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Aggregation without separability: Composite commodity
theorems in quantity-space
*
Daniele Moro
`
Istituto di Economia Agro-alimentare, Universita Cattolica, 29100, Piacenza, Italy Received 16 May 2000; accepted 11 November 2000
Abstract
In this paper, we provide an obvious extension of the Composite Commodity Theorems to the case of inverse demands, where quantities and not prices can be used for aggregation. First, the Hicks–Leontief commodity theorem is extended to the quantity space, and then its generalised stochastic version is presented. 2001 Elsevier Science B.V. All rights reserved.
Keywords: Inverse demand systems; Composite commodity theorems JEL classification: D11; D12
1. Introduction
The existence of consistent commodity aggregates in demand analysis can be justified by relying on the Hicks–Leontief (HL) composite commodity theorem. This theorem states sufficient, although not necessary, conditions for the existence of commodity aggregates: it is required that prices in the group move in the same proportion. Then, individual goods can be aggregated into broad categories, and expenditure allocation to single groups can be obtained in terms of a (simple) price index for each group.
In empirical demand analysis, the assumption that prices are predetermined at the market level may be unrealistic, at least in some cases; thus, it may be more appropriate to have quantities exogenous, and prices that adjust to clear the market. In these situations, the empirical approach to model consumer response to market changes is to specify an inverse demand system, where prices are endogenous and quantities are predetermined, as opposed to direct demand systems. Barten and Bettendorf (1989) discuss the rationale of resorting to inverse demand systems as a device for
*Corresponding author. Tel.: 139-0523-59-9292; fax: 139-0523-59-9282. E-mail address: dmoro@pc.unicatt.it (D. Moro).
modelling price formation in certain markets, where the causality can be seen as going from quantities to prices.
An obvious extension of the HL composite commodity theorem is that of looking for sufficient conditions for commodity aggregation in the quantity space: when quantities of goods move together, so that the quantity ratios remain constant, these individual goods can be treated as a single composite commodity, and expenditure can be allocated to that group by using a quantity index for each group. However, the assumption of perfect collinearity between prices (quantities) is a really stringent one. Recently, Lewbel (1996) has proposed a generalisation of the HL composite commodity theorem, termed as the Generalised Composite Commodity Theorem (GCCT): the idea is that of relaxing the assumption of perfect collinearity and providing a justification for aggregation only in stochastic terms. The requirement of the theorem is that the distribution of the ratio between individual prices and group price indexes is independent of the group price index. This allows sufficient (although not necessary) conditions for aggregation, without imposing any structure on preferences.
In this note we provide an obvious extension of the GCCT to the case of inverse demands, where quantities and not prices can be used for aggregation. Thus the HL commodity theorem is first extended to the case of inverse demands, and then its generalised version is presented.
2. The Hicks–Leontief theorem in the quantity space
Consider individual preferences that can be represented over quantities of goods, q, by means of a direct utility function u q , with the usual properties; alternatively, preferences can be defined overs d normalised prices, r5p /y, where p is a vector of prices and y is income, giving a dual representation
of preferences by means of an indirect utility function vs dr . Inverse demands can be obtained by
solving the following primal problem:
minvs dr s.t. rq#1
r
The solution to this problem gives (Marshallian) inverse demands r*5r* q , with prices as a functions d of quantities.
To derive a demand system, we normally start from a parameterization of a representation of preferences; in the case of inverse demand systems, it is convenient to start from either the direct utility function, exploiting the Hotelling–Wold identity (Anderson, 1980; Weymark, 1980), or from the distance function, exploiting the Shephard–Hanoch lemma.
The distance (or transformation) function d u, q is implicitly defined from the direct utility functions d
u q as u q /d u, qs d f s dg5u: thus, it is the amount by which q must be divided to bring it on the
indifference curve u (see Deaton and Muellbauer, 1980).
d u, qs d5min rq
f
uv(r)#ug
r
Thus, empirical inverse demand function can be obtained either by solving the optimization problem or by specifying a functional form for a preference representation and then applying the derivative property (for example, the Hotelling–Wold identity on the direct utility function or the Shephard–Hanoch lemma on the distance function, giving compensated inverse demand).
To illustrate the composite commodity theorem in the quantity space, we can partition goods in two groups; then we write the indirect utility function as vsr , r , where r indicates normalised prices ofd
1 2 2
goods with constant quantity ratios. Thus the distance function can be retrieved from the following problem: This distance function shares all the properties of a ‘proper’ distance function: it is increasing, homogeneous of degree one and concave in q and1 u, and decreasing in u. This will prove the HL commodity theorem in the quantity space (see Deaton and Muellbauer, 1980: pp. 120–122).
3. Proof of the HL commodity theorem in the quantity-space
9
lim d* u, q ,s 1 ud5d* u, q ,s 1 u9d given that lim d u, q , qs 1 2d5d u, q , qs 1 2d hu→u9 q2→q92
By applying the Shephard–Hanock to the grouped distance function, we see that:
≠d* u, q ,s 1 ud 0
]]]] 5r q2 2
≠u
0
showing that r q is the price of the composite good corresponding to the quantity2 2 u. Thus, with 0
preferences defined over r and r q , we may treat goods in the second group as a single commodity,1 2 2 with quantityu.
