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I was delighted by yesterday’s announcement that Angus Deaton has been awarded the Nobel Prize in Economic Science this year. His contributions have have been many, fundamental, and varied, and I certainly won’t attempt to summarize them here. Suffice to say that the official citation says that the award is “for his contributions to consumption, poverty, and welfare“.
In this earlier post I made brief mention of Deaton’s path-breaking work, with John Muellbauer, that gave us the so-called “Almost Ideal Demand System”.
The AIDS model took empirical consumer demand analysis to a new level. It facilitated more sophisticated, and less restrictive, econometric analysis of consumer demand behaviour than had been possible with earlier models. The latter included the fundamentally important Linear Expenditure System (Stone, 1954), and the Rotterdam Model (Barten, 1964; Theil, 1965).
I thought that readers may be interested in an empirical exercise with the AIDS model. Let’s take a look at it.
First of all,the theoretical model needs to be explained.Let total expenditure on the n goods in the system be
M = Σi (pi qi) , (1)
and let
wi = (pi qi) / M ; i = 1, …, n (2)
denote the “budget share” for the ith good.
The system of demand equations itself is:
M = Σi (pi qi) , (1)
and let
wi = (pi qi) / M ; i = 1, …, n (2)
denote the “budget share” for the ith good.
The system of demand equations itself is:
wi = αi + βi [log(M) – log(P)] + Σj γij log(pj) + εi ; i = 1, …., n (3)
The overall price index, P, is defined by the following translog specification:
The overall price index, P, is defined by the following translog specification:
log(P) = α0 + Σj αj log(pj) + 0.5 Σi Σj γij log(pi) log(pj) . (4)
(All of the summations in equations (1) to (4) run from i = 1 to n; or from j = 1 to n.)
Notice that once (4) is substituted into (3), each of the n equations in the latter system is highly non-linear in the parameters of the model.
In practice, a value for α0 is usually pre-assigned, and there are various ways of choosing an “optimal” value (see Michalek and Keyzer, 1992).
One of the things that I like about empirical exercises such as the one that follows is that they illustrate how the underlying microeconomic theory can be incorporated explicitly into the formulation of the econometric model, and the subsequent estimation and testing.
Notice that once (4) is substituted into (3), each of the n equations in the latter system is highly non-linear in the parameters of the model.
In practice, a value for α0 is usually pre-assigned, and there are various ways of choosing an “optimal” value (see Michalek and Keyzer, 1992).
One of the things that I like about empirical exercises such as the one that follows is that they illustrate how the underlying microeconomic theory can be incorporated explicitly into the formulation of the econometric model, and the subsequent estimation and testing.
(This stands in contrast with a lot of other empirical work that we encounter – see this post.)
Specifically, for the AIDS model there are various restrictions on the parameters that we have to consider:
Engel aggregation requires that
Σk αk = 1 ; Σk βk = 0 ; Σk γkj = 0 ; for all j = 1, …., n (5)
These restrictions will be satisfied automatically, a long as the individual expenditures add up to total expenditure in the sample.
Homogeneity requires that
Σkγik = 0 ; for all i = 1, …., n ; (6)
and Slutsky symmetry requires that
Engel aggregation requires that
Σk αk = 1 ; Σk βk = 0 ; Σk γkj = 0 ; for all j = 1, …., n (5)
These restrictions will be satisfied automatically, a long as the individual expenditures add up to total expenditure in the sample.
Homogeneity requires that
Σkγik = 0 ; for all i = 1, …., n ; (6)
and Slutsky symmetry requires that
γij = γji ; for all i, j = 1, …., n . (7)
(In equations (5) and (6), the summations run from k = 1 to n.)
The homogeneity and symmetry restrictions are testable, and can be imposed, as appropriate.
We’re going to estimate an AIDS model for beer, wine, and spirits (numbered in that order), using annual time-series data for the U.K. over the period 1955 to 1985 inclusive. The data are available on the data page for this blog, and they come from Selvanathan, 1995, p.124). I’m going to use the ‘micEconAids’ package for R (Henningsen, 2015) for estimation and hypothesis testing, and my R code is on the code page for this blog.
