<p>So I did a little bit of digging around on this and it seems to be a result from the indirect  <a href="https://en.wikipedia.org/wiki/Stone%E2%80%93Geary_utility_function" rel="nofollow noreferrer">Stone-Geary utility function</a>. Not 100% sure on this but i'm pretty sure<sup>**</sup>.</p>

<p>Recall that the stone-geary utility function is defined as the following:
<span class="math-container">$$U(x_1,...,x_n)=\prod_{i=1}^n(x_i-\gamma_i)^{\beta_i}$$</span></p>

<p>Note that the marshallian demands for stone-geary preferences are defined as:
<span class="math-container">$$x^*_i=\gamma_i+\frac{\beta_i}{p_i}\left(m-\sum_{j=1}^np_j\gamma_j\right) \forall\ \ i\neq j$$</span></p>

<p>subbing this result into our utility function to obtain our indirect utility function we get:
<span class="math-container">$$V(x^*_1,...,x^*_n)=\prod_{i=1}^n(x^*_i-\gamma_i)^{\beta_i}$$</span>
<span class="math-container">$$V(p_1,...,p_n,m)=\prod_{i=1}^n\left(\gamma_i+\frac{\beta_i}{p_i}\left(m-\sum_{j=1}^np_j\gamma_j\right)-\gamma_i\right)^{\beta_i}$$</span>
<span class="math-container">$$V(p_1,...,p_n,m)=\prod_{i=1}^n\left(\frac{\beta_i}{p_i}\left(m-\sum_{j=1}^np_j\gamma_j\right)\right)^{\beta_i}$$</span></p>

<p>Multiplying both sides by <span class="math-container">$\prod_{i=1}^np_i^{\beta_i}$</span> and <span class="math-container">$\prod_{i=1}^n\frac{1}{\beta_i^{\beta_i}}$</span> we get:</p>

<p><span class="math-container">$$\prod_{i=1}^n\frac{1}{\beta_i^{\beta_i}}\prod_{i=1}^np_i^{\beta_i}V=\prod_{i=1}^n(m-\sum_{j=1}^np_j\gamma_j)^{\beta_i}$$</span></p>

<p>recalling that throughout the literature <span class="math-container">$\beta_0$</span> has been defined as an "unestimatiable parameter" let: <span class="math-container">$\prod_{i=1}^n\frac{1}{\beta_i^{\beta_i}}=\beta_0$</span> and having <span class="math-container">$V(p_1,...,p_n,m)$</span> fixed at some level of utility <span class="math-container">$u$</span> we therefore have:</p>

<p>Therefore:
<span class="math-container">$$\beta_0\prod_{i=1}^n p_i^{\beta_i}u=\prod_{i=1}^n(m-\sum_{j=1}^np_j\gamma_j)^{\beta_i}$$</span></p>

<p>notice how this is very similar to <span class="math-container">$b(p)$</span> as defined by varian<sup>*</sup>. thus it would represent a function which is essentially cobb-douglas preferences centered on income levels above subsistence (stone-geary) <em>in terms of money</em> (not goods).</p>

<p>This shows how spending shaped by prefernces when our consumer is no longer trying to eke out a living.</p>

<p>If we were to take a second order taylor approximation of the RHS around <span class="math-container">$p_j$</span> (note that we cover all of our prices in our system) we get:</p>

<p><span class="math-container">$$\beta_0\prod_{i=1}^n p_i^{\beta_i}u=\log(m)-a_0-\sum_ia_i\log p_i-\frac{1}{2}\sum_i\sum_j\delta^*_{ij}\log p_i\log p_j$$</span></p>

<p>Which is in turn:
<span class="math-container">$$\beta_0\prod_{i=1}^n p_i^{\beta_i}u=\log\left(\frac{m}{P}\right)$$</span></p>

<p>where
 <span class="math-container">$$\log(P)=a_0+\sum_ia_i\log p_i+\frac{1}{2}\sum_i\sum_j\delta^*_{ij}\log p_i\log p_j$$</span></p>

<p>The reason why we use the LHS term is used instead of the RHS term is because of the construction of the AIDS system as an expenditure function which requires that utility to be apart of it by definition.</p>

<blockquote>
  <p>TL;DR This was an attempt to understand why we use a parameter that we end up subbing out from the Almost Ideal Demand System.  </p>
</blockquote>

