we must have D[F(x),x°] — f(x°)/G[F(x)] for some functions / and G. Since F satisfies conditions I, D must satisfy conditions IV and it is evident that / can be taken to be neoclassical and G can be taken to be a monotonically increasing, continuous function of one variable with G(u) > 0 if u > u = -F(Oat). Since D[F(x),x] — I — f(x)/G[F(x)] for every x > Oat, we have G[F(x)] — f(x), a positive, increasing, concave, linearly homogeneous and continuous function for x ^> Oat- Thus F is homothetic.qed Finally, we note that if F is neoclassical and (14) holds, then: (i) all quantity indexes coincide and equal the value of the aggregator function evaluated at the period 1 quantities x1 divided by the value of F evaluated at the period 0 quantities x°; i.e., we have (41) Qm(x0,x1,x) = Qk(p0,P1,x0,x1,x) = Qa(x°,x1,p)=F(x1)/F(x0) for all x On and p On', (ii) all price indexes coincide and equal the ratio of unit costs for the two periods; i.e., we have (42) Pk(p°,p1,x) = Pm(p0,p1,x°,x1,x) = c(p1)/c(p0) for all x On', and (iii) the expenditure ratio for the two periods is equal to the product of the price index times the quantity index: (43) p1Tx1/p0Tx° = [c^/c^Wix^/Fix0)]. 4. Other Approaches to Index Number Theory During the period 1875-1925, perhaps the main approach to index number theory was what Frisch [1936] called the 'atomistic' or 'statistical' approach. This approach assumed that all prices are affected proportionately (except for random errors) by the expansion of the money supply. Therefore, it does not matter which price index was used to measure the common factor of proportionality, as long as the index number contains a sufficient number of statistically independent price ratios. Proponents of this approach were Jevons and Edge-worth but the approach was rather successfully attacked by Bowley [1928] and Keynes. For references to this literature, see Frisch [1936; 2-5]. A 'neostatistical' approach has been initiated by Theil [I960]. For the case of two observations, Theil's best linear price and quantity indexes Pq, Pi, Qo, Qi are the solution to the following constrained least squares problem: 4 min y ef subject to (u) (i) pOTa;0 = PoQo + ei, (ii) p0T x1 = P0Q1 + e2 (iii) p1Tx° = PiQ0 + e3, (iv) p1Tx1 = P1Q1 + e4: and one other normalization such as Pq — 1 is required. As usual, p° and p1 are the price vectors for the two periods while x° and x1 are the corresponding quantity vectors. Pq and Pi are scalars which are interpreted as the price level in periods 0 and 1 respectively while Qo and Q\ are the quantity levels for the two periods. Finally, the e4 are regarded as errors. Kloek and de Wit [1961] suggested a number of modifications to Theil's approach; they suggested (44) for the case of two observations, but with the following three sets of additional normalizations: (1) Pq — 1, e\ — 0, (2) Pq — 1, e\ + — 0, and (3) Pq — 1, e\ — 0, — 0. Stuvel [1957] and Banerjee [1975] have suggested similar 'neostatistical' index number formulae: Stuvel's index numbers P\/Pq and Qi/Qo can be generated by solving (44) subject to the additional normalizations Pq — 1, e\ — 0, — 0 and e2 — e^. The other major approach to index number theory is the test or axiomatic approach, initiated by Irving Fisher [1911] [1922]. The test approach assumes that the price and quantity indexes are functions of the price and quantity vectors pertaining to two periods, say P(p°,p1,x°,x1) and QijPjP1,x°,x1). Tests are a prior 'reasonable' properties that the functions P and Q should possess. However, several researchers (e.g. Frisch [1930], Wald [1937], Samuel-son [1974a], Eichhorn [1976] [1978a], Eichhorn and Voeller [1976]) have shown that not all a priori reasonable properties for P and Q can be consistent with each other; i.e. there are various impossibility theorems. Moreover, if one works with a restricted set of tests which are consistent, the resulting family of index number formulae is often not uniquely determined. However, it turns out that the economic and test approaches to index number theory can be partially reconciled. In the following two sections, we shall assume explicit functional forms for the underlying aggregator function plus the assumption of cost minimizing behavior on the part of the consumer or producer. We shall show that certain functional forms for the aggregator 198 Essays in Index Number Theory 7. The Economic Theory 199 function can be associated with certain functional forms for index number formulae. Many of the resulting index number formulae (e.g. Fisher's [1922] ideal formula) have been suggested as desirable in the literature on the test approach to index number theory. 5. Exact Index Number Formulae Suppose we are given price and quantity data for two periods, p°, p1, x° and x1. A price index P is defined to be a function of prices and quantities, P(p°, p1, x°, x1), while a quantity index Q is defined to be another function of the observable prices and quantities for the two periods, Q(p°,p1,x°,x1). Given either a price index or a quantity index, the other function can be defined implicitly by the following equation (Fisher's [1922] weak factor reversal test): (45) P(p°,p\x0,x1)Q(p0,p\x0,x1)=p1Tx1/p0Tx°; i.e., the product of the price index times the quantity index should equal the expenditure ratio between the two periods. Assume that the producer or consumer is maximizing a neoclassical19 aggregator function / subject to a budget constraint during the two periods. Under these conditions, it can be shown that the consumer (or producer) is also minimizing cost subject to a utility (or output) constraint and that the cost function C which corresponds to / can be written as (46) C\f(x),p]=f(x)c(p) for x > On and p On where c(p) = minx{pTx : f(x) > 1, x > On} is f's unit cost function.20 A quantity index Q{p°tP1, x°, x1) is defined to be exact for a neoclassical aggregator function / if, for every p° 3> On, p1 3> Oat,21 xt 3> Oat a solution to the aggregator maximization problem maxI{/(i) : prTx < prTxr, x > On} — f(xr) > 0 for r — 0,1, we have (47) Q(p0,p\x°,x1) = f(x1)/f(x0). Thus in (47), the price and quantity vectors {p°, p1, x°, x1) are not regarded as completely independent variables — on the contrary, we assume 19/ is positive, linearly homogeneous and concave over the positive orthant and is extended to the nonnegative orthant 51 by continuity. 20Recall (6) with G(u) = u. The function c is also neoclassical. 