# The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

# Divisia Index

• Charles R. Hulten
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_350

## Abstract

The Divisia index, it its modern application, is a continuous-time index related to an underlying economic structure via a potential function. Under certain conditions, the index can retrieve important characteristics of the underlying structure using prices and quantities alone, without full knowledge about the structure itself. The Divisia index is widely used in theoretical discussions of productivity analysis, and has important applications elsewhere. In practice, it is approximated by discrete–time superlative indexes, like the Tornqvist, or by chain indexes. Older applications of the Divisia stressed its discrete-time axiomatic properties.

## Keywords

Aggregation Chain indexes Continuous–time indexes Discrete–time indexes Divisia index Divisia, F. Duality Path dependence Production functions Productivity (measurement problems) Solow, R. Törnqvist index

## JEL Classifications

C43 C80 E01
The Divisia index is a continuous-time index number formula due to François Divisia (1925–6) that has been widely used in theoretical discussions of data aggregation and the measurement of technical change. It is defined with respect to the time paths of a set of prices [P1(t), … , PN(t)] and commodities [X1 (t),…,XN (t)]. Total expenditure on this group of commodities is given by:
$$Y(t)={P}_t(t){X}_1(t)+\dots +{P}_N(t){X}_N(t).$$
(1)
With dots over variables indicating derivatives with respect to time, total differentiation of (1) yields:
$$\frac{\dot{Y}(t)}{Y(t)}=\sum\limits_{i=1}^{i=N}\frac{P_i(t){X}_i(t)}{Y(t)}\frac{{\dot{P}}_i(t)}{P_i(t)}+\, \sum\limits_ {i=1}^{i=N}\frac{P_i(t){X}_i(t)}{Y(t)}\frac{{\dot{X}}_i(t)}{X_i(t)}.$$
(2)
The growth rates of the Divisia price and quantity indexes are the respective weighted averages of the growth rates of the individual Pi (t) and Xi (t), where the weights are the components’ shares in total expenditure. The levels of these indexes are obtained by line integration over the trajectory followed by the individual prices and quantities over the time interval [0, T]. For the quantity index, the line integral has the following form:
$${I}_q\left(0,T\right)=\exp \left\{\int \left[\sum \limits_{i=1}^N\frac{P_i(t){X}_i(t)}{\sum {P}_j(t){X}_j(t)}\frac{{\dot{X}}_i(t)}{X_i(t)}\right]\right\}=\exp \left\{{\int}_r\varphi (X)\mathrm{d}X\right\},$$
(3)
where φ is a vector-valued function whose arguments are Pi (t)/Y (t), prices are assumed to be a function of the Xi, and Γ is the curve described by Xi. A similar expression characterizes the Divisia price index (for a more extensive discussion of Divisia line integrals, see Richter 1966; Hulten 1973; Samuelson and Swamy 1974).

The value of the index defined by (3) depends on the solution of the line integral. This can be obtained by identifying a ‘potential function’ Φ whose partial derivatives are the vector-valued function φ, that is, φ = ∇Φ. Writing Φ = log F function, the value of the index can be shown to equal F[X(T)]/F[X(0)], implying that the index is unique only up to a scalar multiple.

In economic terms, the solution to (3) is associated with some underlying economic relationship among the variables being indexed. Assume, for example, there is a constant returns to scale production function F(X) and Fi = λPi (Fi denotes the partial derivative of F with respect to Xi and λ is a factor of proportionality). Then the function log F can serve as the requisite potential function for (3), and in this particular case, the Divisia index of inputs can be interpreted as the ratio of output at time T to output at time zero.

If the form of the potential function is known a priori, the value of the index could be computed directly from the function F. However, the rationale for the Divisia index is that it provides a way of obtaining the ratio F(X(T))/F(X(0)) by using data on prices and quantities alone, without direct knowledge of F. Intuitively, this is possible because, under sufficiently restrictive assumptions, information about the slope of the function F (as estimated by relative prices) over the path followed by the inputs is sufficient to characterize F up to a scalar multiple.