<hr>

<p><sub>*One may ask, that this isn't entirely true since Deaton and Muellbauer (1980) define <span class="math-container">$\log\{b(p)\}=log\{a(p)\}+\beta_0\prod_{i=1}^n p_i^{\beta_i}$</span> however since this goes away as a result of the definitions used in the structure of the PIGLOG system, id think its alright.</sub></p>

<p><sub>** see Castellón's,Boonsaeng's and Carpio's
 paper <a href="https://www.researchgate.net/profile/Carlos_Carpio2/publication/241750470_Demand_System_Estimation_in_the_Absence_of_Price_Data_an_Application_of_Stone-Lewbel_Price_Indices/links/0c960532c6a65a32dc000000/Demand-System-Estimation-in-the-Absence-of-Price-Data-an-Application-of-Stone-Lewbel-Price-Indices.pdf" rel="nofollow noreferrer">Demand System Estimation in the Absence of Price Data: an Application of
Stone-Lewbel Price Indices</a> page 6</sub></p>

CC BY-SA 4.0
2019-01-20T20:27:22.447Z
2019-01-25T01:09:57.637Z
2019-01-25T01:09:57.637Z
EconJohn
<p>In the original article by <em>Deaton and Muellbauer</em> (1980), <a href="http://www.princeton.edu/~deaton/downloads/An_Almost_Ideal_Demand_System.pdf" rel="nofollow noreferrer">An Almost Ideal Demand System</a>, they characterize the expenditure (cost) function for the class of PIGLOG preferences as:</p>

<p><span class="math-container">$$\log c(u, p) = (1-u) \log \{a(p)\} + u \log \{b(p)\}\tag{1}$$</span></p>

<p>Where they go on to describe in intuitive terms:</p>

<blockquote>
  <p>" With some exceptions (see the Appendix), <span class="math-container">$u$</span> lies between <span class="math-container">$0$</span> (subsistence) and <span class="math-container">$1$</span> (bliss) so that the positive linearly homogeneous functions <span class="math-container">$a(p)$</span> and <span class="math-container">$b(p)$</span> can be regarded as the costs of subsistence and bliss, respectively. " (p. 313)</p>
</blockquote>

<p>Some algebra gets you equations <span class="math-container">$(2)$</span> and <span class="math-container">$(3)$</span> in the paper, which are analogous forms of the equations you have above.</p>

CC BY-SA 4.0
2019-01-14T01:17:41.263Z
I thought that specification was only for the PIGLOG system
EconJohn
CC BY-SA 4.0
2019-01-14T03:43:39.330Z
I am not super familiar with the system, so I am not sure myself. I may take a closer look later.
Kitsune Cavalry
CC BY-SA 4.0
2019-01-14T03:45:24.707Z
I think you might be right. what i meant by the question is that I want to know what $\beta_0 \prod p_i ^{\beta_i}$ is.
EconJohn
CC BY-SA 4.0
2019-01-15T11:02:14.263Z
Well I am not historian of economic thought, but my supervisor explained me that AIDS was coined by Deaton and Muellbauer in the 1980's before the of outbreak of the deases with the same acronym. After this the "Almost Ideal Demand System" became translog (Jorgenson, 1982) and both forms are nested within the PIGLOG class.
Bertrand
CC BY-SA 4.0
2019-01-14T01:05:08.453Z
2019-01-14T01:05:08.453Z
Kitsune Cavalry
<p>In Hal Varian's Microeconomic analysis (page 213) he discusses the <a href="https://aae.wisc.edu/aae705/References/Deaton_AIDS.pdf" rel="nofollow noreferrer">Almost Ideal demand system</a>. There he describes the AIDS system as follows:</p>

<blockquote>
  <p>The almost ideal demand system (AIDS) has an expenditure function of the following form:
  <span class="math-container">$$e(\boldsymbol{p},u)=a(p)+b(p)u$$</span>
  where
  <span class="math-container">$$a(p)=a_0+\sum_ia_i\log p_i+\frac{1}{2}\sum_i\sum_j\gamma^*_{ij}\log p_i\log p_j $$</span>
  <span class="math-container">$$b(p)=\beta_0\prod p_i^{\beta_i}$$</span>
  <span class="math-container">$$...$$</span></p>
</blockquote>

<p>So I know <span class="math-container">$a(p)$</span> is a second order approximation of <span class="math-container">$p$</span> on our expenditure function, however i am having difficulty understanding what exactly <span class="math-container">$b(p)$</span> is.</p>