21Sometimes p° and p1 are restricted to a subset of the positive orthant. that (p°, x°) and (p1, x1) satisfy the following restrictions in order for the price and quantity vectors to be consistent with 'utility' maximizing behavior during the two periods: (48) pT > Oat, xt > Oat, f(xr) = max{/(x) : prTx < prTxr, x > QN} > 0; r = 0,1. x If / is neoclassical, then, using (46), it can be verified that (48) implies (49) and vice versa: (49) pr » Oat, xr » Oat, prTxr = f(xr)c(pr) = C(f(xr),pr) > 0; r = 0,1. Now we are ready to define the notion of an exact price index. A price index P(p°,p1,x°,x1) is defined to be exact for a neoclassical aggregator function / which has the dual unit cost function c, if for every (p°,x°) and (p1^1) which satisfies (48) or (49), we have (50) P(p0,p1,x°,x1) = c(p1)/c(p0). Note that if Q is exact for a neoclassical aggregator function /, then Q can be interpreted as a Malmquist, Allen or implicit Koniis quantity index (recall (41)), and the corresponding price index P defined implicitly by Q via (45) can be interpreted as a Koniis or implicit Malmquist price index (recall (42)). Some examples of exact index number formulae are presented in the following theorems. Before proceeding with these theorems, it is convenient to develop some implications of (48) and (49). If / is neoclassical, (48) is satisfied, and / is differentiable at x° and x1, then (51) pr/prTxr = Vf(xr)/xrTVf(xr)=Vf(xr)/f(xr); r = 0,1. The first equality in (51) follows from the Hotelling [1935; 71], Wold [1944; 69-71], [1953; 145] identity22 while the second equality follows from Euler's Theorem on linearly homogeneous functions, f(xr) — xrTV/(xr). Also if / is neoclassical, (49) holds and f's unit cost function c is differentiable at p° and p1, then (52) xr/prTxr = VpC[f(xr),pr]/C[f(xr),pr] = Vc(pr)/c(pr); r = 0,1. The first equality in (52) follows from Shephard's [1953; 11] Lemma while the second equality follows from (49). 22 Alternatively, the first equality in (51) is implied by the Kuhn-Tucker conditions for the concave programming problem in (48) upon eliminating the Lagrange multiplier for the binding constraint prTx < prTxr. The nonnegativity constraints x > On are not binding because we assume the solution xr On- 200 Essays in Index Number Theory 7. The Economic Theory 201 Theorem 19. (Koniis and Byushgens [1926; 162], Pollak [1971a], Samuelson and Swamy [1974; 574]): The Paasche and Laspeyres price indexes, Pp(p°,px, x°,xx) = p1Jx1 /p0Jx1 and Pl(p0,PX, x1) = p1Tx° /p0Tx°, and the Paasche and Laspeyres quantity indexes, Qp (p°, p1, x°, x1) = p1Tx1 /p1Tx° and Ql (p°, p1, x°,xx) = p0Jx1 / p0Tx°, are exact for a Leontief [1941] aggregator function, f(x) = miiii{xi/bi : i — 1,..., N}, where x = (xi,... ,xn)t > On and & = (&!,..., &at)t ^ Oat is a vector of positive constants. Proof: If / is the Leontief or fixed coefficients aggregator function defined above, then its unit cost function is c(p) = pTb for p ^ Oat. Now assume (49). Then PL=p1Tx°/p0Tx° = p1T[Vc(p°)/c(p0)] using (52) = p1Tb/c(p°) since Vc(p°) = b ee dp1)/c(p°). Similarly, Pp^p^x'/p^x1 = l/(p°Ta;1/p1Ta;1) = l/^lVc^1)/^1)]] using (52) — c(p1)/p0Tb since Vc(p1) — b ee ctf)/C(jP). Thus Pl and Pp are exact price indexes for /, and thus the corresponding quantity indexes, Qp and Ql, defined implicitly by the weak factor reversal test (45), are exact quantity indexes for /.qed Theorem 20. (Pollak [1971a] Samuelson and Swamy [1974; 574]): The Paasche and Laspeyres price and quantity indexes are also exact for a linear aggregator function, f(x) = aTx where aT = (ai,... ^ Oat is a vector of fixed constants. Proof: Assume (48).23 Then Ql ee p^x'/p^x0 = x1T{\/f(x°)/f(x0)} using (51) — x1Ta/f(x°) since V/(x) — a ^iV)//(A 23Note that the definition of exactness requires xT 3> Oat and xT is a solution to the appropriate aggregator maximization problem. Thus it can be seen that p° must be proportional to a. Similarly, Qp — f(x1)/f(x°) and so Ql and Qp are exact for the linear aggregator function / defined above. Thus the corresponding price indexes, Pp and Pl, defined implicitly by the weak factor reversal test (45) are exact price indexes for / and its corresponding unit cost function, c(p) = minx{pTx : aTx > 1, x > On} — niini{pi/ai : i — 1,..., iV}.qed The above theorems show that more than one index number formula can be exact for the same aggregator function, and one index number formula can be exact for quite different aggregator functions. Theorem 21. (Koniis and Byushgens [1926; 163-166], Afriat [1972b; 46], Pollak [1971a], Samuelson and Swamy [1974; 574]): The family of geometric price indexes defined by Pg{p°, p1, x°, x1) = Yii=i(Pi / PiY1 (where for i — 1, 2,..., N, st ee mi(si,s}), s° = p^x® /p0T x°, s] = p\x\lp1Txx and rrii is any function which has the property mj(s, s) = s) is exact for a Cobb-Douglas [1928] aggregator function f defined by n n a, — 1. (53) f(x) — ao Y[ XT, where ao > 0, ai > 0,..., ajv > 0, i=l The family of geometric quantity indexes, n QgO/V^V1) = Hixl/x0^, Sl ee m^, si) i=l is also exact for the aggregator function defined by (53). Proof: If / is Cobb-Douglas and (48) holds, then for r — 0,1, differentiating (53) yields rdf(xr) dx. -/f(xr) = a4 = xrlprJpr xr using (51) Thus s' — sj — ai — Si — rrii(s', sj) and n n Pcip^p1^0^1) ee n(Pi/p?)si= i=l n i=l n i=l i=l since it can be verified by Lagrangian techniques that the Cobb-Douglas function defined by (53) has the unit cost function n n z(p) — k Wp'i1 where k — 1/ao a"1 ■ i=l i=l 202 Essays in Index Number Theory 7. The Economic Theory 203 Thus Pa is exact for /. Similarly n n i=l i=l n n i=l i=l and so is also exact for / defined by (53).qed Theorem 22. (Byushgens [1925], Koniis and Byushgens [1926; 1971], Frisch [1936; 30], Wald [1939; 331], Afriat [1972b; 45] [1977], Pollak [1971a] and Diew-ert [1976a; 132]):24 Irving Fisher's [1922] ideal quantity index and the corresponding price index Ppip^p^x^x1) = (p1Tx1/p0Tx1)1/2(p1Tx°/p0Tx0)1/2 = (PPPL)V2=p1Tx1/p0Tx0QF(p0,p1,x°,x1) are exact for the homogeneous quadratic function f defined by (54) f(x) = {xTAxf2, xeS where A is a symmetric N x N matrix of constants and S is any open, convex subset of the nonnegative orthant 51 such that f is positive, linearly homogeneous and concave over this subset.25 Proof: We suppose that the following modified version of (48) holds:26 (55) pT > Oat, xt > Oat, f(xr) = max{/(x) : prTx < prTxr, x e S}; r = 0,1. 24Samuelson [1947; 155] states that S. Alexander also derived this result in an unpublished Harvard paper. 25/ can be extended to the nonnegative orthant as follows. Because (xTAx)1/2 is linearly homogeneous, S can be taken to be a convex cone. Extend / to S, the closure of S, by continuity. Now define the free disposal level sets of / by L(u) = {x : x > x', f(x') > u, x' g S} for u > 0. The extended / is defined as f(x) = maxu{w : x g L(u), u > 0} for x > On- 26The nonnegativity constraints x > On have been replaced by x e S. Because we assume that S is an open set and we assume that xr g S, the constraints x g S are not binding in (55). Since only the budget constraints prTx < prTxr will be binding in the concave programming problems defined in (55), the Hotelling-Wold relations (51) will also hold, since the / defined by (54) is differentiable. Thus pr/prTxr = \/f(xr)/f(xr) for r = 0,1 by (51) = ^{xrTAxr)-1'22Axr/{xrTAxr)1'2 differentiating (54) (56) =Axr/xrTAxr, and QHpWV1) = [x^tf/p^x^/x^tf/p^x1)]1'2 = [xlT(Ax°/x™Ax°)/x{1T(Axl/xi-TAxl)Y'2 using (56) = (x^Ax^/ix^Ax0)1/2 since x1TAx° = x0TAxx = /(x^/fix0) using (54). Thus Qp and the corresponding implicit price index PF (p°, p1, x°, x1) = p^x1 /p0Tx°Qp (p°, p1, x°, x1) = /(a;1)^1)//^0)^0)!