Essential goods: How does one restrict the utility function?

Question

I understand that solutions on boundary of the set under consideration when doing constrained optimization are often problematical. Usually it is said that we assume that goods are essential to insure an interior solution.

But are there not a set of conditions that can be imposed on the utility function for example - rather that stipulating it as a primitive - to insure that the solution is interior?

I realize that this offcourse depends on the particular program under consideration. So to be concrete I am primarily interested in the standard program where

$$\max_x \ u(x) \\ s.t. \ \ px \leq w$$

where $p>>0$ and $w>0$.

Answer 1

In order to have $x^*_j>0$, it is possible to impose either
(i) a condition on the marginal utility: $\lim_{x_j\rightarrow0} \partial{U}/\partial{x_j}(x)=+\infty$
(ii) or an inequality $x_j \geq a_j>0$ where $a_j$ is interpreted as a subsistence level of $x_j$
Often the utility function is reparameterized and written $U(x-a)$ with the constraint $X:=x-a\geq 0$, which ensures that $x^*\geq a$ at the inner optimum.

Example: in the Cobb-Douglas case with subsistence levels, we have $$ U(X_1,X_2)=U(x_1-a_1,x_2-a_2)=(x_1-a_1)^\alpha(x_2-a_2)^\beta $$

Provided that the income $m$ is high enough, the Marshallian demands are given by:

$$x^*_1 = a_1 + \frac{\alpha}{\alpha+\beta}\frac{m-p_1a_1-p_2a_2}{p_1}\\ x^*_2 = a_2 + \frac{\beta}{\alpha+\beta}\frac{m-p_1a_1-p_2a_2}{p_2}. $$