<p>What is <span class="math-container">$b(p)$</span>?</p>

CC BY-SA 4.0
2019-01-14T08:05:09.107Z
$\textbf{b(p)}$ represents how much expenditures need to increase if one wants to achieve a higher utility level. Ideally it is also a second or third order approximation in $p$.
Bertrand
CC BY-SA 4.0
2019-01-14T23:13:56.083Z
@Bertrand so $b(p)$ is just an expenditure function a HOD(1) in $u$?
EconJohn
CC BY-SA 4.0
2019-01-15T10:46:42.510Z
Yes $b(p)$ is HOD1 but in $p$.
Bertrand
CC BY-SA 4.0
2019-01-15T10:51:38.033Z
By the way, it is quite surprising to me than Varian calls this the AIDS. I re-read Deaton and Muellbauer (1980, p. 66) and this is actually "the Linear Expenditure System", while AIDS is rather (DM, p. 75) $log(e(p,u)) = a_0(p) + b_0(p)u$.
Bertrand
<microeconomics><consumer-theory>
what is $b(p)$ in the Almost Ideal Demand System?
EconJohn
# what is $b(p)$ in the Almost Ideal Demand System?

(Asked by: username_0 on 2019-01-14)

In Hal Varian's Microeconomic analysis (page 213) he discusses the [Almost Ideal demand system](https://aae.wisc.edu/aae705/References/Deaton_AIDS.pdf). There he describes the AIDS system as follows:

> The almost ideal demand system (AIDS) has an expenditure function of the following form:
> $$e(\boldsymbol{p},u)=a(p)+b(p)u$$
> where
> $$a(p)=a\_0+\sum\_ia\_i\log p\_i+\frac{1}{2}\sum\_i\sum\_j\gamma^\*\_{ij}\log p\_i\log p\_j $$
> $$b(p)=\beta\_0\prod p\_i^{\beta\_i}$$
> $$...$$

So I know $a(p)$ is a second order approximation of $p$ on our expenditure function, however i am having difficulty understanding what exactly $b(p)$ is.

What is $b(p)$?

**Tags:** microeconomics, consumer-theory
> **Comments:**
> **username_1**: $\textbf{b(p)}$ represents how much expenditures need to increase if one wants to achieve a higher utility level. Ideally it is also a second or third order approximation in $p$.  
> **username_0**: @Bertrand so $b(p)$ is just an expenditure function a HOD(1) in $u$?  
> **username_1**: Yes $b(p)$ is HOD1 but in $p$.  
> **username_1**: By the way, it is quite surprising to me than Varian calls this the AIDS. I re-read Deaton and Muellbauer (1980, p. 66) and this is actually "the Linear Expenditure System", while AIDS is rather (DM, p. 75) $log(e(p,u)) = a_0(p) + b_0(p)u$.  

---

## 2 answers

### Answer by username_0 on 2019-01-20. Score: 1 | Accepted answer

So I did a little bit of digging around on this and it seems to be a result from the indirect [Stone-Geary utility function](https://en.wikipedia.org/wiki/Stone%E2%80%93Geary_utility_function). Not 100% sure on this but i'm pretty sure\*\*.

Recall that the stone-geary utility function is defined as the following:
$$U(x\_1,...,x\_n)=\prod\_{i=1}^n(x\_i-\gamma\_i)^{\beta\_i}$$

Note that the marshallian demands for stone-geary preferences are defined as:
$$x^\*\_i=\gamma\_i+\frac{\beta\_i}{p\_i}\left(m-\sum\_{j=1}^np\_j\gamma\_j\right) \forall\ \ i\neq j$$

subbing this result into our utility function to obtain our indirect utility function we get:
$$V(x^\*\_1,...,x^\*\_n)=\prod\_{i=1}^n(x^\*\_i-\gamma\_i)^{\beta\_i}$$
$$V(p\_1,...,p\_n,m)=\prod\_{i=1}^n\left(\gamma\_i+\frac{\beta\_i}{p\_i}\left(m-\sum\_{j=1}^np\_j\gamma\_j\right)-\gamma\_i\right)^{\beta\_i}$$
$$V(p\_1,...,p\_n,m)=\prod\_{i=1}^n\left(\frac{\beta\_i}{p\_i}\left(m-\sum\_{j=1}^np\_j\gamma\_j\right)\right)^{\beta\_i}$$

