Chapter 3 Theory at Work
The authors discuss the application of four different models on demand analysis. They started with Stone’s analysis in 1954 on using single equation methodology to measure elasticities.
The single equation model:
The single equation model is formulated as follows:
\[\begin{equation} log q_{i} = \alpha_{i} + e_{i}logx + \sum_{k=1}^{n}e_{ik}logp_{k}, \end{equation}\]where \(e_{i}\) is the total expenditure elasticity, and \(e_{ik}\) is the cross-price elasticity of \(k\)th price on the \(i\)th demand.
Using British data for 48 food categories over the years 1920-38, Stone used this equation to estimate demand functions. He found that among different categories, equations for orange and bananas show controversial results:
“According to the oranges equation, bananas are a substitute for oranges, but according to the bananas equation, the pair are complements”.
These results suggest that an alternative approach must be adopted to measure complementarity and substitutability.
The linear expenditure system:
To overcome above problem, Stone (1954) studies a general linear formulation of demand as in the following form:
\[\begin{equation} p_{i}q_{i} = \beta x + \sum_{j=1}^{n}\beta_{ij}p_{j}. \end{equation}\]The authors note that the linear expenditure system is the only form of above equation that satisfies theoretical restrictions of adding up, homogeneity, and symmetry. In this case, above equation can be written as:
\[\begin{equation} p_{i}q_{i} = p_{i}\gamma_{i} + \beta_{i}(x - \sum p_{k}\gamma_{k})$ with $\sum \beta_{k}=1. \end{equation}\]The interpretation of this equation is as follows:
“The committed expenditures \(p_{i}\gamma_{i}\) are bought first, leaving a residual, “supernumerary expenditure” \(x - \sum p_{k}\gamma_{k}\), which is allocated between the goods in the fixed proportions \(\beta_{i}\). Hence, apart from the subsistence expenditure \(\sum p_{k}\gamma_{k}\), total outlay is divided in a constant pattern between the commodities.”
The authors explain the limitations of the linear expenditure system for practical purposes.
- A constant elasticity functional form is used to satisfy adding up condition by allowing only unitary elasticities.
- Linearity of above equation introduces some contradictions such as having an inferior good, but violating concavity, or holding concavity, but not allowing two goods to be complements, i.e., results in “every good must be a substitute for every other good”.
The authors write an important note here:
“These restrictions do not mean that the model cannot be applied in practice, only that its application must be restricted to those cases where its limitations are not thought to be serious. Even then, care must be taken in interpreting the results and a careful distinction drawn between properties of the model, imposed a priori, and proporties of the data. For example, no one can claim to have “discovered” that goods are substitutes from results obtained using the linear expenditure system.”
A new model of demand:
In this subsection, the authors explains the Almost Ideal Demand System proposed by Deaton and Muellbauer (1980). They started with the following equation:
\[\begin{equation} w_{i}=\alpha_{i} + \beta_{i}logx, \end{equation}\]where \(w_{i}\) is the budget share for good \(i\) and \(x\) is the total expenditure spent on all goods.
Then, they define a cost function as follows:
\[\begin{equation} log c(u,p) = a(p) + ub(p), \end{equation}\]where \(p\) is the price and \(u\) is the utility of an individual.
The authors state that when the following two equations substituted into above cost equation, the budget shares \(w_{i}\) can be derived from \(\partial log c/ \partial logp_{i} = w_{i}\):
\(a(p) = \alpha_{0} + \sum\alpha_{k}logp_{k} + \frac{1}{2}\sum_{k}\sum_{l}\gamma_{kl}^{*}logp_{k}logp_{l}\)
\(b(p)=\beta_{0}\Pi p_{k}^{\beta_{k}}\).
After substitution for \(u\), the budget share equation becomes:
\[\begin{equation} w_{i}=\alpha_{i} + \sum_{j}\gamma_{ij}logp_{j} + \beta_{i}log(x/P), \end{equation}\]where \(P\) is price index defined by
\[\begin{equation} log P = \alpha_{0} + \sum\alpha_{k}logp_{k}+\frac{1}{2}\sum_{k}\sum_{l}\gamma_{kl}logp_{k}logp_{l}. \end{equation}\]The theoretical restrictions applied to budget share equation are:
Adding up restriction: for all \(j\), \(\sum_{k}\alpha_{k}=1\), \(\sum_{k}\beta_{k}=0\), \(\sum_{k}\gamma_{kj}=0\).
Homogeneity restriction: for all \(j\), \(\sum_{k}\gamma_{jk}=0\).
Symmetry restriction: \(\gamma_{ij}=\gamma_{ji}\).
The authors test homogeneity and symmetry conditions on postwar British annual data from 1954-74. They observe that they reject homogeneity for food, clothing, housing, and transport, which they interpret in the following way:
“…, the imposition of homogeneity generates positive serial correlation in the residuals. This suggests the converse proposition, that the rejection of homogeneity may be caused by an inappropriate specification of the dynamics of bahvior….This suggests that time trends and possibly lagged dependent variables are omitted variables in the static model and that the price coefficients are biased by their omission.”
Then, the authors continue on testing symmetry condition.
“The imposition of this [symmetry] led to a further but less sharp drop in likelihood especially when the model was estimated in first differences. Hence, on these data, it is unclear whether we must reject symmetry as a restriction beyond homogeneity.”
Lastly, the authors make a conclusion form their work.
“We have looked at different models, each embodying different approximations, and these have been fitted to different data sets from several countries, but the same conclusions have repeatedly emerged. Demand functions fitted to aggregated time series data are not homogeneous and probably not symmetric.”
An assessment:
In the final words of this chapter, the authors emphasize the difficulty of calculating cross-price elasticities on time-series data.
They also note that the Rotterdam, translog, and AIDS models are examples of how to built demand systems, “at least over broadly defined groups of commodities”. But they admit that “these models produce a conflict with the theory”. “The restrictions of homogeneity and symmetry, basic to the assumptions of a linear budget constraint and the axioms of choice, are consistently rejected by the data”.
They note that “…But it is much more than a question of finding models that fit well. The real challenge is one of intellectual honesty; we must construct models that are fundamentally credible as representations of the behavior and phenomena we are trying to understand.”
The authors discuss this topic in detail listing related problems as follows:
Aggregation over commodities.
Working with aggregated data as if it is related to a single consumer. The authors say that “there is no general reason to suppose that this is valid.”
Deriving demands as a function of prices: “This only makes sense as long as prices are set exogenously (say by manufacturers) and quantities supplied elastically to meet whatever demand emerges.”