# Relative prices in demand system estimation

Context/ Setup:

Microeconometrics offers many tools to study features of the demand for different goods/ groups of goods, such as the well known Almost Ideal Demand System (AIDs). The AIDs model imposes some restrictions on the parameters of the demand model derived from demand theory. Suppose you don't want to impose the restrictions right from the beginning, but you want to estimate a demand model with a simple equation-by-equation OLS.

Some notation:

• $$j = 1, \dots, J$$ indexes the goods/ groups of goods,
• $$w_j$$ is the budget share of $$j$$,
• $$p_j$$ is the price of $$j$$ and $$m$$ the total expenditure of a household.

The simple model estimated by equation-by-equation OLS:

$$w_j = \beta_0 + \beta_1 p_j + \beta_2 m + \epsilon$$.

Question:

In a more sophisticated model I want to consider the effect of prices from other goods/ groups of goods on $$w_j$$. I think for that purpose the construction relative prices is appropriate. But how to compute these releative prices and integrate them in the simple model above? In the Deaton and Muellbauer (1980) paper on AIDs I found some guidance, but wasn't that sure how to apply it.

• A possibility is described here: economics.stackexchange.com/questions/50316/… Mar 14, 2022 at 9:51
• Thanks that is helpful. So in my case the ratio would be something like that: $p_j / \sum_{i \neq j} p_i$?
– timm
Mar 14, 2022 at 10:01
• Usually researchers like Blundell of Deaton consider a consumer price index for normalization (which weights the elementary prices according to their importance). Mar 14, 2022 at 10:07
• And I guess, that $\text{log}(m/\text{CPI})$ should be considered rather than just $m$ in the above model.
– timm
Mar 14, 2022 at 14:48
• The Stone price index is usually used in the literature instead of the CPI. See Deaton and Muellbaur (1980).
– timm
Mar 19, 2022 at 16:51

After having done some further research, I think imposing the homogeneity restriction on the coefficients associated to the price variables is a way of creating relative prices (see Ref1). The homogeneity restriction implies that consumers base their descisions on relative prices and are not influenced by absolute changes of equal size of the nominal variables in the demand system model.

Hence, the model might look like this for a demand system with $$J$$ goods and good indexed with $$2$$ as numeraire good:

$$w_i = \alpha_i + \beta_{1, i} (log(p_1) - log(p_2)) + \dots + \beta_{J,i}(log(p_J) - log(p_2)) + \gamma_i log(m/P^*) + \epsilon_i.$$

This can be rearranged by applying a calculation rule of the logarithm to:

$$w_i = \alpha_i + \beta_{1, i} (log(p_1/p_2)) + \dots + \beta_{J,i}(log(p_J/p_2)) + \gamma_i log(m/P*) + \epsilon_i.$$

Whereby the model structure is obtained from the following post: Ref2. The idea is that the homogeneity striction of $$\sum_j \beta_{j,i} = 0$$ can be translated to the stated model structure. The choice of the price variable, whichs is subtracted from the $$j-1$$ others is arbitrary. Note: $$P^*$$ is the Stone price index.

Edit

As @Betrand points out in the comments it may make more sense to put the prices in relation with a general price index instead of picking an arbitrary price as basis of comparison. Hence, it should be:

$$w_i = \alpha_i + \beta_{1, i} (log(p_1) - log(P^*)) + \dots + \beta_{J,i}(log(p_J) - log(P^*)) + \gamma_i log(m/P*) + \epsilon_i.$$

This specification seems to be used in some demand system models (at least it looks similiar to that used in Stone (1954)).

• A coefficient for the income effect seems to be missing. Do you have a definition for $P^*$, and an explanation for why $p_2$ is used for normalizing prices and not $P^*$? A symmetric treatment of all prices is intuitively somewhat less arbitrary. Mar 19, 2022 at 17:01
• Sure, $P*$ is the Stone price index as used in Deaton and Muellbaur (1980). Can you elaborate on the "symmetric treatment of all prices"?
– timm
Mar 19, 2022 at 17:44