Microeconomic foundation of discrete choice model

Question

(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.

Answer 1

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.