4. The generalised Hicks–Leontief commodity theorem in the quantity space
The HL commodity theorem requires that (some) quantities will show perfect collinearity through time; of course this is a very stringent requirement for aggregation, and actual data usually do not satisfy such conditions. Then, the generalised HL composite commodity theorem in the quantity space (GCCT-Q) relaxes the assumption of perfect collinearity by allowing the quantity ratio to vary through time; thus the requirement for the GCCT-Q is that the distribution of the ratio of the individual quantity to the group quantity index be independent of the group quantity index (Lewbel, 1996).
To illustrate the GCCT-Q let q and wi i5r q be quantities and budget shares; furthermore, definei i
Q and WI I5oi[I w as group quantity indexes and budget shares; finally define hi i5ln q , Hi i5ln Q ,I and ri5ln qi2ln Q .I
Given this, the HL commodity theorem in the quantity space can be re-stated as: if the vectorr is constant, then the vector of group budget shares W is a solution to the consumer optimisation problem. Here, we present its generalisation.
As we said, the idea is that of relaxing the assumption of perfect collinearity and providing a justification for aggregation only in stochastic terms. Consider (inverse) demand functions in share form, wi5g his d1e , with E ei
s
iuhd
50; then, assume as in Lewbel (1996) that the functions g h areis d rational, and that the distribution of r is independent of H.At this point, we define group expenditures equations as WI5G (H )I 1e , with E eI
s
IuHd
50, that isG is the conditional expectation ofI oi[I w given the vector of aggregate quantity indexes. Further,i
*
Independence between the distribution of relative quantitiesr and H will ensure that the aggregate share equation in terms of group indexes H is equal to the conditional expectation over the sum of individual share equations. Furthermore assume that the matrix CV, whose element is
*
1
symmetry and negativity , if the individual inverse demand functions g h are rational (i.e. they areis d derived from an optimisation problem and satisfy conditions for integrability). This is the Generalised Composite Commodity Theorem in the quantity space (GCCT-Q).
5. Proof of the GCCT-Q
A formal proof of the GCCT-Q can be given following Lewbel (1996).
Proof of adding-up.
Proof of symmetry. Define the log form of the Antonelli matrix for the grouped demand functions as
the matrix with elements:
where a is the element of the log form of the Antonelli matrix of individual inverse demandij functions. If and only if the matrix [CV ] is symmetric, then, repeating all the steps backward, weI J
1
may show that the log form of the Antonelli matrix for the grouped demand functions is symmetric, that is AI J5A .JI
Proof of negative-semidefiniteness. The proof follows the outline of the proof of symmetry. This
˜ ˜
negativity property pertains to negative semidefiniteness of the matrix [A ], where AI J I J5AI J1
˜
demand functions follows from negative semidefinitess of the matrix
F G
aij , and from negativej
˜
semidefiniteness of the matrix [CV ], since the matrix CVI J
f
I Jg
is positive semidefinite by construction, although this is a sufficient but not necessary condition for negative semidefiniteness of the matrix˜
A . h
f g
I JThe empirical implications of the GCCT have been addressed in Lewbel (1996) and Davis et al. (2000), where it is shown how to test the independence assumption; such testing procedure is related to the statistical property of the economic variables; Davis et al. (2000) discuss at length the statistical implications of such procedure.
6. Concluding remarks
In this brief note, we have provided conditions for justifying aggregation among goods in inverse demand systems, without involving any structure on preferences. Extending the Hicks–Leontief (HL) composite commodity theorem to the case of inverse demand, we see that aggregation among goods is justified when quantities move in fixed proportion, thus quantity ratios remain constant.
possible in real world, we also show how this theorem can be re-cast in stochastic terms, following the suggestion of Lewbel (1996). We then have the generalised composite commodity theorem in the quantity space (GCCT-Q), with the HL composite commodity theorem being a special case of the GCCT-Q.
Quantity aggregation can be useful in empirical work, and this approach can be extended to the production context.
References
Anderson, R.W., 1980. Some theory of inverse demand for applied demand analysis. European Economic Review 14, 281–290.
Barten, A.P., Bettendorf, L.J., 1989. Price formation of fish: an application of an inverse demand system. European Economic Review 33, 1509–1525.
Blackorby, C., Primont, D., Russell, R.R., 1978. Duality, Separability and Functional Structure: Theory and Economic Applications. North-Holland, New York.
Davis, G.C., Lin, N., Shumway, R., 2000. Aggregation without separability: tests of the United States and Mexican agricultural production data. American Journal of Agricultural Economics 82, 214–230.
Deaton, A., Muellbauer, J., 1980. Economics and Consumer Behavior. Cambridge University Press, Cambridge.
Lewbel, A., 1996. Aggregation without separability: a generalized composite commodity theorem. American Economic Review 86, 524–543.