/^1)//^0)] ^ing (49) = cip^/cip0) are exact for the aggregator function / defined by (54) where c is the unit cost function which is dual to /.qed The set S which occurs in (54) will be nonempty if we take A to be a symmetric matrix with one positive eigenvalue (and the corresponding eigenvector is positive) while the other eigenvalues of A are zero or negative. For example, take A — aaT where a 3> Oat is a vector of positive constants. In this case, S can be taken to be the positive orthant and f(x) = (xTa,aTx)1/2 — aTx, a linear aggregator function. Thus the Fisher price and quantity indexes are also exact for a linear aggregator function. The above example shows that the matrix A in (54) does not have to be invertible. However if A^1 does exist, then, using Lagrangian techniques, it can be shown27 that c(p) = (pTA~1p)1/2 for p g S* where S* is the set of positive prices where c{p) is positive, linearly homogeneous and concave. See Pollak [1971a] and Afriat [1972b; 45]. 204 Essays in Index Number Theory 7. The Economic Theory 205 6. Superlative Index Number Formulae The last example of an exact index number formula is very important for the following reason: unlike the linear aggregator function aTx or the geometric aggregator function defined by (53), the homogeneous quadratic aggregator function f(x) = (xT Ax)1/2 can provide a second order differential approximation to an arbitrary, linearly homogeneous, twice continuously differentiable aggregator function, i.e. (xTAx)xl2 is a flexible functional form.28 Thus if the true aggregator function can be approximated closely by a homogeneous quadratic, and the producer or consumer is engaging in competitive maximizing behavior during the two periods, then the Fisher price and quantity indexes will closely approximate the true ratios of unit and output (or utility). Note that it is not necessary to econometrically estimate the (generally unknown) coefficients which occur in the A matrix, only the observable price and quantity vectors are required. Diewert [1976a; 117] defined a quantity index Q to be superlative29 if it is exact for an aggregator function / which is capable of providing a second order differential approximation to an arbitrary twice continuously differentiable linearly homogeneous aggregator function. Thus Theorem 22 implies that Fisher's ideal index number formula Qp is superlative. Theorem 23. (Koniis and Byushgens [1926; 167-172], Pollak [1971a], Diewert [1976a; 133-134]): Irving Fisher's ideal price and quantity indexes, Pf andQF, are exact for the aggregator function which is dual to the unit cost function c defined by (57) c(p) = (pTBp)1/2 28/ is a flexible functional form if it can provide a second order (differential) approximation to an arbitrary twice continuously differentiable function /* at a point x*. f differentially approximates /* at x* iff (i) f{x*) — f*{x*), (ii) V/(x*) = Vf*(x*) and (iii) V2f(x*) = V2f*(x*), where both / and /* are assumed to be twice continuously differentiable at i* (and thus the two Hessian matrices in (iii) will be symmetric). Thus a general flexible functional form / must have at least 1 + N + N(N + l)/2 free parameters. If / and /* are both linearly homogeneous, then f*(x*) — x*TVf*(x*) and V2f*(x*)x* — 0N, and thus a flexible linearly homogeneous functional form / need have only N + N(N - l)/2 = N(N + l)/2 free parameters. The term 'differential approximation' is in Lau [1974; 184]. Diewert [1974b; 125] or [1976a; 130] shows that (xTAx)xl2 is a flexible linearly homogeneous functional form. 29The term is due to Fisher [1922; 247] who defined a quantity index Q to be superlative if it was numerically close to his ideal index, Qp. where B is a symmetric matrix of constants and S* is any convex subset of 51 such that c is positive, linearly homogeneous and concave over S* .30 Proof: Assume that (49) is satisfied where p°, p1 e S*, c is defined by (57) and / is the aggregator function dual to this c. Then, since c is differentiable, (52) also holds. Thus we have P^V1,^1) = {j>1Tx1/P^x1)1'2{j>1TxQ/frxQ)1'2 = b0TVc(p1)/c(p1)]-1/2[p1TVc(p°)/c(p°)]1/2 using (52) = (p0TBP1/P1TBP1)-1'2(p1TBP0/p0TBP0)1'2 differentiating (57) = (p1TV)1/2/(P°TV)1/2 since p0TBp1 = p1TBp0 = c(p1)/c(p°) using (57). Thus Pp and the corresponding implicit quantity index QF(p°,p1,x°,x1)= p^x^p^x^Pp(p°, p1, x°, x1) =/(^WV/^WMp1)/^0)] using (49) = /(^)//(^0) are exact for the unit cost function defined by (57).qed The set S* which occurs in (57) will be nonempty if we take B to be a symmetric matrix with one positive eigenvalue (and the corresponding eigenvector is a vector with positive components) while the other eigenvalues of B are zero or negative. For example, take B = bbT where b Oat is a vector of positive constants. In this case, S* can be taken to be the positive orthant and c(p) — {pTbbTp)1/2 — pTb, a Leontief unit cost function. Thus the Fisher price and quantity indexes are also exact for a Leontief aggregator function.31 This example shows that the / and c defined by Theorem 23 do not have to coincide with the / and c defined in Theorem 22. However, Qp and Pp are exact for both classes of functions. Of course, if B~x or A~x exist, then the / and c defined in Theorem 22 coincide with the / and c defined in Theorem 23 (for a subset of prices and quantities at least). A price index P is defined to be superlative if it is exact for a unit cost function c which can provide a second order differential approximation to an 30 The aggregator function / which is dual to c defined by (57) can be constructed using the local duality techniques explained in Blackorby and Diewert [1979]. 31This fact was first noted by Pollak [1971a]. 206 Essays in Index Number Theory arbitrary twice continuously differentiable unit cost function. Since the c defined by (57) can provide such an approximation, Theorem 23 implies that Pp is a superlative price index. If P is a superlative price index and Q is the corresponding quantity index defined implicitly by the weak factor reversal test (45), then we define the pair of index number formulae (P, Q) to be superlative. Similarly, if Q is a superlative quantity index and P is the corresponding implicit price index defined by (45), then the pair of index number formulae (P, Q) is also defined to be superlative. Before defining some additional pairs of superlative indexes, it is necessary to note the following result. If EN 2 — 1 atzt l_yN N 2 ^-^i=l ^-^7=1 l2J ^2 is a quadratic function defined over an open convex set S, then for every z°, z1 g S, the following identity is true: (58) /V) - /V) = ^[V/V) + v/V)]T(^ - z°) where V/*(zr) is the gradient vector of /* evaluated at zr, r — 0,1. The above identity follows simply by differentiating /* and substituting the partial derivatives into (58).32 Now define the Tornqvist [1936] price and quantity indexes, Pq and Qq: N (59) (60) Poip^p^x^x^^nipi/p") (s?