1. Review of Consumer Theory and Duality

Consumer Welfare

1. Review of consumer theory and duality

We have represented consumer theory by the maximization problem n V(p, r) = Maxx {u(x) : p  x  r, x    }, (1) which has for solution the Marshallian demand functions x*(p, r), where V(p, r) = u(x*(p, r)) is the indirect utility function.

Consider the expenditure minimization problem n E(p, U) = Minx {p  x: u(x)  U, x    }, (2) which has for solution the Hicksian demands xc(p, U), where E(p, U) is the expenditure c function which satisfies Shephard's lemma E/pi = xi , i = 1, …, n. The expenditure function measures the smallest amount of money the household is willing to pay facing price p to reach the household utility level U. As such, it is a willingness-to-pay measure.

Note: The expenditure minimization problem (2) is very similar to the cost minimization problem we analyzed earlier. In the single output case, the main difference is that f(x)  q replaces u(x)  U in (2), where f(x) is the production function and q is output. Otherwise, all analytical results obtained in the context of cost minimization apply to the expenditure minimization problem (2).

 Duality between V and E: Under non-satiation, and given r > 0, we have xc(p, U) = x*(p, E(p, U)), (3a) and x*(p, r) = xc(p, V(p, r)), (3b)

implying that E(p, V(p, r)) = r, (4a) and V(p, E(p, U)) = U. (4b)

Expressions (4a)-(4b) show the indirect utility and expenditure functions are inverse functions of each other.

T n Consider a reference commodity bundle g = (g1, …, gn)  R satisfying g  0, g  0. Define the benefit function

b(x, U, g) = Max {: u(x -  g)  U, (x -  g)  0}, (5) where b(x, U, g) is the benefit function. It measures the quantity of the bundle g that the household must give up to obtain x starting with the utility level U. In the case where the bundle g is worth one unit of money (i.e. where p  g = 1), then the benefit function has a simple and intuitive interpretation: it measures the household willingness-to-pay to reach x starting with utility level U.

1 Recall that b(x, U, g) is concave in x if u(x) is a quasi-concave function. The concavity of the benefit function is noteworthy. It means that the benefit function b(x, U, g) exhibits diminishing marginal values with respect to x.

 Duality between E and b: Assume that p  g = 1. Then

E(p, U) = -Maxx {b(x, U, g) - p  x: x  0}. (6)

Equation (6) involves the maximization of the net benefit function [b(x, U, g) - p  x]. This has two attractive characteristics. It is intuitive: it states that consumers should try to choose consumption goods x so as to maximize the benefit b, net of the cost of purchasing the goods, p  x. And it is a "nice" unconstrained maximization problem, with an objective function that is concave in x, and linear in prices p.

Under differentiability, the FOC for an interior solution to (6) are b/x = p. This simply states that, at the optimum, marginal benefit,  b/  x , must equal marginal cost, p. And the benefit function being concave, this is a necessary and sufficient condition to identify a global solution to the maximization problem in (6).

 If x > 0, then

u(x) = Minp {V(p/1): p  x  1, p  0}, (7) which has for solution the price dependent Marshallian demand functions p*(x, 1).

 If x > 0, then

b(x, U) = Minp {p  x - E(p, U): p  g = 1, p  0}, (8) which has for solution the price dependent Hicksian demand functions pc(x, U), with b/x = pc(x, U) from the envelope theorem.

 If u(x) is non-satiated in g and x > 0, then b(x, U) = 0 is equivalent to u(x) = U. (9)

The duality relationships are summarized in Figure 2.

Figure 2: Duality when u(x) is quasi-concave

(5) Direct utility Benefit function function u(x) b(x, U, g) (9)

(1) (7) (2) (6) (8)

Indirect utility (4b) Expenditure function function V(p, r) (4a) E(p, U)

2 Figure 2 shows that that each of the function u(x), V(p, r), b(x, U, g) and E(p, U) can be recovered from any of the others. These are mathematical relationships. However, they provide some flexibility in conducting economic analysis (depending on the data available and the questions being investigated…).

2. Implications for Welfare analysis We have seen that the direct and indirect utility functions characterize Marshallian household behavior. This is relevant in empirical analysis of household behavior since Marshallian behavior x*(p, r) is observable. Alternatively, the expenditure and benefit functions are relevant in the characterization of Hicksian household behavior. They hold household utility at some given level U. Since utility is typically not observable, these functions cannot be used directly in the empirical analysis of household behavior (although they can be used indirectly through duality; see the AIDS specification). However, the expenditure and benefit functions provide the foundation for welfare analysis.

