<p>After having done some further research, I think imposing the <strong>homogeneity restriction</strong> on the coefficients associated to the price variables is a way of creating relative prices (see <a href="https://www.sv.uio.no/econ/english/about/facts-and-figures/nobel-prize-winners/haavelmo/published-works/files/TH1951a.pdf" rel="nofollow noreferrer">Ref1</a>). 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.</p>
<p>Hence, the model might look like this for a demand system with <span class="math-container">$J$</span> goods and good indexed with <span class="math-container">$2$</span> as <strong>numeraire good</strong>:</p>
<p><span class="math-container">$w_i = \alpha_i + \beta_{1, i} (log(p_1) - log(p_2)) + \dots + \beta_{J,i}(log(p_J) - log(p_2)) + \gamma_i log(m/P^*)  + \epsilon_i.$</span></p>
<p>This can be  rearranged by applying a calculation rule of the logarithm to:</p>
<p><span class="math-container">$w_i = \alpha_i + \beta_{1, i} (log(p_1/p_2)) + \dots + \beta_{J,i}(log(p_J/p_2)) + \gamma_i log(m/P*)  + \epsilon_i.$</span></p>
<p>Whereby the model structure is obtained from the following post: <a href="https://stackoverflow.com/questions/58441299/r-force-regression-coefficients-to-add-up-to-1">Ref2</a>. The idea is that the homogeneity striction of <span class="math-container">$\sum_j \beta_{j,i} = 0$</span> can be translated to the stated model structure. The choice of the price variable, whichs is subtracted from the <span class="math-container">$j-1$</span> others is arbitrary. Note: <span class="math-container">$P^*$</span> is the Stone price index.</p>
<p><strong>Edit</strong></p>
<p>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:</p>
<p><span class="math-container">$w_i = \alpha_i + \beta_{1, i} (log(p_1) - log(P^*)) + \dots + \beta_{J,i}(log(p_J) - log(P^*)) + \gamma_i log(m/P*)  + \epsilon_i.$</span></p>
<p>This specification seems to be used in some demand system models (at least it looks similiar to that used in Stone (1954)).</p>

CC BY-SA 4.0
2022-03-19T17:01:06.690Z
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.
Bertrand
CC BY-SA 4.0
2022-03-19T17:44:34.163Z
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
CC BY-SA 4.0
2022-03-19T16:50:23.893Z
2022-03-21T14:14:16.447Z
2022-03-21T14:14:16.447Z
timm
<p><strong>Context/ Setup:</strong></p>
<p>Microeconometrics offers many tools to study features of the demand for different goods/ groups of goods, such as the well known <strong>Almost Ideal Demand System</strong> (AIDs). The AIDs model imposes some <strong>restrictions on the parameters</strong> 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 <strong>equation-by-equation OLS</strong>.</p>
<p>Some notation:</p>
<ul>
<li><span class="math-container">$j = 1, \dots, J$</span> indexes the goods/ groups of goods,</li>
<li><span class="math-container">$w_j$</span> is the budget share of <span class="math-container">$j$</span>,</li>
<li><span class="math-container">$p_j$</span> is the price of <span class="math-container">$j$</span> and <span class="math-container">$m$</span> the total expenditure of a household.</li>
</ul>
<p>The simple model estimated by equation-by-equation OLS:</p>
<p><span class="math-container">$w_j = \beta_0 + \beta_1 p_j  + \beta_2 m + \epsilon$</span>.</p>
<p><strong>Question:</strong></p>
<p>In a more sophisticated model I want to consider the effect of prices from other goods/ groups of goods on <span class="math-container">$w_j$</span>. I think for that purpose the construction <strong>relative prices</strong> is appropriate. <em>But how to compute these releative prices and integrate them in the simple model above?</em> In the Deaton and Muellbauer (1980) paper on AIDs I found some guidance, but wasn't that sure how to apply it.</p>

CC BY-SA 4.0
2022-03-19T16:51:08.797Z
The Stone price index is usually used in the literature instead of the CPI. See Deaton and Muellbaur (1980).
timm
CC BY-SA 4.0
2022-03-14T09:51:13.920Z
A possibility is described here: https://economics.stackexchange.com/questions/50316/what-is-a-simple-demand-function-that-allows-for-different-price-and-income-elas/50323#50323
Bertrand
CC BY-SA 4.0
2022-03-14T10:01:22.963Z
Thanks that is helpful. So in my case the ratio would be something like that: $p_j / \sum_{i \neq j} p_i$?
timm
CC BY-SA 4.0
2022-03-14T10:07:20.447Z
Usually researchers like Blundell of Deaton consider a consumer price index for normalization (which weights the elementary prices according to their importance).
Bertrand
CC BY-SA 4.0
2022-03-14T10:26:25.820Z
I see. In the three goods case, would one consider: $p_1/p_2$ and $p_1/p_3$ if one studies the demand for a good index by $1$? And how should the consumer price index (CPI) be included in this case? I guess something like $p_1 / \text{CPI}$ is wrong, as it does not account for the individual price relationships.
timm
CC BY-SA 4.0
2022-03-14T14:48:06.927Z
And I guess, that $\text{log}(m/\text{CPI})$ should be considered rather than just $m$ in the above model.
timm
<microeconomics><demand><demand-system-estimation>
Relative prices in demand system estimation
timm
# Relative prices in demand system estimation

(Asked by: username_0 on 2022-03-14)

**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.

**Tags:** microeconomics, demand, demand-system-estimation

> **Comments:**
> **username_1**: A possibility is described here: https://economics.stackexchange.com/questions/50316/what-is-a-simple-demand-function-that-allows-for-different-price-and-income-elas/50323#50323  
> **username_0**: Thanks that is helpful. So in my case the ratio would be something like that: $p_j / \sum_{i \neq j} p_i$?  
> **username_1**: Usually researchers like Blundell of Deaton consider a consumer price index for normalization (which weights the elementary prices according to their importance).  
> **username_0**: I see. In the three goods case, would one consider: $p_1/p_2$ and $p_1/p_3$ if one studies the demand for a good index by $1$? And how should the consumer price index (CPI) be included in this case? I guess something like $p_1 / \text{CPI}$ is wrong, as it does not account for the individual price relationships.  
> **username_0**: And I guess, that $\text{log}(m/\text{CPI})$ should be considered rather than just $m$ in the above model.  
> **username_0**: The Stone price index is usually used in the literature instead of the CPI. See Deaton and Muellbaur (1980).  

---

## 1 answers

### Answer by username_0 on 2022-03-19. Score: 1

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](https://www.sv.uio.no/econ/english/about/facts-and-figures/nobel-prize-winners/haavelmo/published-works/files/TH1951a.pdf)). 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](https://stackoverflow.com/questions/58441299/r-force-regression-coefficients-to-add-up-to-1). 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)).

> **Comments:**
> **username_1**: 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.  
> **username_0**: Sure, $P*$ is the Stone price index as used in Deaton and Muellbaur (1980). Can you elaborate on the "symmetric treatment of all prices"?  

---

