How are these maximization problems same? This question is from Nicholson-Snyder's "Microeconomic Theory",page 200.(e-book here.)

Compare$$V^*(p_1,\dots,p_n,I_1,\dots,I_m)= \max_{x_1,…,x_n}
\left[U(x_1,…,x_n)\text{ such that}\sum_{i\in r}p_ix_i\leq I_r;r=1,\dots,m\right]$$and$$ \max_{I_1,\dots,I_m}V^*\text{ such that }\sum_{r=1}^mI_r=I ,\tag {ii} $$ to the utility maximization problem $$\max_{x_i}U(x_1,\dots,x_n) \text{ such that } \sum_{i=1}^np_ix_i\leq I.\tag{iii}$$
  Without any further restrictions, these two maximization processes will yield the same result; that is,
  Equation ii is just a more complicated way of stating
  Equation iii.

I can't understand why these are equivalent equations. I even can't understand the notations, what do variables beneath max mean?
 A: The variables $x_1, \ldots, x_n$ are consumption of goods $1, \ldots, n$, respectively. Instead $I_1,\ldots,I_m$ is a partition of the budget $I$ for the $m$ groups of goods. For example with $n=3$ goods, $m=2$ groups and $I=10$ total budget, we could partition the goods into $\{\{1,2\},\{3\}\}$ and decide to allocate $I_1 = 5$ "dollars" to the first group, and $I_2 = I - I_1 = 5$ dollars to the second. The general question is whether there is such a (non-trivial, i.e., $m > 1$) partition such that the demand function decomposes as in equation (i) (the important thing to note here is that demand for goods in group $r$ only depends on prices of other goods in group $r$ and budget $I_r$, but not on anything else).
To see that the two-stage maximation problem is equivalent to (iii), note that the constraints in the unnumbered equation imply the constraint in (iii) (why?). Therefore $V^*(p_1,\ldots,p_n,I_1,\ldots,I_m) \leq \max_{x_i}U(x_1,\dots,x_n)$, for any $I_1, \ldots, I_m$. (Note that the partition into $m$ groups is fixed throughout.)
As for the reverse inequality, suppose that $(x_1^*, \ldots, x_n^*)$ solves (iii). Without loss of generality (given the usual assumptions we make on $U$), the constraint in (iii) holds with equality. Let $I_r^* = \sum_{i\in r} p_ix_i^*$. Then $V^*(p_1,\ldots,p_n,I_1^*,\ldots,I_m^*) \geq U(x_1^*, \ldots, x_n^*)$, because the bundle $(x_1^*, \ldots, x_n^*)$ is feasible. Finally $\max_{I_1,\dots,I_m}V^* \geq V^*(p_1,\ldots,p_n,I_1^*,\ldots,I_m^*)$, and the constraint is satisfied with equality because of our previous assumption. (Of course both $\geq$ hold with equality, but we only needed this direction.) 