2.1. Measuring the welfare effects of price changes Consider a situation where a household faces a change in consumer prices from (pa, ra) to (pb, rb). We would like to evaluate the effects of this change on the welfare of the household. A natural choice is to rely on the expenditure function E(p, U) defined in (2). We have seen that the expenditure E(p, U) measures the household willingness-to- pay to reach a utility level U when facing p. This suggests measuring the welfare effects of a change from (pa, ra) to (pb, rb) by the function W(pa, pb, U) = rb - E(pb, U) - ra + E(pa, U), (10) which reflects the change in household income net of expenditures, holding U constant.

The household is made better off (worse off) by the change from (pa, ra) to (pb, rb) if W(pa, pb, U) > 0 (< 0). Intuitively, the household is better off (worse off) when its income net of expenditure increases (decreases) while holding its utility constant at U.

Note from (10) that W/pa = E/pa. Also, from calculus, if E(p, U) is differentiable, W(pa, pb, U) = rb - ra + E(pa, U) - E(pb, U)

pib n = rb - ra -  i1  (E/pi) dpi, pia

pib n c = rb - ra -  i1  xi (p, U) dpi, (11) pia from Shephard's lemma. Equation (11) is an important result. It states that welfare change can be measured exactly from the integral of Hicksian demand functions xc(p, U).

To make use of (11) in welfare analysis, two issues must be settled: 1/ how to choose U c in (11)?; and 2/ how to measure xi (p, U)?

3 Two options have been commonly used in the choice of U.

 We can choose U = V(pa, ra). Then, the welfare measurement given in (11) is called compensating variation (CV), where CV ≡ W(pa, pb, V(pa, ra)) = rb - ra + E(pa, V(pa, ra)) - E(pb, V(pa, ra)). This amounts to choosing the situation before the change as the reference situation in welfare analysis. As such, the compensating variation (CV) is the maximum amount of money the household would be willing to pay to face a change from (pa, ra) to (pb, rb) while remaining at the initial utility level U = V(pa, ra).

 Or we can choose U = V(pb, rb). Then, the welfare measurement given in (11) is called the equivalent variation (EV), where EV ≡ W(pa, pb, V(pb, rb)) = rb - ra + E(pa, V(pb, rb)) - E(pb, V(pb, rb)). This amounts to choosing the situation after the change as the reference situation in welfare analysis. As such, the equivalent variation (EV) is the amount of money the household would be willing to receive to avoid facing a change from (pa, ra) to (pb, rb) while remaining at the subsequent utility level U = V(pb, rb).

In general, CV  EV. In addition, xc(p, U) being typically unobservable, we need to find some simple way of measuring the right-hand side in (11). One possibility is to replace xc(p, U) in (11) by the Marshallian demand x*(p, r). If we do this, we obtain

pib n CS = rb - ra -  i1  xi*(p, r) dpi, (12) pia which is called consumer surplus (CS). In the case a single price decrease in p1 from p1a to p1b, this illustrated in figure 3.

Figure 3: Case of a single price decrease p1

Marshallian demand x *(p, y) 1

p1a D A

p1b C B

Figure 3 shows that the consumer surplus CS is measured as the area ABCD: it is the area to the left of the Marshallian demand curve and between the two prices pa and pb.

In general, CV  CS  EV. However, there is one situation where they all become identical. This is the situation where price changes occur for commodities that exhibit zero income effects (i.e., where xi*/r = 0). In this case, from the Slutsky equation c (xi*/p = xi /pj – (xi*/r) xj*), we know that xi*/r = 0 implies that Marshallian c price effects are equal to Hicksian price effects: xi*/p = xi /p. Then, (11) and (12) become identical for any choice of U. This gives the following important result:

4 If price changes take place for commodities exhibiting zero income effects

(  xi */  r = 0), then CS = CV = EV.

In such a situation, equation (12) provides a convenient way to conduct empirical welfare analysis.

However, in the presence of income effects (xi*/r  0), consumer surplus CS no longer provides an exact welfare measure for (11). This is illustrated in figure 4 in the context of positive income effects under a single price decrease: p1 decreases from p1a to p1b.

