Microeconomic foundation of discrete choice model (1) Does the following result in a "valid" (in the sense of being consistent with the economic theory) market demand function?
A consumer $i$ maximizes a utility $u_{ij}$ in choosing one of J alternatives, $j=1,..,J$:
$u_{ij} = v_j - \alpha p_j + \epsilon_{ij}$
where $v_j$ is the utility of alternative $j$ without the effect of price (i.e. $-\alpha p_j$) and the logit error term $\epsilon_{ij}$.
The market demand results then as the choices of all consumers.

Is in accordance with economic theory to follow (1) and not assume a budget constraint but let the ultility of an alternative be directly affected by price?  And additionally assume that a consumer chooses one alternative (i.e., a corner solution follows directly from this assumption).  Does (1) result a  "valid" (in the sense of being consistent with the economic theory) market demand function ?

(2) Typically the theory of the microeconomic foundation of logit choice model (e.g.
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1718571)  assumes that consumers maximize a utility function without the price effect but subject to a budget constraint. With a linear utility function, the maximization problem results in a corner solution of a consumer choosing only one choice alternative.
This setup (as outlined in the cited paper) leads than to the market demand.
Your first equation is an indirect utility function. The consumer chooses only one good and we assume that he can afford it. For simplification, the indirect utility discards the budget $m$ (often assumed be the same for all $i$).
Here, your demand function is perfectly inelastic. If the individual demand would not have been perfectly inelastic, the form would have been like $V=a-s(p)$, where $s'$ is strictly concave and increasing.

See, for example, Palma/Anderson/Thisse.