Here are the basic results, which are self-explanatory:
We see that the total expenditure (“income”) elasticities suggest that beer is a necessity, while spirits and (especially) wine are both luxury goods. It would be a good idea to recall that the data are for U.K. for the period 1955 to 1985! Each good has own-price elasticities of demand that are negative, as expected. Beer is own-price inelastic, while wine and spirits are own-price elastic. The compensated price elasticities suggest, among other things, that each of the three beverages are substitutes.
The “summary” command in my code yields some additional results:
(In equations (5) and (6), the summations run from k = 1 to n.)
The homogeneity and symmetry restrictions are testable, and can be imposed, as appropriate.
We’re going to estimate an AIDS model for beer, wine, and spirits (numbered in that order), using annual time-series data for the U.K. over the period 1955 to 1985 inclusive. The data are available on the data page for this blog, and they come from Selvanathan, 1995, p.124). I’m going to use the ‘micEconAids’ package for R (Henningsen, 2015) for estimation and hypothesis testing, and my R code is on the code page for this blog.
In the following application, we’ll assume weak separability of the underlying utility function, so “Total Expenditure” (M) will be the total expenditure on the three types of alcoholic beverages. The sample means of the budget shares for three goods are 91.2%, 6.6%, and 2.2% for beer, wine, and spirits, respectively. The last of the expenditure share is fairly constant over the sample period, while those for beer and wine decrease from 94.1% to 88.3%, and increase from 3.9% to 9.6% respectively.
Here is my R code to obtain a value for α0, and to estimate the 3-equation system with both the homogeneity and symmetry restrictions imposed on the parameters.
Here is my R code to obtain a value for α0, and to estimate the 3-equation system with both the homogeneity and symmetry restrictions imposed on the parameters.
Here are the basic results, which are self-explanatory:
We see that the total expenditure (“income”) elasticities suggest that beer is a necessity, while spirits and (especially) wine are both luxury goods. It would be a good idea to recall that the data are for U.K. for the period 1955 to 1985! Each good has own-price elasticities of demand that are negative, as expected. Beer is own-price inelastic, while wine and spirits are own-price elastic. The compensated price elasticities suggest, among other things, that each of the three beverages are substitutes.
The “summary” command in my code yields some additional results:
Recalling that these results are for an AIDS model in which both the homogeneity and symmetry restrictions have been imposed, we had better test to see if these restrictions are supported by the data. Here is some R code to facilitate this:
The results are:
- We should reject the symmetry restrictions when they are added to the homogeneity restrictions. ( p = 0.0003).
- We should reject the homogeneity restrictions against the alternative of no restrictions, other than Engel aggregation. (p = 4*10-6).
- We should reject the (joint) symmetry and homogeneity restrictions in favour of no restrictions. other than Engel aggregation. (p = 3*10-8).
In short, we should remove the homogeneity and symmetry restrictions. Here are the estimation results when we do this:
Once again, the total expenditure (“income”) elasticities suggest that beer is a necessity, while spirits and wine are both luxury goods. Each good has own-price elasticities of demand that are negative, as expected. Now both beer and wine are own-price inelastic, while spirits is own-price elastic. The compensated price elasticities still suggest that each of the three beverages are substitutes.
The rest of the results are:
Let’s explore these results for the unrestricted model a little further.
There are other things that can be done with the ‘micEconAids’ package in R. It’s a great resource, and is just one of the packages available from the ‘micEcon’ project.
So, congratulations to Angus Deaton on his Nobel Prize, and let’s not forget the many seminal contributions that he made to consumer demand theory, beyond the AIDS model.
References
Barten, A. P., 1964. Consumer demand functions under conditions of almost additive preferences. Econometrica, 32, 1-38.
Deaton, A. and J. Muellbauer, 1980. An almost ideal demand system. American Economic Review, 70, 312-326.
Henningsen, A., 2015, Demand analysis with the almost ideal demand system: Package ‘micEconAids’, CRAN Repository.
Michalek, J. and M. A. Keyzer, 1992. Estimation of a two-stage LES-AIDS consumer demand system for eight EC countries. European Review of Agricultural Economics, 19, 137-163.
Selvanathan, E. A., 1995. Data-analytic techniques for consumer economics. In E. A. Selvanathan and K. W. Clements (eds.), Recent Developments in Applied Demand Analysis:Alcohol, advertising and global consumption. Springer, Berlin.
Stone, R., 1954. Linear expenditure systems and demand analysis: An application to the pattern of British demand. Economic Journal, 64, 511-527.
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