Figure 4: Case of a single price decrease

Hicksian demand xc(p, U ) a p1 Hicksian demand xc(p, U ) b

p1a F A C Marshallian demand x*(p, y) p1b E D B

x1b* x1a* x1

Figure 4 shows that CV = area ADEF, EV = area CBEF, and CS = area ABEF. It also shows that EV > CS > CV. This gives the following result. In the context of positive income effects and price decreases, EV > CS > CV.

c In addition, from the Slutsky equation, we have ln(xi*)/ln(pi) - ln(xi /ln(pi) = c -ln(xi*)/ln(r) (pi xi*/r). Thus, ln(xi*)/ln(pi) - ln(xi /ln(pi)  0 whenever

ln(xi*)/ln(r) (pi xi*/r)  0. This means that we expect the own price slope of the Marshallian demand xi*(p, r) to be "close" to the own price slope of the Hicksian demand c xi (p, U) when the income elasticity ln(xi*)/ln(r) is small and/or the budget share (pi xi*/r) is small. Under such circumstances, from figure 4, we expect CS to provide a good approximation to either CV or EV. This gives the following result.

If price changes occur for commodities with low income elasticity,

(  ln(xi *)/  ln(r) = small) and/or small budget shares (pi xj*/r = small), then CV  CS  EV.

5 This establishes conditions under which consumer surplus CS can provide a good approximation to either CV or EV.

. Linking Marshallian and Hicksian demands: An Example Consider the case where the expenditure function E(p, U) takes the form ln(E(p, U)) = a(p) + b(p)/[U-1 - c(p)], where n n n a(p) = 0 +  j1 j ln(pj) + 0.5  i1  j1 ij ln(pi) ln(pj), with ij = ji for j  i, n ln(b(p)) = ln(0) +  j1 j ln(pj), n c(p) =  j1 cj ln(pj), and the parameters satisfy

ij = ji for all i ≠ j, n i1 i = 1, n n i1 ij (= i1 ji) = 0, for all j = 1, …, n, n i1 i = 0, n  j1 cj = 0. Using Shephard’s lemma, the corresponding Hicksian expenditure shares are c c Si (p, U)  pi xi /E = ln(E)/ln(pi), -1 -1 2 = a(p)/ln(pi) + b(p)/ln(pi)/[U - c(p)] + [c(p)/ln(pi)] b(p)/[U - c(p)] , n -1 -1 2 = i +  j1 ij ln(pj) + i b(p)/[U – c(p)] + ci b(p) /[U - c(p)] , i = 1, …, n. From duality, solving ln[E(p, V)] = ln(r) for V under the above specification yields V-1 = c(p) + b(p)/[ln(r) – a(p)], Using the duality relationship x*(p, r) = xc(p, V(p, r)), this gives the Marshallian budget shares

Si*(p, r)  pi xi*/r n 2 = i +  j1 ij ln(pj) + i [ln(r) – a(p)] + ci [1/b(p)] [ln(r) – a(p)] , i = 1, …, n. This shows that the above specification (called the "quadratic almost ideal demand system" or QAIDS) exhibits income effects through ln(r) as well as [ln(r)]2. The above Marshallian budget shares can be estimated, yielding parameter estimates that can be used to recover the expenditure function E(p, U) and to conduct welfare analysis…

2.2. Measuring the welfare effects of quantity changes n Consider a situation where a household faces a change in consumption x    from xa to xb. We would like to evaluate the effects of this change on the welfare of the household. A natural choice is to rely on the benefit function b(x, U, g) defined in (5). We have seen that the benefit function b(x, U) measures the quantity of the bundle g that the household must give up to obtain x starting from the utility level U. In the case where the bundle g is worth one unit of money (i.e. where p  g = 1), the benefit function measures the household willingness-to-pay to reach x starting from utility level U. This suggests measuring the welfare effects of a change from xa to xb by the function W'(xa, xb, U) = b(xb, U, g) - b(xa, U, g), (13)

6 which reflects the change in household benefit, holding U constant.

The household is made better off (worse off) by the change from xa to xb if W'(xa, xb, U) > 0 (< 0). Intuitively, the household is said to be better off (worse off) when its benefit increases (decreases) while holding utility constant at U.

Note from (13) that W'/xb = b/xb = marginal benefit. Also, when p  g = 1 and if b(x, U, g) is differentiable in x, W'(xa, xb, U) = b(xb, U, g) - b(xa, U, g)

x b n =  i1  (b/xi) dxi xa

x b n c =  i1  pi (x, U) dxi, (14) xa since b/x = pc(x, U) from the envelope theorem applied to (8). The functions pc(x, U) are Hicksian price-dependent demands. They can be interpreted as the shadow prices of the bundles x. Equation (14) is an important result. It states that welfare change can be assessed from the integral of Hicksian price-dependent demands. This is illustrated in figure 5 in the case a single quantity change in x1 from x1a to x1b.

Figure 5: Case of a single quantity change p1

Price dependent Hicksian demand p c(x, U) 1 A

D C x1a x1b x1

Figure 5 shows that household welfare changes can be measured by the area ABCD: it is the area below the price-dependent demand curve and between the two quantities xa and xb.

