Necessary and unnecessary parameter restrictions for CDES demand systems

In a 1975 Econometrica paper, Giora Hanoch proposed the so-called CDES functional form of a consumer demand system. Though not very popular in econometric applications, this functional form is a good choice for modelling consumer demand in CGE analysis, because it is a good compromise between the desire for flexibility on the one hand and parametric parsimony on the other. It allows to choose own-price and income elasticities subject to the restriction that the so-called expansion parameters must all be nonnegative and not all zero, and that the so-called substitution parameters must not exceed one. If one believed in the literature, one furthermore would have to impose the restriction that the substitution parameters must all lie on the same side of zero. The main purpose of this paper is to show that this restriction is unnecessary. Any sign pattern of substitution parameters can be admitted without destroying global regularity of the demand system. For proving this result, the paper defines the CDES system in a form differing slightly from the form suggested by Hanoch; instead of a sum of power functions the implicit form is defined as a sum of Box-Cox transforms. The paper covers a full analysis of the properties of the function; it discusses the flexibility of the form showing that even without the unnecessary restriction it is still not flexible enough to reproduce any desired regular pattern of own-price and income elasticities. An optimisation approach to parameter calibration subject to the required sign constraints may still be needed. It is demonstrated that calibrations with and without the mentioned unnecessary restriction can lead to entirely different parameters and different behaviour of the calibrated function. Differences between calibrations with and without the unnecessary restriction are illustrated using the numerical example in Huff et al. (1997). The final Section of the paper shows that the form presented in this paper is (almost) equivalent to the original Hanoch form. “Almost” means that, with measure zero in parameter space, data can be such that they allow a representation in our, but not in Hanoch’s form. Apart from this “non-generic” case, the original Hanoch form and our form with a suitable parameter transformation define the same demand system.