Re: st: quaids: how demographics variables change the expenditure function?

["Brian P. Poi" <bpoi@stata.com>]

On 05/09/2013 10:39 PM, Gary Mena wrote:
> Hello,
>
> although quaids command has provided me with a lot of help in my attempt to estimate compensating and equivalent variation, I've realized that demographics variables were very important when modelling demand systems.
>
> However, I haven't been able to understand how the quaids command handles demographics variables (I understand this command uses Ray's (1983) expenditure function scaling technique but it isn't explicit ly documented how).This is important given that if the expenditure function changes, then the welfare estimates will change.
>
> Could someone please indicate me how does the expenditure function change with the demographics option in quaids command?
>
> Thanks!
>
> Gary Mena L. 		 	   		

Gary,

The demographics alter expenditures in two ways.  Say we have a single demographic variable, nkids, the number of children in a household, and our consumption categories include food and clothing among other things.  Let's define a "reference" household as a household with nkids = 0.  Then for a household with nkids = 2, total expenditures on all goods will increase by virtue of the fact that more food must be bought, more clothes must be bought, etc.  Second, the pattern of consumption of the household with nkids = 2 may differ from a household with nkids = 0.  Kids outgrow clothing relatively quickly, so that household might spend a larger fraction of total expenditures on clothing than a household with nkids = 0.

Mathematically, say we have an expenditure function e(p,u).  To incorporate demographics z, Ray's (1983) method writes the expenditure function as

   e(p,z,u) = eR(p,u) * m(z) * phi(p,z,u)

Here, eR(p,u) denotes the "reference" household with all the demographic variables equal to zero.  eR(p,u) is the standard expenditure function for an AIDS or QUAIDS model.

m(z) is a function of the demographic variables and accounts for the first effect of demographic variables I mentioned above.  It shifts total expenditures based on the demographic variables z.

phi(p,z,u) accounts for changes in consumption patterns due to the demographic variables z.

I'm not even going to try to retype the formulas for m(z) and phi(p,z,u) here, but they are in my (2012) Stata Journal article.  How did I come up with the function for phi()?  Lots of mind-numbing algebra and trial-and-error until I managed to get expenditure share functions that look like natural extensions of the QUAIDS model without demographics, in the same way that Ray's expenditure shares accord with the AIDS model.

You asked about computing compensated variation (CV) and equivalent variation (EV).  Because we are accounting for demographics now, both CV and EV will be functions of the demographic variables.  For example, you can compute average CV's and EV's for households with no kids, 1 kid, 2 kids, and so on.  The only difference is that now you must work with the expenditure function e(p,z,u) defined above rather than e(p,u) without demographics.  The algebra is messy but definitely feasible.

The same is true for the elasticities: they also depend on the demographic variables.  In the SJ article I show a simple example where I compute elasticities separately for urban and rural consumers.  You can do the same for any other demographic variable as well.


I hope this helps.

   -- Brian Poi
   -- bpoi@stata.com

References
----------
Ray, R. (1983). Measuring the costs of children: an alternative approach. Journal of Public Economics, 22, 89--102.

Poi, B. (2012). Easy demand-system estimation with quaids. Stata Journal, 22(3), 443--446.
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