As above, this raises two issues in welfare analysis: 1/ how to choose U?; and 2/ how to measure empirically the right-hand side in (14) (e.g., by using the Marshallian price-dependent demand p*(x) instead of pc(x, U))?

Two convenient options can be used in the choice of U.  We can choose U = u(xa). Then, the welfare measurement given in (14) is called compensating variation (CV'), where CV' ≡ W'(xa, xb, u(xa)) = b(xb, u(xa), g) - b(xa, u(xa), g). This amounts to choosing the situation before the change as the reference situation in welfare analysis. As such, when p  g =1, the compensating variation (CV') is the maximum amount of money the

7 household would be willing to pay to face a change from xa to xb while remaining at the initial utility level U = u(xa).

 Or we can choose U = u(xb). Then, the welfare measurement given in (11) is called the equivalent variation (EV'), where EV' ≡ W'(xa, xb, u(xb)) = b(xb, u(xb)) - b(xa, u(xb)). This amounts to choosing the situation after the change as the reference situation in welfare analysis. As such, when p  g = 1, the equivalent variation (EV') is the amount of money the household would be willing to receive to avoid facing a change from xa to xb while remaining at the subsequent utility level U = u(xb).

In general, CV'  EV'. In addition, pc(x, U) being typically unobservable, we need to find some simple way of measuring the right-hand side in (14). One possibility is to replace pc(x, U) in (14) by the Marshallian price-dependent demand p*(x). If we do this, we obtain the consumer surplus measure

x b n CS' =  i1  pi*(x) dxi. (15) xa

In the case a single quantity change (where x1 increases from x1a to x1b), this illustrated in figure 6.

Figure 6: Case of a single quantity increase p1

Price-dependent Marshallian demand p *(x) 1 p1a A

D C x1a x1b x1

Figure 6 shows that the consumer surplus CS' is measured as the area ABCD: it is the area below the price-dependent Marshallian demand curve and between the two quantities x1a and x1b.

In general, CV'  CS'  EV'. To investigate the relationships between these expressions, note from duality that p*(x) = pc(x, u(x)). Under differentiability, this gives ∂p*/∂x = ∂pc/∂x + (∂pc/∂U)(∂u/∂x).

c c c This shows that ∂pi*/∂xj = ∂pi /∂xj when ∂pi /∂U = 0. It means that when ∂pi /∂U = 0, Marshallian quantity effects become identical to Hicksian quantity effects: ∂pi*/∂xj = c ∂pi /∂xj. In this situation, (14) and (15) become identical for any choice of U. This implies that CS' = CV' = EV' when quantity changes take place for commodities that exhibit

8 c c c ∂pi /∂U = 0. However, when ∂pi /∂U ≠ 0, then ∂pi*/∂xj ≠ ∂pi /∂xj, and CS' no longer provides an exact welfare measure for (14).

. Linking Marshallian and Hicksian price-dependent demands: An Example Consider the following specification for the benefit function b(x, U, g) b(x, U, g) = (x) - (x)/[U-1 - (x)], where (x) > 0. The benefit function satisfies (∂b/∂x)  g = 1 for all x and U. This implies (/x)  g = 1, (/x)  g = 0, and (/x)  g = 0. In addition, when the reference bundle g is chosen such that p  g = 1, using the envelope theorem in (8) implies that the marginal benefit equals the price-dependent Hicksian demands: b/x = pc(x, U). This yields c 2 2 pi (x, U) = /xi - /xi [U/(1 - U (x))] - (/xi) (x) U /[1 - U (x)] , i = 1, …, n. Solving b(x, U) = 0 yields U = u(x). Thus, U/[1 - U (x)] = (x)/(x). From * c duality, the price-dependent Marshallian demands are pi (x) = pi (x, u(x)). Given p  g = 1, it follows that * 2 pi (x) = /xi - /xi [(x)/(x)] - (/xi) [(x) /(x)]]. Let N N N (x) = 0 +  j1 j xj +  j1  k1 ½ jk xj xk, N (x) = exp(0 +  j1 j xj), N (x) =  j1 j xj, with

jk = kj for all j  k, as symmetry restrictions. Then, with (/x)  g = 1, (/x)  g = 0, and (/x)  g = 0 holding for all x imply the following restrictions N  j1 j  gj = 1, N  j1 jk  gj = 0, k = 1, ..., N, (using the symmetry restrictions), N  j1 j  gj = 0, and N  j1 j  gj = 0. Given p  g = 1, it follows that the price-dependent Marshallian demands for the i-th good is * N 2 pi (x) = i +  k1 ik xk - i [(x)] - i [(x) /(x)], i = 1, …, n. These price-dependent Marshallian demands can be estimated, yielding parameter estimates that can be used to recover the benefit function b(p, U, g) and to conduct welfare analysis…

