# what is $b(p)$ in the Almost Ideal Demand System?

In Hal Varian's Microeconomic analysis (page 213) he discusses the Almost Ideal demand system. 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 $$b(p)$$?

• $\textbf{b(p)}$ 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 Jan 14 at 8:05
• @Bertrand so $b(p)$ is just an expenditure function a HOD(1) in $u$? – EconJohn Jan 14 at 23:13
• Yes $b(p)$ is HOD1 but in $p$. – Bertrand Jan 15 at 10:46
• 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 Jan 15 at 10:51

In the original article by Deaton and Muellbauer (1980), An Almost Ideal Demand System, 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.

• I thought that specification was only for the PIGLOG system – EconJohn Jan 14 at 1:17
• I am not super familiar with the system, so I am not sure myself. I may take a closer look later. – Kitsune Cavalry Jan 14 at 3:43
• 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 Jan 14 at 3:45
• 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 Jan 15 at 11:02

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. 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 page 6