Measurement in Economics pp 649-670 | Cite as

# Separability and the Existence of Aggregates

Conference paper

## Abstract

If a continuous real-valued function Φ in n variables x where F and f are continuous real-valued functions, then the function is said to be (continuously)

_{1},...,x_{n}has the representation$$ \Phi ({x_1},...,{x_n}) = F(f({x_{1,...,}}{x_k}),{x_{k + 1}},...,{x_n}) $$

(1.1)

*separable*in the subset of variables x_{1},...,x_{k}. The concept of separability is intimately tied with the concept of aggregation and indices: If Φ is a function which gives the interrelationship among n economic quantities x_{1},...,x_{n}, then a function f may serve as an aggregate for the quantities x_{1},...,xk when and only when (1.1) holds; moreover, f may qualify as a quantity index for x_{1},...,x_{k}if it has certain desired properties such as monotonicity (see Eichhorn [4] for a methodological definition of economic indices).## Keywords

Boundary Point Economic Index Dimension Zero Quantity Index General Topological Space
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

- [1]V.I. ARNOL’d: Representation of Continuous Functions of Three Variables by the Superposition of Continuous Functions of Two Variables,
*Amer. Math. Soc. Transl.*28, 1963, 61–147.Google Scholar - [2]C. BLACKORBY/ D. PRIMONT/ R.R. RUSSELL: DUALITY, SEPARABILITY, AND FUNCTIONAL STRUCTURE, Elsevier North-Holland, New York, 1978.Google Scholar
- [3]G. DEBREU: Topological Methods in Cardinal Utility Theory, in MATHEMATICAL METHODS IN THE SOCIAL SCIENCES, K. Arrow, S. Karlin, and P. Suppes (eds.), Stanford University Press, Standford, 1959, 17–26.Google Scholar
- [4]W. EICHHORN: What is an Economic Index? An Attempt of an Answer, in THEORY AND APPLICATIONS OF ECONOMIC INDICES, W. Eichhorn, R. Henn, O. Opitz, and R.W. Shephard (eds.), Physica-verlag, Würzburg, 1978, 3–40.Google Scholar
- [5]W.M. GORMAN: The Structure of Utility Functions,
*Rev. Econ. Studies*35, 1968, 367–390.CrossRefGoogle Scholar - [6]K. KURATOWSKI: TOPOLOGY Vol. 1, new edition, Academic Press, New York and London, 1968.Google Scholar
- [7]K. KURATOWSKI: TOPOLOGY Vol. 2, new edition, Academic Press, New York and London, 1968.Google Scholar
- [8]K. MAK: On Separability: Functional Structure,
*J. Econ. Theory*40, 1986, 250–282.CrossRefGoogle Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1988