3. Index numbers Index numbers are commonly used for two reasons:

9 1/ as relative welfare measures (e.g., cost of living index reflecting the welfare effects of price changes, or standard of living index reflecting the welfare effects of quantity chnages); and 2/ as means of generating price and quantity indexes for a commodity group (e.g., food, capital, etc.). Although these two motivations are quite different, they are unified by their common objective of “summarizing economic information about a group of commodities.” For relative welfare measures, the group of commodities includes all commodities relevant to an economic agent. For price and quantity index, the group of commodities involves only a subset of all the commodities relevant to a decision-maker (e.g., food, capital, etc.).

Note: In the case of a subset of commodities, the theoretical justification for using a price index requires imposing weak separability restrictions on the cost function, profit function, or expenditure function. For example, a weakly separable expenditure function (or cost function) implies the existence of a “price aggregator function” that summarizes all the information related to the prices of commodities within the given subset. Then, a price index is simply an empirical measurement of the corresponding price aggregator function over the subset of commodities in the expenditure-cost function. Similarly, in the case of a subset of commodities, the theoretical justification for using a quantity index requires imposing weak separability restrictions on the utility function, shortage function, distance function, or production function. Indeed, a weakly separable utility function (or shortage function, or distance function, or production function) implies the existence of a “quantity aggregator function” that summarizes all the information related to the quantities of commodities within the given subset. Then, a quantity index is simply an empirical measurement of the corresponding quantity aggregator function over the subset of commodities in the relevant utility-distance-production function. Note that justifying the existence of both a price aggregator function and a quantity aggregator function for a subset of commodities requires the weak separability of both the expenditure-cost function, and the utility-distance- production function. This implies stronger restrictions than just the weak separability of the utility-distance-production function. Indeed, it requires the homothetic separability of the utility-distance-production function. Note that this may appear rather restrictive…

3.1. Price index For simplicity, we discuss the price index in the context of household consumption (with the understanding that all the arguments could be presented similarly in the context of firm production). The analysis is based on the expenditure function E(p, U) = minx{p  x: u(x)  U, x  0}. We consider a situation where the prices p change from pa to pb.

3.1.1. The true price index Definition: The true price index (P) is P(pa, pb, U) = E(pb, U)/E(pa, U).

10 This shows that a price index is a unit-free measure of relative cost-expenditure change, reflecting the effects of changing prices that keep household utility U constant. It satisfies P > (=, or <) 1 whenever the price change tends to increase (maintain, or decrease) cost or expenditure. It means that (P-1)100 is the percentage change in cost or expenditure due to changing prices, keeping utility U constant.

Definition: The Divisia price index (DP) is ln(pb ) a b n i c DP(p , p , U) = exp[ i1 a wi (p, U) dln(pi)], ln(p ) i or ln(pb ) n i c ln(DP) =  i1 a wi (p, U) dln(pi), ln(p ) i c c where wi (p, U) = pi xi (p, U)/E(p, U) = the i-th Hicksian expenditure share.

Proposition 1: The true price index P and the Divisia price index DP are equivalent: DP(pa, pb, U) = P(pa, pb, U).

Proof: The definition of P gives ln(P) = ln E(pb, U) – ln E(pa, U). But, under differentiability, ln(pb ) b a n i ln E(p , U) – ln E(p , U) =  i1 a (ln E(p, U)/pi) dpi, ln(p ) i ln(pb ) n i =  i1 a (ln E(p, U)/ln(pi)) dln(pi). ln(p ) i Shephard’s lemma gives c c ln E/ln(pi) = wi (p, U) = pi xi (p, U)/E(p, U) = the i-th Hicksian cost share. It follows that ln(pb ) b a n i c ln(P) = ln E(p , U) – ln E(p , U) =  i1 a wi (p, U) dln(pi) = ln(DP). ln(p ) i

3.1.2. The Laspeyres price index Definition: The Laspeyres price index (LP) is LP(pa, pb, xa) = (pb  xa)/(pa  xa).

Note: The Laspeyres price index can be alternatively written as a b a n b a n a a LP(p , p , x ) = ( i1 pi xi )/( i1 pi xi ) n a a n a a b a = i1 {[pi xi )/(  j1 pj xj )] (pi /pi )} n a a b a = i1 [wi(p , x ) (pi /pi )] n where wi(p, x) = pi xi/(  j1 pj xj) is the “i-th commodity share”.

