Nice question, a nice effort of imagination.
Just some general observations and suggestions, as the answer could be prohibitively long and time-consuming, if one had to made all the calculations and verify all the cases.
1) All your budget constraints describe a price discount on quantities, as the overall price paid for each unit of a good will decrease as the quantity of the good increases. This is true also for case $1$.
Actually, the overall price of one unit of good $i$, let's call it $P_i^O$ is given by:
$$P_i^O= P_i^a+ \frac{P_i^v}{x_i}\;\;\;\;\;\;\;(1)$$
which evidently decreases as the quantity $x_i$ increases.
From an intuitive point of view, how can this case be considered? Let's write the budget constraint $BC_1$ in the following way:
$$\sum_{i=1}^n \left( P_i^a+ \frac{P_i^v}{x_i} \right) x_i = \sum_{i=1}^n P_i^a x_i+ \sum_{i=1}^n P_i^v \leq M$$
The last sum is a term independent of the quantity $x_i$, an amount paid independently of the quantity of the good.
I suggest that it can be seen as a 'ticket', as if a fixed sum paid when you enter the shop of the good you want to buy, and obviously this is a price (per unit) that is lesser the greater the quantity you buy.
From a formal, mathematical, point of view this case is completely analogous to the usual linear budget constraint, so the formal treatment is the same.
2) The budget constraint $BC_2$ of course doesn't allow solutions in which some good is not bought. In $BC_2$, $0$ solutions must be excluded, and this can be immediately seen observing that there is the term
$$\frac{P_i^v}{x_i^2}x_i= \frac{P_i^v}{x_i}$$
which is not defined for $x_i=0$.
3) The budget constraints $BC_1$ and $BC_3$ look very different because the first one gives you a linear budget constraint, as $x_i$ in the denominator in the second term in the sum simplifies with the $x_i$ that multiplies it, in $BC_3$ there isn't a simplification like that, so we have a very different, not linear function, the overall price is not a linear function of the quantity as in $BC_1$.
And analogously for $BC_2$ and $BC_4$, they use very different functions.
4) You wrote
- Are there any mathematical hardships when solving for basic utility functions trio (cobb-doug; subst; compl)?
- Is there some more efficient alternative than what I tried here?
You have tried not linear forms of functions relating negatively the overall price $P_i^O$ to quantity, as in equation $(1)$. Why not to try the simplest linear form?
$$P_i^O= P_i^a- P_i^vx_i.\;\;\;\;(2)$$
Anyway, of course, the possible analytical intricacies of the various budget constraints can be ascertained only making the calculations with various cases of utility functions, verifying if nothing prohibitively cumbersome occurs.
What I could suggest to frame the problem is to use a generic form for the quantity discount, that is a general differentiable function $p_i$ relating negatively the overall price $P_i^O$ of a good $i$ to its quantity, that is
$$P_i^O= p_i(x_i)$$
with $p'_i <0,$ for all $i=1,…,n$.
(Your price functions in your $BC$ are particular cases of this $p_i$).
Let's formulate the problem of consumer's maximization with these $p_i$ using Lagrange multipliers (the budget constraint taken as an equality). I limit myself to two variables.
The problem is
$$max \;\;\;\;U(x_1,x_2)$$
s. t.
$$p_1 (x_1) x_1+p_2(x_2)x_2=M,$$
where $U$ is a differentiable utility function with the usual properties.
The Lagrangian is:
$$\mathcal {L}= U(x_1,x_2)+ \lambda (p_1 (x_1) x_1+p_2(x_2)x_2-M)$$
This first order conditions are:
$$\frac {\partial U(x_1,x_2)}{\partial x_1}+\lambda (p_1(x_1)+p’(x_1) x_1=0.\;\;\;\;(3)$$
$$\frac {\partial U(x_1,x_2)}{\partial x_2}+\lambda (p_2(x_2)+p’(x_2) x_2=0.\;\;\;\;(4)$$
$$p_1 (x_1) x_1+p_2(x_2)x_2-M=0$$.
Eliminating $\lambda$ from $(3)$ and $(4)$ we have:
$$\frac {\frac {\partial (U(x_1,x_2)}{\partial x_1}}{\frac {\partial (U(x_1,x_2)}{\partial x_2}}=\frac{p_1(x_1)+p’(x_1) x_1}{p_2(x_2)+p’(x_2) x_2}\;\;\;\; (5)$$
The conditions $(5)$ are not very different from the usual first order conditions
$$\frac {\frac {\partial (U(x_1,x_2)}{\partial x_1}}{\frac {\partial (U(x_1,x_2)}{\partial x_2}}=\frac{p_1}{p_2}\;\;\;\; (6)$$
the difference is that the second side of equation $(5)$ is not the ratio of prices as in $(6)$, but the ratio of the derivatives of $p_i (x_i) x_i$ with respect to quantities $x_i$ (but nothing surprising, except the fact that these conditions are not so elegant as usual).
The first side of $(5)$ is the same as in the classical problem of consumer’s maximization, the ratio of marginal utilities, so I can’t see analytical difficulties in writing these first order conditions in the case of differentiable functions as a Cobb Douglas.