+4)/2 2=1 N Qoip^p^x^x^^Hixl/x^ where p° > Oat, p1 > Oat, x° > Oat, x1 > Oat, s° = p°x°/p0Tx° and s} = p\x\/p1Tx1 for i — 1, 2,..., N. Theorem 24. (Diewert [1976a; 119]): Qo is exact for the homogeneous translog aggregator function f defined as33 it„- In .t„- 4- 2 (61) In f(x) = qq + y ^ a,In x,■ + ^ y ^ y ^ ctij In x,In Xj, itS 320n the other hand if /* satisfies (58) for all z°, z1 e S, then Diewert [1976a; 138] (assuming that /* is thrice differentiable) and Lau [1979] (assuming that /* is once differentiable) show that /* must be a quadratic function. 33This functional form is due to Christensen, Jorgenson and Lau [1971] and Sargan [1971]. 7. The Economic Theory 207 where Ei=i ai ~ 1> aij ~ aji f°r a^ i> J> EjLi a«i ~ 0 f°r i — 1, ■ ■ ■, N and S is an open convex subset of 51 such that f is positive and concave over S (the above restrictions on the a's ensure that f is linearly homogeneous). Proof: Assume that the producer or consumer is engaging in maximizing behavior during periods 0 and 1 so that (55) holds. Now define z4 = In xrA for r — 0,1 and i (V2) EiLiE^Li «2j-2-j 1, 2,..., N. If we define f*(z) = a0 + E?=i "i* + where the a's are as defined in (61), then, since /* is quadratic in z, we can apply the identity (58). Since d.r(zr)/dz3 = d\n.f(xr)/d\nx3 = [x^ / f(xr)][df(xr)/dx3] for r — 0,1 and j — 1,..., N, (58) translates into the following identity involving the partial derivatives of the / defined by (61): In/(a;1)-In/(a;0) 2 ^ 1 2 N dfix1) df(x°) [Vta.ln/^1)- dxi f(x°) dx -Vm^ln/C^Klna;1 - In a;0) 2 ^2 = 1 Xj Pj p1Txx p0Tx° \n(x\/xl) using (51). Infix1)//^0) Therefore fix1)/f(x°) = Hixl/xW'l+'W -e Qoipoy^^x1). qed N Define the implicit Tornqvist price index, Po(p°, p1, x°, x1) = p1Tx1 /[p0Tx° x Qo(p°tP1 , x°, x1)]. Since Qo is exact for the homogeneous translog / defined by (61), and since the homogeneous translog / is a flexible functional form (it can provide a second order differential approximation to an arbitrary twice continuously differentiable linearly homogeneous aggregator function), (Pq, Qq) is a superlative pair of index number formulae. Theorem 25. (Diewert [1976a; 121]):34 P0 defined by (59) is exact for the translog unit cost function c defined as (62) In c(p) = a*0 12_^=1aijln Piln pj > p^s* 34Theil [1965; 71-72] virtually proved this theorem; however, he did not impose linear homogeneity on c(p) defined by (62), which is required in order for (52) to be valid. 208 Essays in Index Number Theory 7. The Economic Theory 209 where J]^Ii at — 1> atj — aji f°r a^ h 3, SjLi atj — 0 for i — 1,..., N and S* is an open, convex subset of 51 such that c is positive and concave over S*. Proof: Assume that the producer or consumer is engaging in cost minimizing behavior during periods 0 and 1 and thus we assume that (49) and its consequence (52) hold, with p0,^1 g S*. Since lnc(p) is quadratic in the variables Zi = lnpj, we can again apply the identity (58) which translates into the following identity involving the partial derivatives of the c defined by (62): In c(p ) - In c(p ) = - 2_^l=1 p\ dc(p1) pf dc(p°) c(p1) dpi c(p°) dpi (In p] — lnp°) lncO?1)/^0) 1 2 i=i p\A p1T x1 Pixi p0Tx° MpI/Pi) using (52). Therefore c(p1)/c(p°) — Pq(p°,p1,x°,x1) using definition (59) qed. Now define the implicit Tornqvist quantity index, Qo(p°, p1, x°, x1) = p1T x1 /p0Tx°P0(p0 ,px, x°, x1). Since Pq is exact for the flexible functional form defined by (62), (P$, Q$) is also a superlative pair of index number formulae. It should be noted that the translog unit cost function is in general not dual to the homogeneous translog aggregator function defined by (61) (except when all oiij — 0 — a*j and a\ — a*, in which case (61) and (62) reduce to the Cobb-Douglas functional form). Thus far, we have found three pairs of superlative index number formulae: (Pf,Qf), (PqiQo) and (Pq,Qo). In turns out that there are many more such formulae. For r ^ 0, define the quadratic mean of order r aggregator function35 fr as (63) fr(x) = (^2i=1 =1 ^i^M72) 7 ' x e S where S is an open subset of 51 where fr is neoclassical, and define the quadratic mean order r unit cost function3® cr as (64) Crip) ^i=l r/2 r/2 i=i bijPi P l/r pes* 35An ordinary mean of order r (see Hardy, Littlewood and Polya [1934]) is defined as Fr(x) = (X)j=i aixTi)X^T f°r x On where m > 0 and X)j=i ai — 1. Note that kFrix) where k > 0 is the constant elasticity of substitution functional form (see Arrow, Chenery, Minhas and Solow [1961]) so that fr defined by (63) contains this functional form as a special case. 36See Denny [1974] who introduced cr to the economics literature. where S* is an open subset of 51 where cr is neoclassical. For r ^ 0, define the following price and quantity indexes: (65) -r/2 -r/2 -l/r -l/r where p°, p1, x°, x1 On, — Pi%i/p0Tx° and s} — p\x\/p1Tx1 for i — 1,2,...,N. It can be shown37 (in a manner analogous to the proof of Theorem 22), that for each r ^ 0, Qr defined by (65) is exact for fr defined by (63). Similarly, it can be shown38 (in a manner analogous to the proof of Theorem 23), that Pr defined by (65) is exact for cr defined by (64). Since it is easy to show (cf. Diewert [1976a; 130] that fr and cr are flexible functional forms for each r ^ 0, it can be shown that (Pr,Qr) and iPr,Qr) are pairs of superlative index number formulae for each r ^ 0, where p x /p° x° Pr and Pr = p1Tx1 /p0Tx°Qr. Note that P% — Pf (Fisher's ideal price index) and Q2 — Qf (Fisher's ideal quantity index) so that (P2,<52) — (P2,Q2) — iPF,Qr)-Moreover, it can be shown that the homogeneous translog aggregator function defined by (61) is a limiting case of fr defined by (63) as r tends to zero (similarly, the translog unit cost function defined by (62) is a limiting case of cr as r tends to zero)39 and that Qo defined by (60) is a limiting case of Qr as r tends to 0 while Pq defined by (59) is a limiting case of PT as r tends to 0.40 Given such a multiplicity of superlative indexes, the question arises: which index number formula should be used in empirical applications? The answer appears to be that it doesn't matter, provided that the variation in prices and quantities is not too great going from period 0 to period 1. This is because it has been shown41 that the functions Pr and Ps differentially approximate each other to the second order for all r and s, provided that the derivatives are evaluated at any point where p° — p1 and x° — x1: i.e. we have Prip0^1^0^1) = Psip°,p1,x°2x1), VPrip0^1^0^1) = VP^p0,))1,!0,!1) and V2Pr(p°,p1, x°, x1) — V2Ps(p°, p1, x°, x1) for all r and s, provided that P 0 _ p1 3> Oat and x° — x1 3> Oat. VPr stands for the AN dimensional vector of first order partials of Pr, V2Pr stands for the AN matrix of second order 37See Diewert [1976a; 132]. 38See Diewert [1976a; 133-134]. 39See Diewert [1980; 451]. 40 See Khaled [1978; 95-96]. 41 See Diewert [1978b] who utilizes the work of Vartia [1976a] [1976b]. Vartia [1978] provides an alternative proof. 