The consumer price index (CPI) is a Laspeyres price index over all commodities consumed by households. Note that it does not require information about the quantities xb observed after the price change.

11 Note: For general price increases (pb  pa) the Laspeyres price index LP gives an upward biased estimate of the true price index P. Indeed, we have P – 1 = [E(pb, U) – E(pa, U)]/E(pa, U) pb n i a = [ (E(p, U)/pi) dpi]/E(p , U) i1 pa i pb n i c a = [ xi (p, U) dpi]/E(p , U), (from Shephard’s lemma) i1 pa i n c a b a a [ i1 xi (p , U) (pi – pi )]/E(p , U), (since Hicksian demands slope downward) = [pb  xa – pa  xa]/[pa  xa] if U = u(xa) = LP – 1. However, there are two situations where this upward bias vanishes: 1/ when all prices increase proportionally, in which case they have no effect on demand since the Hicksian demands xc(p, U) are homogenous of degree zero in prices p; 2/ when all goods are consumed in fixed proportions with zero Allen elasticities of substitution, implying the absence of price effects on the Hicksian demands xc(p, U).

3.1.3. The Paasche price index Definition: The Paasche price index (PP) is PP(pa, pb, xb) = (pb  xb)/(pa  xb).

Note: The Paasche price index can be alternatively written as a b b n b b n a b PP(p , p , x ) = ( i1 pi xi )/( i1 pi xi ) n a b n a b b a = i1 {[pi xi )/(  j1 pj xj )] (pi /pi )} n a b b a = i1 [wi(p , x ) (pi /pi )] n where wi(p, x) = pi xi/(  j1 pj xj) is the “i-th commodity share”.

Note that the Paasche price index requires information about the quantities xb observed after the price change. To the extent that price information is easier to obtain than quantity information, the Paasche price index is typically more difficult to implement empirically than the Laspeyres price index.

3.1.4. The Fisher price index Definition: The Fisher price index (FP) is FP(pa, pb, xa, xb) = [LP(pa, pb, xa) PP(pa, pb, xb)]1/2 = {[(pb  xa)(pb  xb)]/[(pa  xa)(pa  xb)]}1/2.

Note that the Fisher price index requires information about both quantities xa and xb (i.e., both before and after the price change).

12 3.1.5. The Tornquist-Theil price index Definition: The Tornquist-Theil price index (TP) is a b n c a c b b a TP(p , p , U) = exp{ i1 (1/2)[wi (p , U) + wi (p , U)][ln(pi ) – ln(pi )]}, or n c a c b b a ln(TP) = i1 (1/2)[wi (p , U) + wi (p , U)][ln(pi ) – ln(pi )], c c where wi (p, U) = pi xi (p, U)/E(p, U) = the i-th Hicksian expenditure share.

Note that the Tornquist-Theil price index TP can be interpreted as an approximation to the Divisia price index DP. Also, calculating TP requires information about expenditures both before and after the price change.

3.2. Quantity index For simplicity, we discuss the quantity index in the context of household consumption (with the understanding that all the arguments could be presented similarly in the context of firm production). Consider the distance function D(x, U) that is dual to the household utility function u(x), where D(x, U) = 1 is the implicit solution to u(x/D) = U. The distance function D(x, U) is linear homogenous and increasing in x. This distance function plays the same role for quantity index as the expenditure function for price index (as discussed above). In a way similar to Shephard’s lemma, it satisfies c D(x, U)/xi = pi (x, U), c where pi (x, U) is the “Hicksian shadow price” of the i-th commodity. From the linear homogeneity of D(x, U) in x, Euler equation gives n i1 (D/xi) xi = D(x, U) = 1, or n c i1 pi (x, U) xi = 1.

Let x = (x1, …, xn) is a vector of commodities, and p = (p1, …, pn) is a vector of corresponding prices. If we want to evaluate a standard of living index, then U = u(x) is the household utility function. Alternatively, if we want to obtain a quantity index for a subset of commodities, then U = u(x) is an aggregator function for the subset of commodities, aggregator function that is an argument of the household utility function under weak separability. We consider a situation where the quantities x change from xa to xb.

3.2.1. The true quantity index (sometimes called the Malmquist index) Definition: The true quantity index (Q) is Q(xa, xb, U) = D(xb, U)/D(xa, U).

This shows that a quantity index is a unit-free measure of relative distance, reflecting the effects of changing quantities that keep household utility U constant. It satisfies Q > (=, or <) 1 whenever the quantity change tends to increase (maintain, or decrease) distance. It means that (1 – Q)100 is the percentage change in distance due to changing quantities, keeping utility U constant.

