Estimating Demand Elasticities and Compensating Variation

A student posed the following question:

Hi Respected Sir,
I am writing a report on Compensating Variation (CV) in case of more than two goods say 8 goods. My question is how can we estimate it in case of more than 2 goods.
Another Question is How can we write the equations for 8 commodities in Almost Ideal Demand System (AIDS) to calculate Own , Cross and Expenditure Elasticities of demand.
In which software both of the methods can be calculated?
I am waiting for your reply.
Thanks in anticipation

My response:

This question is more econometrics than it is intermediate micro.  So I am only going to provide a partial response and stick to the economic theory part, though I must say I'd only teach what I discuss below at the graduate level. 

The traditional approach to consumer choice is to start with preferences, specified by a utility function, u, and combine that with a budget constraint, that depends on a price system p = (p1,p2,...,pn) that specifies the prices of each good or service, and the consumers income y.  Together the price system and income determine the budget set.  The consumer's choice, or demand, or optimum, call it x*(u,p,y) solves the problem of maximizing utility subject to the budget constraint.   

There is an alternative approach called the duality approach, which is useful conceptually and in laying the foundation for the econometrics.  Two value functions are determined.  One, measured in dollar terms, is called the Expenditure Function.  It is analogous to the Cost Function developed in the theory of the firm.  The expenditure function maps indifference curves (or when there are more than two goods, level sets of the utility function) and price systems into an expenditure level - the least expenditure it takes to reach that indifference curve at the given prices.   The other value function, measured in utility terms, is called the Indirect Utility Function.  It maps budget sets into utility levels.  Alternatively, it gives the utility attained at the consumer's choice. 

One of the powerful results from duality theory is that you can recover the consumer demand's from these value functions.  The compensated demands (these measure the substitution effect only) are given by the first partial derivatives of the Expenditure Function with respect to the specific price.  The ordinary demands can be obtained in a similar way from the Indirect Utility Function, though the result, known as Roy's Identity, is a bit more complicated.   

Let me close with the little I know about the Almost Ideal Demand System.  Deaton and Mulbauer start with the Expenditure Function, express it in log form, and then linearize it locally, assuming it is some average of the expenditure at subsistence (the worst possible point) and bliss (the best possible point).  This makes it suitable for estimation. 

Good luck on your paper.