210 Essays in Index Number Theory 7. The Economic Theory 211 partials of Pr, etc. The quantity indexes Qr and Qs similarly differentially approximate each other to the second order for all r and s, provided that prices and quantities are the same for the two periods. These results are established by straightforward but tedious calculations — moreover, the assumption of optimizing behavior on the part of the consumer or producer is not required in order to derive these results. Diewert [1978b] also shows that the Paasche and Laspeyres price indexes, Pp and Pl, differentially approximate each other and the superlative indexes, Pr and Ps, to the first order for all r and s, provided that prices and quantities are the same for the two periods. Thus if the variation in prices and quantities is relatively small between the two periods, the indexes Pl, Pp, Pr and Ps will all yield approximately the same answer. Diewert [1978b] argues that the above results provide a reasonably strong justification for using the chain principle when calculating official indexes such as the consumer price index or the GNP deflator, rather than using a fixed base, since in using the chain principle the base is changed every year, and thus the changes between p° and p1 and x° and x1 will be minimized, leading to smaller discrepancies between Pl and Pp, and even smaller discrepancies between the superlative indexes Pr and Ps-42 However, in some situations (e.g. in cross country comparisons or when decennial census data are being used) there can be considerable variation in the price and quantity data going from period (or observation) 0 to period (or observation) 1, in which case the indexes Pr and Ps can differ considerably. In this situation, it is sometimes useful to compare the variation in the N quantity ratios (xj/x®) to the variation in the N price ratios (pj/Pi)- If there is less variation in the quantity ratios than in the price ratios, then the quantity indexes Qr defined by (66) are share weighted averages of the quantity ratios and will tend to be more stable than the implicit indexes Qr. On the other hand, if there is less variation in the price ratios than in the quantity ratios (the more typical case), then the price indexes Pr defined by (65) are share weighted averages of the price ratios (p\ /pf) and will tend to be in closer agreement with each other than the implicit price indexes Pr. Thus, in the first situation, we would recommend the use of (Pr, Qr) for some r,43 while in the second situation we would recommend the use of (Pr, Qr) for some r.44 Notice The chain principle can also be justified from the viewpoint of Divisia indexes; see Wold [1953; 134-139] and Jorgenson and Griliches [1967]. 43If (x\/x1) = k > 0 for all i, then (Pr,Qr) = (p1Tx1 /p0Tx°k, k) for all r, and the use of (Pr, Qr) can be theoretically justified using Leontief's [1936; 54-57] Aggregation Theorem. 44If {pl/pfj = k > 0 for all i, then (Pr,Qr) = (k,p1Tx1 /p0Tx°k) for all r, and the use of (Pr,Qr) can be theoretically justified using Hicks' [1946; 312- that the Fisher index, (Pp,Qp) — (P^tQ?) — (^2,^2) can be used in either situation. A further advantage for the Fisher formulae (Pp,Qp) is that Qp is consistent with revealed preference theory: i.e., even if the true aggregator function / is nonhomothetic, under the assumption of maximizing behavior, Qp will correctly indicate the direction of change in the aggregate when revealed preference theory tells us that the aggregate is decreasing, increasing or remaining constant (cf. Diewert [1976a; 137]). Recall also that Qp is consistent both with a linear aggregator function (perfect substitutability) and a Leontief aggregator function (no substitutability). No other superlative index number formula Qr or Qr, r ^ 2, has the above rather nice properties. We conclude this section by showing that some of the above superlative index number formulae are also exact for nonhomothetic aggregator functions. Theorem 26. (Diewert [1976a; 122]): Let the functional form for the cost function C(u,p) be a general translog defined by (66) \nC(u,p) = a0 + }_^l=1 ailnPi + ^ Z^l=1 Z^j=1^%i lnPtlnPj EN 1 2 Si In pi Inu + -£o(ln u) i—i 2 where the parameters satisfy the following restrictions: at = 1; 7^ = for all 1, j; V 7,?=0 fori = 1,2, ...,N, and V 5t = 0. ^—^ j—1 —^i—1 Let (u°,p°) and (u1^1) belong to a (u,p) region where C(u,p) satisfies conditions II where u° > 0, u1 > 0, p° 3> Oat, p1 3> Oat and the corresponding quantity vectors are x° = \/pC(u°,p0) > Oat and x1 = VpC(m1,p1) > Oat respectively. Then (68) PofrV,*0,*1) = cvy)/cv,p°) where Pq is the Tornqvist price index defined by (59) and the reference utility level u* = (u^u1)1/2. Proof: For a fixed u*, \nC(u*,p) is quadratic in the variables Zi = hipi and thus we may apply the identity (53) to obtain InCVy) -lnC(u*,p°) 313] Composite Commodity Theorem. See also Wold [1953; 102-110], Gorman [1953; 76-77] and Diewert [1978a; 23]. 212 Essays in Index Number Theory 7. The Economic Theory 213 VN, \\p\\nC(u*y)/dPi] + [p^nCiu^p^/dpA [\np\ - lnPl°) Z-'2—1 L J = o E , b^lnC^1^1)/^] + [rfdlnCVy)/^] (In- lnPl°) where the equality follows upon evaluating the derivatives of C and noting that 21nw* — lnw1 + lnw°, = lnP0(p0,P1,*V1) using the definitions of x°, x1 and Pq and equations (52).qed Note that the right hand side of (68) is the true Koniis price index which corresponds to the general translog cost function defined by (66), evaluated at the reference utility level of u*, the square root of the product of the period 0 and 1 utility levels, u° and u1. We note that the translog cost function can provide a second order differential approximation to an arbitrary twice continuously differentiable cost function. Theorem 27. (Diewert [1976a; 123-124]): Let the aggregator function F be such that F's distance function D is the translog distance function defined by 1nD(u,x) = 1nC(u,x) where C is defined by (66) and (67). Let (u°,x°) and (u1^1) belong to a (u,x) region where D(u,x) satisfies conditions IV where u° > 0, u1 > 0, x° > Oat, x1 > Oat, D(u°,x°) = 1, D^x1) = 1 and the corresponding vectors of normalized prices are p°/p0Tx° = \/xD(u°, x°) > On andp^/p^x1 = Va,.Z?(m1, x1) > Oat respectively.45 Then (69) hljPy,!*,!1) = Diu^x^/Diu^x0) where Qq is the Tornqvist quantity index defined by (60) and the reference utility level u* = (u^u1)1/2. Proof: For a fixed u*, ]nD(u*,x) is quadratic in the variables Zi = lnxj and thus we may apply the identity (58) to obtain In £>(«*, x1) - In D(u*,x°) 1 2 E■_! [[xldlnDiu^x^/dx,] + [x?d]nD(u*,x0)/dxi]\(]iixl -lnx?) ^Yl1*-! [H^D^^^/dx,] + [x1d\nD{u°,x°)/dxi^[ Hxj/x®) 45These assumptions imply that xT is a solution to the aggregator maximization problem maxx{F(x) : prTx — prTxr, x > Oat} — F(xr) = ur for r — 0,1 where F is locally dual (cf. Blackorby and Diewert [1979]) to the translog distance function D defined above. where the equality follows upon evaluating the derivatives of D and noting that 2 In u* — In u1 + In u° 1 2 ^ EL [täPi/P^Diu1^1)] + [x^/p^x^iu^x0)}] lnfoVz?) 