13 Definition: The Divisia quantity index (DQ) is ln(xb ) a b n i c DQ(x , x , U) = exp[ i1 a wi (x, U) dln(xi)], ln(x ) i or ln(xb ) n i c ln(DQ) =  i1 a wi (x, U) dln(xi), ln(x ) i c c n c where wi (x, U) = pi (x, U) xi/D(x, U) = the i-th Hicksian share (since i1 pi (x, U) xi = D = 1).

Proposition 1: The true quantity index Q and the Divisia quantity index DQ are equivalent: DQ(xa, xb, U) = Q(xa, xb, U).

Proof: The definition of TQ gives ln(Q) = ln D(xb, U) – ln D(xa, U). But ln(xb ) b a n i ln D(x , U) – ln D(x , U) =  i1 a ln D(x, U)/ln(xi) dln(xi). ln(x ) i c Given D(x, U)/xi = pi (x, U), we have c c ln D/ln(xi) = wi (x, U) = pi (x, U) xi/D(x, U) = the i-th Hicksian share. It follows that ln(xb ) b a n i c ln(TQ) = ln D(x , U) – ln D(x , U) =  i1 a wi (x, U) dln(xi) = ln(DQ). ln(x ) i

3.2.2. The Laspeyres quantity index Definition: The Laspeyres quantity index (LQ) is LQ(xa, xb, pa) = (pa  xb)/(pa  xa).

Note: The Laspeyres quantity index can be alternatively written as a b a n a b n a a LQ(x , x , p ) = ( i1 pi xi )/( i1 pi xi ) n a a n a a b a = i1 {[pi xi )/(  j1 pj xj )] (xi /xi )} n a a b a = i1 [wi(p , x ) (xi /xi )] n where wi(p, x) = pi xi/(  j1 pj xj) is the “i-th commodity share”.

3.2.3. The Paasche quantity index Definition: The Paasche quantity index (PQ) is PQ(xa, xb, pb) = (pb  xb)/(pb  xa).

Note: The Paasche quantity index can be alternatively written as a b b n b b n b a PQ(x , x , p ) = ( i1 pi xi )/( i1 pi xi ) n b a n b a b a = i1 {[pi xi )/(  j1 pj xj )] (xi /xi )} n b a b a = i1 [wi(p , x ) (xi /xi )]

14 n where wi(p, x) = pi xi/(  j1 pj xj) is the “i-th commodity share”.

3.2.4. The Fisher quantity index Definition: The Fisher quantity index (FQ) is FQ(xa, xb, pa, pb) = [LQ(xa, xb, pa) PQ(xa, xb, pb)]1/2 = {[(pa  xb)(pb  xb)]/[(pa  xa)(pb  xa)]}1/2.

3.2.5. The Tornquist-Theil quantity index Definition: The Tornquist-Theil quantity index (TQ) is a b n c a c b b a TQ(x , x , U) = exp{ i1 (1/2)[wi (x , U) + wi (x , U)][ln(xi ) – ln(xi )]}, or n c a c b b a ln(TQ) = i1 (1/2)[wi (x , U) + wi (x , U)][ln(xi ) – ln(xi )], c c where wi (x, U) = pi (x, U) xi /D(x, U) = the i-th Hicksian expenditure share.

Note that the Tornquist-Theil quantity index TQ can be interpreted as an approximation to the Divisia quantity index DQ.

3.3. Superlative index numbers Definition: A superlative price (quantity) index is a true index associated with a flexible expenditure (distance) function (i.e. a function that does not impose a priori restrictions on the Allen elasticities of substitution).

Examples of expenditure (or distance) functions that are “flexible” are the quadratic, the “generalized Leontief”, and the translog. It also includes the class of functions that are “geometric mean of order k”.

Definition: A “geometric mean function of order k” is f(z) = [(zk/2)T A (zk/2)]1/k, for some k  0, where z is a (n1) vector, and A is a (nn) symmetric matrix.

Note: Under homothetic preferences, the expenditure function is E(p, U) = h(u) c(p). Then, if c(p) is “geometric mean of order k”, it gives c(p) = [(pk/2)T A (pk/2)]1/k = a “flexible form” for any k  0, = a translog cost function as k  0 = (p1/2)T A (p1/2) = a “generalized Leontief” cost function if k = 1, = [(p)T A (p)]1/2 if k = 2.

Proposition: The Tornquist-Theil price (quantity) index is a superlative price (quantity) index because it is a true price (quantity) index associated with a translog expenditure (distance) function.