Multiplying both sides by $\prod\_{i=1}^np\_i^{\beta\_i}$ and $\prod\_{i=1}^n\frac{1}{\beta\_i^{\beta\_i}}$ we get:

$$\prod\_{i=1}^n\frac{1}{\beta\_i^{\beta\_i}}\prod\_{i=1}^np\_i^{\beta\_i}V=\prod\_{i=1}^n(m-\sum\_{j=1}^np\_j\gamma\_j)^{\beta\_i}$$

recalling that throughout the literature $\beta\_0$ has been defined as an "unestimatiable parameter" let: $\prod\_{i=1}^n\frac{1}{\beta\_i^{\beta\_i}}=\beta\_0$ and having $V(p\_1,...,p\_n,m)$ fixed at some level of utility $u$ we therefore have:

Therefore:
$$\beta\_0\prod\_{i=1}^n p\_i^{\beta\_i}u=\prod\_{i=1}^n(m-\sum\_{j=1}^np\_j\gamma\_j)^{\beta\_i}$$

notice how this is very similar to $b(p)$ as defined by varian\*. thus it would represent a function which is essentially cobb-douglas preferences centered on income levels above subsistence (stone-geary) *in terms of money* (not goods).

This shows how spending shaped by prefernces when our consumer is no longer trying to eke out a living.

If we were to take a second order taylor approximation of the RHS around $p\_j$ (note that we cover all of our prices in our system) we get:

$$\beta\_0\prod\_{i=1}^n p\_i^{\beta\_i}u=\log(m)-a\_0-\sum\_ia\_i\log p\_i-\frac{1}{2}\sum\_i\sum\_j\delta^\*\_{ij}\log p\_i\log p\_j$$

Which is in turn:
$$\beta\_0\prod\_{i=1}^n p\_i^{\beta\_i}u=\log\left(\frac{m}{P}\right)$$

where
$$\log(P)=a\_0+\sum\_ia\_i\log p\_i+\frac{1}{2}\sum\_i\sum\_j\delta^\*\_{ij}\log p\_i\log p\_j$$

The reason why we use the LHS term is used instead of the RHS term is because of the construction of the AIDS system as an expenditure function which requires that utility to be apart of it by definition.

> TL;DR This was an attempt to understand why we use a parameter that we end up subbing out from the Almost Ideal Demand System.

---

\*One may ask, that this isn't entirely true since Deaton and Muellbauer (1980) define $\log\{b(p)\}=log\{a(p)\}+\beta\_0\prod\_{i=1}^n p\_i^{\beta\_i}$ however since this goes away as a result of the definitions used in the structure of the PIGLOG system, id think its alright.

\*\* see Castellón's,Boonsaeng's and Carpio's
paper [Demand System Estimation in the Absence of Price Data: an Application of
Stone-Lewbel Price Indices](https://www.researchgate.net/profile/Carlos_Carpio2/publication/241750470_Demand_System_Estimation_in_the_Absence_of_Price_Data_an_Application_of_Stone-Lewbel_Price_Indices/links/0c960532c6a65a32dc000000/Demand-System-Estimation-in-the-Absence-of-Price-Data-an-Application-of-Stone-Lewbel-Price-Indices.pdf) page 6


---

### Answer by username_2 on 2019-01-14. Score: 2

In the original article by *Deaton and Muellbauer* (1980), [An Almost Ideal Demand System](http://www.princeton.edu/~deaton/downloads/An_Almost_Ideal_Demand_System.pdf), they characterize the expenditure (cost) function for the class of PIGLOG preferences as:

$$\log c(u, p) = (1-u) \log \{a(p)\} + u \log \{b(p)\}\tag{1}$$

Where they go on to describe in intuitive terms:

> " With some exceptions (see the Appendix), $u$ lies between $0$ (subsistence) and $1$ (bliss) so that the positive linearly homogeneous functions $a(p)$ and $b(p)$ can be regarded as the costs of subsistence and bliss, respectively. " (p. 313)

Some algebra gets you equations $(2)$ and $(3)$ in the paper, which are analogous forms of the equations you have above.

> **Comments:**
> **username_0**: I thought that specification was only for the PIGLOG system  
> **username_2**: I am not super familiar with the system, so I am not sure myself. I may take a closer look later.  
> **username_0**: I think you might be right. what i meant by the question is that I want to know what $\beta_0 \prod p_i ^{\beta_i}$ is.  
> **username_1**: Well I am not historian of economic thought, but my supervisor explained me that AIDS was coined by Deaton and Muellbauer in the 1980's before the of outbreak of the deases with the same acronym. After this the "Almost Ideal Demand System" became translog (Jorgenson, 1982) and both forms are nested within the PIGLOG class.  

---