46 using pr/pr xr — VxD(ur, xr), r — 0,1 = hiQ0(p0,p1,x0,x1) using D^jX1) — 1, D(u°,x°) — 1 and the definition of Qo-qed Note that the right hand side of (69) is the Malmquist quantity index which corresponds to the translog distance function, evaluated at the reference utility level u* — (vPu1)1/2. Theorem 27 provides a fairly strong justification for the use of Qq in empirical applications, since the translog distance function can differentially approximate an arbitrary twice continuously differentiable distance function to the second order.47 However, the Fisher ideal index Q2 can be given a similar strong justification in the context of nonhomothetic aggregator functions.48 7. Historical notes and additional related topics Our survey of the economic theory of index numbers is based on the work of Koniis [1924], Frisch [1936], Allen [1949], Malmquist [1953], Pollak [1971a], Afriat [1972a] [1972b] [1977] and Samuelson and Swamy [1974]. The results noted in Sections 2 and 3 are either taken directly from or are straightforward modifications of results obtained by the above authors, except that in many cases we have weakened the original author's regularity conditions.49 46This identity is due to Shephard [1953; 10-13] and Hanoch [1978a; 116]. 47Let D be a distance function which satisfies certain local regularity properties and let F be the corresponding local aggregator function, and C be the corresponding local cost function. Blackorby and Diewert [1979] show that if D differentially approximates D* to the second order, then F differentially approximates F*, and C differentially approximates C* to the second order where F* and C* are dual to D*. 48See Diewert [1976b; 149]. 49Our regularity conditions can be further weakened: for all of the results in Sections 2 and 3 which do not involve the Malmquist quantity index, we need only assume that F be continuous and be subject to local nonsatiation (it turns out that the corresponding C will still satisfy conditions II). Also Theorems 11, 12, 14 and 16 can be proven provided that F be only continuous from above and increasing. 214 Essays in Index Number Theory 7. The Economic Theory 215 The reader will have noted that many of the proofs in Sections 2 and 3 use arguments that are used in revealed preference theory. For further material on the interconnections between revealed preference theory and index number theory, see Leontief [1936], Samuelson [1947; 146-163], Allen [1949], Diewert [1976b], Vartia [1976b; 144] and Afriat [1977]. There is extensive literature on the measurement of real output or real value added that is analogous to our discussion on the measurement of utility or real input: see Samuelson [1950a], Bergson [1961], Moorsteen [1961], Fisher and Shell [1972b; 49-113] (the last three references make use of a quantity index analogous to the Malmquist index), Samuelson and Swamy [1974; 588-592], Sato [1976b], Archibald [1977] and Diewert [1980]. Background material on the duality between cost, production or utility, and distance or deflation functions can be found in Shephard [1953] [1970], McFadden [1978a], Hanoch [1978a], Blackorby, Primont and Russell [1978], Diewert [1974a] [1978c], Deaton [1979] and Weymark [1980]. Turning now to Sections 5 and 6, for theorems which prove converses to Theorems 19 to 25 under various regularity conditions, see Byushgens [1925], Koniis and Byushgens [1926], Pollak [1971a], Diewert [1976a] and Lau [1979]. Sato [1976a] shows that a certain index number formula (which was defined independently by Vartia [1974]) is exact for the CES aggregator function defined by (63) with = 0 for i ^ j for all r, while Lau [1979] develops a partial converse theorem. In Theorem 22, preferences were assumed to be represented by the transformed quadratic function, {xTAx)1/2. The assumption that preferences can be represented, at least locally, by a general quadratic function of the form a,Q + aTx + l/2xTAx has a long history in economics, perhaps starting with Bennet [1920]. Other authors who have approximated preferences quadrati-cally, in addition to those mentioned in Theorem 22, include Bowley [1928], Hotelling [1938], Hicks [1946; 331-333], Kloek [1967], Theil [1967; 200-212] [1968], and Harberger [1971]. Kloek and Theil utilize quadratic approximations in the logarithms of prices and quantities and they obtain results which are related to Theorems 25 and 26 above. Kloek [1967] shows that the Tornqvist price index Po(p°,p1, x0^1) approximates the true Koniis price index Pk(p°\pX,um) to the second order where um, an intermediate utility level, is defined implicitly by the equation C(um, p°)/C(u°, p°) = C(u1, p1)/C(um, p1) and C is the true cost function. On the quantity side, Kloek [1967] shows that the implicit Tornqvist quantity index Qo(p°, p1, x°, x1) approximates the true Allen quantity index QA(x°, x1^™) = C[F{x1),pm] / C[F(x°), pm\ to the second order where pm = {p™ ,p™, ■ ■ ■ , P™)T', an intermediate price vector, is defined by p™ = (PiPiY^2, i — 1,..., N and F is the aggregator function dual to the true cost function C. On the other hand, Theil [1968] shows that Po(p°,p1,x°,x1) approximates the true Koniis price index Pk(p°, p1 ,u) to the second order where u, an intermediate utility level, is defined as u = G{pm/ym) where G is the indirect utility function dual to the true cost function C,50 pm is Kloek's intermediate price vector defined above and ym = (p0Tx°p1Tx1)1/2 is an intermediate expenditure. Finally, on the quantity side, Theil [1967] [1968] proves Kloek's result (i.e. that Qo(p° ,px ,x° ,xx) approximates Qa(x°, x1 ,pm) to the second order) and in addition, shows that the direct Tornqvist quantity index Qo(p°,px,x°,xx) also approximates Qa(x°, x1^"1) to the second order. It should be noted that index number theory and consumer surplus analysis are closely related. Thus the Paasche-Allen quantity index Qa(x°, xx,px) = C[F{xx),px]/C[F{xfi),px}, is closely related to Hicks' [1941-42; 128] [1946; 40-41] compensating variation in income,^1 C[F(x1), p1] — C[F(x°), p1], and the Laspeyres-Allen quantity index, Qa(x°, xx,p°) = C[F(x1),p°]/C[F(x°),p°], is closely related to Hicks' [1941-42; 128] [1946; 331] equivalent variation in income, C[F(x1),p°] — C[F(x°),p0]. Thus the various bounds we developed for index numbers in the previous section have counterparts in consumer surplus analysis. Hicks [1941-42] and Samuelson [1947; 189-202] emphasized the interconnection between index number theory and consumer surplus measures. For additional results and references to the literature on consumer surplus, see Hotelling [1938], Samuelson [1942], Harberger [1971], Silberberg [1972], Hause [1975], Chipman and Moore [1976] and Diewert [1976b]. The attractiveness of the Malmquist quantity index Qm (x° , x1, x) does not seem to have been noted in the applied welfare economics literature, although the closely related concept inherent in Debreu's [1951] coefficient of resource utilization has been recognized. Perhaps in the future there will be more applications of the Kloek-Theil approximation results, or of Theorem 27 above which shows that the Tornqvist quantity index Qq is numerically equal to a certain Malmquist index. Another type of price and quantity index which we must mention is the Divisia [1925] [1926; 40] index (which is perhaps due to Bennet [1920; 461]). The Bennet-Divisia justification for these indexes proceeds as follows. Regard (x\,... ,xm)t = x and (p±,... ,pn)t = p as functions of time, x(t) and p(t) for i — 1,..., N. Now differentiate expenditure with respect to time and we 50G(pm/ym) = max„{w : C(u,pm/ym) < 1} = m&xx{F(x) : (pm/ym)Tx < l,x > Oat} where C is the cost function and F is the aggregator function. 