Proof: We present the proof only for a price index associated with the translog expenditure function n n n ln[E(p, U)] = a0 + i1 ai ln(pi) + au ln(U) + (1/2)  j1 i1 aij ln(pi) ln(pj) (B1) n 2 + i1 aiu ln(pi) ln(U) + auu [ln(U)] .

15 For a change in prices from pa to pb, the associated true price index P satisfies ln(P) = ln[E(pb, U)] – ln[E(pa, U)] n b a n n b b = i1 ai [ln(pi ) – ln(pi )] + (1/2) i1  j1 aij ln(pi ) ln(pj ) n n a a n b a – (1/2) i1  j1 aij ln(pi ) ln(pj ) + i1 aiu ln(U) [ln(pi ) - ln(pi )].(B2) Shephard’s lemma gives c c ln(E)/ln(pi) = pi xi (p, U)/E(p, U) = wi (p, U) = the i-th Hicksian cost share. (B3) In the context of the translog expenditure function (B1), we have n ln(E)/ln(pi) = ai +  j1 aij ln(pj) + aiu ln(U). (B4) Let Ek = E(pk, Uk), where k = a, b. Using the above results, the log of the Tornquist-Theil price index TP is n c b b c a a b a ln(TP) = i1 (1/2)[wi (p , U ) + wi (p , U )][ln(pi ) – ln(pi )] n b b a a b a = (1/2) i1 [ln(E )/ln(pi ) + ln(E )/ln(pi )][ln(pi ) – ln(pi )], (from (B3)) n n b b n a = (1/2) i1 [ai +  j1 aij ln(pj ) + aiu ln(U ) + ai +  j1 aij ln(pj ) a b a + aiu ln(U )][ln(pi ) – ln(pi )], (from (B4)) n b a n n b b = i1 ai [ln(pi ) – ln(pi )] + (1/2) i1  j1 aij ln(pj ) ln(pi ) n n a a - (1/2) i1  j1 aij ln(pj ) ln(pi ) n b a b a + (1/2) i1 aiu [ln(U ) + ln(U )][ln(pi ) – ln(pi )] n b a n n b b = i1 ai [ln(pi ) – ln(pi )] + (1/2) i1  j1 aij ln(pi ) ln(pj ) n n a a – (1/2) i1  j1 aij ln(pi ) ln(pj ) n * b a * a b 1/2 + i1 aiu ln(U ) [ln(pi ) - ln(pi )], where U = (U U ) , = ln(P) (from (B2) evaluated at U* = (Ua Ub)1/2).

Proposition: The Fisher price (quantity) index is a superlative index because it is a true index associated with an expenditure (distance) function that is a “quadratic mean of order two”.

Proof: Again, we present a proof only for the price index. Consider the case of homothetic preferences where the expenditure function is E(p, U) = h(U) c(p), and c(p) = (pT A p)1/2 is a quadratic mean of order two (k = 2). Then, Shephard’s lemma gives /p = xc = (pT A p)-1/2 A p, (C1) or A p = xc (pT A p)1/2, (C2) where xc = xc(p, U) is the Hicksian demand function. For a change in price from pa to pb, we have P = E(pb, U)/E(pa, U) = c(pb)/c(pa) (under homothetic preferences) = {[(pb)T A (pb)]/[(pa)T A (pa)]}1/2, (where c(p) = (pT A p)1/2) = {[(pb)T xcb ((pb)T A pb)1/2]/ [(pa)T xca ((pa)T A pa)1/2]}1/2, (from (C2) = {[(pb)T xcb (pb)T xca]/ [(pa)T xca (pa)T xcb)]}1/2, (from (C1) = [(LP) (PP)]1/2

16 = FP.

3.4. Implicit index In general, an index (either price or quantity) can be interpreted as a ratio of expenditures. If so, when applied to the same group of commodities, we expect the following relationship to hold (price index)(quantity index) = [pb  xb]/[pa  xa].

Note that this relationship always holds for the Fisher index since (FP)(FQ) = [(LP) (PP)]1/2 [(LQ) (PQ)]1/2 = [pb  xb]/[pa  xa]. However, it does not always hold for the other indexes. In this case, the above relationship can be treated as an identity. Then, if only one of the two indexes (either price or quantity) is known, the other one can always be obtained from this identity. Obtaining an index in this manner generates an implicit index. A useful rule of thumb is to calculate a “direct index” for the variables (either prices or quantities) that tend to vary “more proportionally”, and calculate an “implicit index” for the variables that tend to vary “less proportionally”.