In page 75 of Mascolell's microeconomic theory, there is a diagram showing some relations. In particular, there is a relation saying $e(p,u) = v(p, e(p,u))$. I wonder if this is a typo? If not, how to interpret it so that it reconciles with the usual relation $v(p,e(p,u)) = u$?
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We can say that $v(p,w)=e(p,v(p,w))$ (or, equivalently, that $e(p,u)=v(p,e(p,u))$ if we are using the "money metric" indirect utility function. You can read about this in section 3.I of MWG. I guess that is what the authors had in mind when they drew figure 3.G.3.
More generally, the relationship between $e$ and $v$ is one of inverses: $e(p,u)$ tells us the mapping $$\text{target utility}->\text{money spent}$$ while $v(p,w)$ tells us the mapping $$\text{money available}->\text{attainable utility}.$$ So $e$ maps from money to utility and $v$ maps from utility back into money.
In words: if we take a given $w$ and feed it into $v$, we will get an answer to the question "what is the most utility I could ever achieve, $\bar{u}$?" Now, if I take that $\bar{u}$ and feed it into the expenditure function I get an answer to the question "what do I have to spend to achieve utility level $\bar{u}$. That answer will be exactly $w$ (so $e$ 'undoes' the transformation applied by $v$, and vice-versa).
That's just a long-winded way of saying that you can obtain the expenditure function by inverting $v$ in its second argument, or obtain $v$ by inverting $e$. This is made explicit in equation 3.E.1 of MWG.
answered Sep 28, 2017 at 8:42