51Hicks' verbal definition of the compensating variation can be interpreted to mean C[F(x°), p1] — C[F(x°), p°], and this interpretation is related to the Laspeyres-Koniis cost of living index. 216 Essays in Index Number Theory 7. The Economic Theory 217 obtain.52 (70) afy" Pi(t)xi(tj\/dt = y]N pl{t)dxl{t)/dt + YjN Xi(t)dpi(t)/dt. lZ-'2 — 1 J/ Z-'2=1 Z-'2—1 Now divide both sides of the above equation through by ~Y^=1Pi{t)xi(t) p(t)Tx(t) and we obtain the identity: N (71) (91n[p(t)Tx(t)]/9t = y si(t)dhixi(t)/&t + 'y sl{t)d\npl{t) / dt -'2—1 ^-'2 — 1 where Si(t) = pi(t)xi(t)/p(t)Tx(t) for i — 1,2,..., N. The term on the left hand side of (70) is the rate of change of expenditures, which is decomposed into a share weighted rate of change of quantities plus a share weighted rate of change of prices. Denote ii(t) = dxi(t)/dt and Pi(t) = dpi(t)/dt and integrate both sides of (70) to obtain (72) lnp(l)Tx(l)/p(0)Tx(0) = J ^"^siitixiiti/xiif) dt dt. The first term on the right hand side of the above equation is defined to be the natural logarithm of the Divisia quantity index, ]n[X(l)/X(0)], while the second term is the logarithm of the Divisia price index, ln[P(l)/P(0)]. The above derivation of the Divisia indexes, X(l)/X(0) and P(1)/P(0), is devoid of any economic interpretation. However, Ville [1951-52], Malmquist [1953; 227], Wold [1953; 134-147], Solow [1957], Gorman [1959; 479] [1970], Jorgenson and Griliches [1967; 253] and Hulten [1973] show that if the consumer or producer is continuously maximizing a well behaved linearly homogeneous aggregator function subject to a budget constraint between t — 0 and t — 1, then P(1)/P(0) — Pk(p(0),p(1),x) (i.e. the Divisia price index equals the true Koniis price index for any reference quantity vector x On) and we can deduce that X(1)/X(0) — Qm(x(0), x(1), x) — Qa(x(0), x(l),p) — 52'The fundamental idea is that over a short period the rate of increase of expenditure of a family can be divided into two parts x and /, where x measures the increase due to change of prices and I measures the increase due to increase of consumption; x is the total of the various quantities consumed, each multiplied by the appropriate rate of increase of price, and I is the total of the prices of commodities, each multiplied by the rate of increase in its consumption' (Bennet [1920; 455]). I is the first term on the right hand side of (70) while x is the second term. Qk(p(0),p(1), x(0), x(1), x) (i.e. the Divisia quantity index equals the Malmquist, Allen, and implicit Koniis quantity indexes for all reference vectors x > Oat and p > Oat). On the other hand, Ville [1951-52; 127], Malmquist [1953; 226-227], Gorman [1970; 7], Silberberg [1972; 944] and Hulten [1973; 1021-1022] show that if the aggregator function is not homothetic, then the line integrals defined on the right hand side of (72) are not independent of the path of integration and thus the Divisia indexes are also path dependent. We have not stressed the Divisia approach to index numbers in this survey since economic data typically are not collected on a continuous time basis. Since there are many ways of approximating the line integrals in (72) using discrete data points, the Divisia approach to index number theory does not significantly narrow down the range of discrete type index number formulae, P(p°,p1, x°, x1) and Q{p ), that are consistent with the Divisia approach. The line integral approach also occurs in consumer surplus analysis; see Samuelson [1942] [1947; 189-202], Silberberg [1972], Rader [1976] and Chipman and Moore [1976]. Divisia indexes and exact index number formulae also play a key role in another area of economics which has a vast literature, namely the measurement of total factor productivity. A few references to this literature are Solow [1957], Domar [1961], Richter [1966], Jorgenson and Griliches [1967] [1972], Gorman [1970], Ohta [1974], Star [1974], Usher [1974], Christensen, Cummings and Jorgenson [1980], Diewert [1976a; 124-129] [1980; 487-498] and Allen [1981]. To see the relationship of this literature to superlative index number formulae, consider the following example: Let ur = ,f{xr) > 0, r — 0,1 be 'intermediate' output produced by a competitive (in input markets) cost minimizing firm where xr On is a vector of inputs utilized during period r, and / is the homogeneous translog production function defined by (61). Letting w° On and w1 On be the vectors of input prices the producer faces during periods 0 and 1, Theorem 24 tells us that (73) /(x^/fix0) =Q0(w°,w\x°,x1) where Qo is the Tornqvist quantity index defined by (60). Using (49), we also have (74) c(wr)f(xr) = wrl xr r = 0, 1 where c(w) is the unit cost function which is dual to f(x). Suppose now that 'final' output is yr = arf(xr), r — 0,1 where ar > 0 is defined to be a technology index for period r. The ratio a1 /a0 can be defined to be a measure of Hicks neutral technical progress.53 Using (73), (75) a1/a0 = (y1 /y°)/[f (x1)/f (x0)} = y'/y^w0, w1, x°, x1). 53 See Blackorby, Lovell and Thursby [1976] for a discussion of the various types of neutral technological change. 218 Essays in Index Number Theory 7. The Economic Theory 219 Thus a1/a0 can be calculated using observable data.54 The unit cost function for y in period r is c(w) /ar. Now suppose the producer behaves monopolisti-cally on his output market and sells his period r output yr at a price pr equal to unit cost times a markup factor mr > 0, i.e. (76) p — m c(w )/a 0,1. Using (76), (77) to1/™0 = (pV^XaVaVKw1)/^0)] = (p1y1/p°y0)/(w1Tx1/wQa'x0) using (74) and (75). Thus the rate of markup change to1/to0 can be calculated by (77), the value of output ratio deflated by the value of inputs ratio, using observable data.55 However, if pure profits are zero in each period, then pryr — wrTxr — [mrc(wr)/ar][arf (xr)] (using (76)) — mrwrTxr (using (74)) so that mr — 1 for r — 0,1. Another area of research which somewhat surprisingly is closely related to index number theory is the measurement of inequality; see Blackorby and Donaldson [1978] [1980] [1981]. Typically, a price or quantity index is not constructed in a single step. For example, in constructing a cost of living index, first food, clothing, transportation and other subindexes are constructed and then they