The Dixit-Stiglitz aggregator is often used in New Keynesian (representative agents) models to form an aggregate indicator from many different indicators. Its general form is

Z = (1/n)[SumZia]1/a for i = 1 to n with (1/n) for the calculation of a “mean”. Zi may be the production of sector i or good i, the consumption of different goods and services, the working time of different agents etc. The coefficient a is mostly a function of an elasticity of substitution e - a = (1-e)/e.

In many articles, a continuous version of this aggregation is used, like Z as the integral of Z(i)adi instead of the sum. That means to assume a continuity of goods, persons or firms etc., i.e one cannot even count or identify them. For me, that is the transition from nonsense to simply stupid economics.

At the first sight, the discrete version above looks like a mix of an arithmetic and geometric mean, but in fact it is a CES function with identical share parameters for all Zi’s. The results are quite far from both means, and they are dependent on the number of the Zi data – in general increasing with n. Therefore, I do not know how to interpret them.

My questions are: Are there arguments in favour of this aggregator? Have you ever seen a sensible application on empirical data?

PS: See enclosed a copy of one page of an article by Ngai/Petrongolo on Gender Gaps and the Rise of the Service Economy, where total labor input is a CES combination of male and female working hours. Let us assume an elasticity of substitution of 2 and equal shares for men and women and that the firm employs both groups for 400 hours. Then (3) reads L = [0.5*4000.5+0.5*4000.5]2 = 400, i.e. the labor input is only half of the hours paid. Poor employer!