I have seen that researchers use different utility function in demand systems estimation such as Stone Geary. What is the role of these utility functions? What are utility function other than stone Geary?
The utility function taken for estimating demand systems depends on several parameters:
The nature of theoretical models and definitions of the variables used in these models are quite different from the nature of data that is available to us for estimation. Standard microeconomic theory tells us that demand is obtained as a result of some form of utility maximization. Naturally, therefore, the structure of demand function would depend heavily on the structure of utility function used to specify the consumer's preferences. Therefore, when we are trying to estimate a demand system, we look at the data we have at hand and the structure of the data, the variables available etc. give us an idea of what kind of demand system would be suitable, and hence, what kind of utility function we should start with.
Sometimes, the nature of analysis we would like to conduct can only be done only with a particular demand system. And therefore, it is natural to start with a utility specification that yields that particular demand system. On other occasions, we might just want to "keep things easy" and therefore choose utility functions which yield "tractable" demand functions. Stone-Geary is a good example of this. Stone Geary utility function yields what is known as the "linear expenditure systems". The analysis can then be carried out in terms of "product-wise expenditures", which is usually what is available to us in the form of data. CES-utility function is another such example.
Aggregation is also an important consideration. We might want aggregate the demand system and be able to say something about the properties of a more macro kind of economic entity. In that case, we don't really have many options, since aggregation requires very strict requirements. We then stick to some usual homothetic utility function (like Cobb-Douglas or Stone-Geary).
Finally, there are demand systems which have all the above listed nice properties. This is indeed what is usually done these days. Deaton and Muellbauer's (Linear/Quadratic - approximated) Almost Ideal Demand System (AIDS) is the state of the art demand system, inspired by the work of Stone and Geary on expenditure systems, as well as Gorman's work on aggregation. It's "easy" to estimate, requires lesser assumptions, has nice(est) aggregation properties, and can be made to fit usually available household-level survey data.
Hope it helps.