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 $b(p)$?
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.