$\begingroup$

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

eq 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.