are combined to form a single cost of living index. Vartia [1974; 39-42] [1976a; 124] [1976b; 84-89] defines an index number formula P(p°,p1,x°,x1) to be consistent in aggregation if the numerical value of the index constructed in two (or more) stages necessarily coincides with the value of the index calculated in a single stage. Vartia [1976b; 90] stresses the importance of the consistency in aggregation property for national income accounting and notes that the Paasche and Laspeyres indexes have this property (as do the geometric indexes Pq and Qa defined in Theorem 21 above). Vartia [1976b; 121-140] exhibits many other index number formulae that are consistent in aggregation. Unfortunately, the two families of superlative indexes, (Pr,Qr) and (PS,QS), are not consistent in aggregation for any r or s. However, Diewert [1978b] using some of Var-tia's results shows that the superlative indexes are approximately consistent in aggregation (to the second order in a certain sense). Additional results are contained in Blackorby and Primont [1980]. Related to the consistency in aggregation property for an index number formula are the following issues which have been considered by Pollak [1975], Primont [1977], Blackorby and Russell [1978] and Blackorby, Primont and Russell [1978; Chapter 9]: (i) under what 54This part of the analysis is due to Diewert [1976a; 124-129]. 55This argument is essentially due to Allen [1981]. Allen also generalized his results to many outputs and to nonneutral measures of technical change. conditions do well defined Koniis cost of living subindexes exist for a subset of the commodity space and (ii) under what conditions can the subindexes be combined into the true overall Koniis cost of living index Pr-? Finally, a related result is due to Gorman [1970; 3] who shows that the line integral Divisia indexes defined above 'aggregate conformably' or are consistent in aggregation, to use Vartia's term. If we are given more than two price and quantity observations, then some ideas due to Afriat [1967] can be utilized in order to construct nonparametric index numbers. Let there be / given price-quantity vectors {pl,xl) where pl On, xl > Oat, i — 1, 2,..., /. Use the given data in order to define Afriat's ijth cross coefficient, Dij = (plTx^/plTxl) — 1 for 1 < i,j < I. Now consider the following linear programming problem in the 21 + 2I2 variables Xi i, j = 1, .. .,1: (78) minimize s,.- subject to (i) (Ü) (üi) XiDij — (f>j — X, > 1; i 1,2, > 0, 4 > 0, ar. > 0; i,3 = 1,2,...,/, and ''•./' ...../• Diewert [1973b]56 shows that if x IS cl solution to (79) m&x{F(x) : plTx < plTxl, x>0N} for i — 1,2,... ,1 where F is a continuous from above aggregator function which is subject to local nonsatiation (so that the budget constraint plTx < plTxl will always hold as an equality for an x which maximizes F{x) subject to the budget constraint), then the objective function in the programming problem (78) will attain its lower bound of zero. On the other hand, Afriat [1967] shows that if the objective function in (78) attains its lower bound of 0 so that X*Dij > 4>j — * for all i and j where A*, * denote solution variables to (78), then the given quantity vector xl is a solution to the utility maximization problem (79) for i — 1, 2,..., I. Moreover Afriat [1967; 73-74] shows that a utility function F* which is consistent with the given data in the sense that F*(xl) — maxa;{P*(a;) : plTx < plTxl; x > On} for i — 1,2,...,I can be defined as F*(x) = mini{F*(x) : i — !,...,/} where (80) F* (x) = <% + X* [(p11 x/p11 xl)-l], i = l,2 56Afriat [1967] has essentially this result. However, there is a slight error in his proof and he does not phrase the problem as a linear programming problem. (78) corrects some severe typographical errors in Diewert's [1973b; 421] equation (3.2). 220 Essays in Index Number Theory 7. The Economic Theory 221 and where the number * and A* are taken from the solution to (78). Afriat notes that this F* is continuous, increasing and concave over the nonnegative orthant and that F*(xl) — * for i — 1,... ,1. Thus if the observed data are consistent with a decision maker maximizing a continuous from above, locally nonsatiated aggregator function F(x) subject to I budget constraints, then the solution to the linear programming problem (78) can be used in order to construct an approximation F* to the true F, and this F* will satisfy much stronger regularity conditions. Diewert [1973b; 424] notes that we can test whether the given data are consistent with the additional hypothesis that the true aggregator function is homothetic or linearly homogeneous by adding the following restrictions to (78): (iv) Aj — 4>i, 1 — 1,..., I. Geometrically, these additional restrictions force all of the hyperplanes defined by (61) through the origin; i.e. F*(0jy) — 0 for all i. Once the linear program (78) is solved, either with or without the additional normalizations (iv), we can calculate F*(xl) — 4>* for all i and thus the quantity indexes F*(xl+1)/F*(xl) can readily be calculated. Diewert and Parkan [1978] calculated these nonparametric quantity indexes using some Canadian time series data57 and compared them with the superlative indexes Q2, Qo and Qq. The differences among all of these indexes turned out to be small.58 The above method for constructing nonparametric indexes is of course closely related to revealed preference theory. Finally, we mention that there is an analogous 'revealed production theory' which allows one to construct nonparametric index numbers and nonparametric approximations to production functions and production possibility sets by solving various linear programming problems:59 see Farrell [1957], Afriat [1972a], Hanoch and Rothschild [1972] and Diewert and Parkan [1983]. References for Chapter 7 Afriat, S.N., 1967. "The Construction of Utility Functions from Expenditure Data," International Economic Review 8, 67-77. Afriat, S.N., 1972a. "Efficiency Estimation of Production Function," International Economic Review 13, 568-598. 57However, slightly different but equivalent normalizations were used. In particular, when the general nonhomothetic problem (78 (i), (ii) and (iii)) was solved, ((78) (iii)) was replaced by Aj > 0 for i — 1,..., /, i = Q2(p1,pI,x1,xI) in order to make the nonhomothetic nonparametric quantity indexes, *+1/(f>*, comparable to Qzip1 ,pl+1, xl, xl+1) for i — 1, 2,1. 58Diewert and Parkan [1978] also investigated empirically the consistency in aggregation issue. Price indexes were constructed residually using (45). 59In the context of production theory, the (output) aggregate F(x) is observable, in contrast to the utility theory context where F(x) is unobservable. Afriat, S.N., 1972b. "The Theory of International Comparisons of Real Income and Prices." In International Comparisons of Prices and Output, D.J. Daly (ed.), National Bureau of Economic Research, New York: Columbia University Press, 13-69. 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