When the objective is to form an index of a subset of inputs – aggregate labour input, for example – the required potential function is a ‘piece’ of a production function. Specifically, if one wants to form a Divisia index of the first M inputs, the production function needs to be weakly separable into a function of these inputs, that is, F{G[X1(t),…,XM(t)], XM+1(t),…,XN(t)}. The function log G serves as the potential function for the line integration (see also Balk 2005).

These considerations apply to Divisia price indexes as well. The relevant potential function is now the factor price frontier Ψ[P1(t),…,Pn(t)]. A basic result of duality theory shows that the partial derivatives of Ψ are proportional to the corresponding Xi(t).

The discussion suggests that the existence of the Divisia index is closely linked to the conditions for consistent aggregation. Furthermore, the required existence of a potential function implies that aggregation cannot proceed with just any set of prices or quantities. There must be an a priori reason for supposing that the variables to be indexed are theoretically related. This is an important characteristic of the Divisia index, one which it shares with the broader class of economic index numbers (in contrast to the non-structural axiomatic approach associated with Irving Fisher 1921; see also Balk 2005). The potential function theorem establishes the conditions under which the Divisia index is an ‘exact’ index number (to use the terminology of Diewert 1976) for some underlying economic structure.

Divisia indexes have the desirable property that they are invariant when the path of integration lies entirely in the same level set of the potential function. That is, ifone input is substituted for another along a given isoquant, the value of the index will not change. However, there is no guarantee of invariance when the path of integration lies across several level sets. This reflects the mathematical property that line integrals are, in general, path dependent.

Path dependence means that the index (3) will generally have a different value for a path β(t) ∈ Γ1 than path α(t)∈Γ , even though the beginning and end points of Γ1 and Γ are identical. This can lead to the following situation: the economy moves along Γ1 from X from X' (which is on a different isoquant); the economy then returns along Γ to the original point X; because of path dependence, the vector of quantities represented by the vector X will have a different Divisia index value after the trip around the composite path, and subsequent circuits will produce still different values. The value of the Divisia index at any point X is thus arbitrary under path dependence. The uniqueness of the Divisia index thus involves path independence.

The condition for path independence is the existence of a homothetic potential function, log F, such that φ = ∇logF, where φ is defined in (3). Given the existence of the potential function, the value of (3) is F(X(T))/F(X(0)), implying path independence since (3) depends only on the end points of the path, X(0) and X(T). Conversely, if (3) is path independent, there exists a potential function log F such that ∇log F = φ. In some applications in productivity analysis, the homotheticity condition must be strengthened to linear homogeneity, but this can be weakened depending on data availability (Hulten 2001, pp. 11–12).

We note, finally, that the Divisia index is defined using time as a continuous variable. Data on prices and quantities typically refer to discrete points in time, and the indexes constructed from them must therefore have a discrete–time form. The continuous-time Divisia index is nevertheless useful, both for informing the structure of these discrete-time indexes (for example, for the determining which variables are conceptually related), and for interpreting the results. The Divisia framework is also appropriate for the theoretical analysis of many economic problems, such as the use of Divisia indexes by Solow (1957) in growth accounting.

One approach to linking discrete and continuous index numbers is to approximate the continuous variables of (2) with their discrete time counterparts. Under the Törnqvist (1936) approach, the growth rates of prices and quantities are approximated by logarithmic differences, and the continuous weights by two period arithmetic averages. The Tornqvist approximation to the growth rate of the Divisia quantity index can then be written:
$$\sum \limits_{i=1}^{i=T}\frac{1}{2}\left[\frac{P_{i,t}{X}_{i,t}}{Y_i}+\frac{P_{i,t-1}{X}_{i,t-1}}{Y_{t-1}}\ \right]\ \left[\log\ {X}_{i,t}-\log {X}_{i,t-1}\right]$$
(4)
A similar approximation applies to the growth rate of the Divisia index of prices.

While the Törnqvist index may be regarded as approximate, Diewert (1976) has shown that it is exact when the underlying potential function has the (continuous) translog form. This result is very important in its own right, but can also be regarded as an important conceptual link between the discrete and continuous–time families of index numbers, given the exact properties of the Divisia index in continuous time.

The continuous Divisia index can also be approximated by using chain indexing procedures (the Divisia index is sometimes regarded as a chain whose links are defined over infinitesimal time periods). Other numerical approximation techniques can also be employed.

## Bibliography

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