Horizontal demand curve If it is assumed that every consumer within a particular region has the same 'valuation' for a certain product, and consumers will purchase no more than one unit of the product, is the market demand curve perfectly elastic (horizontal)?

Any insight would be appreciated.
Let $v_i$ be the valuation that the $i$-th consumer has for the product, and $v^*$ the common valuation of the product, where the $v$'s are measured in monetary units, and are assumed continuous.  Consumers are here assumed binary -they either purchase $0$ or $1$ (i.e. the product is indivisible). We assume that if the valuation is equal to the price, they do want to purchase the product.

Write the indicator function $I\{v_i\geq p\}$, that takes the value $1$ if the valuation is greater or equal than the price (and so the consumer will want to buy the one unit of product. Let there be $N$ consumers. Then, demand can be expressed as

$$Q^{d} = \sum_{i=1}^NI\{v_i\geq p\}$$

and since the valuation is assumed common to all,

$$Q^{d} = N\cdot I\{v^*\geq p\}$$

We see that  if $p\leq v^*$, quantity demanded will be equal to $N$, while if $p&gt; v^*$ quantity demanded will be equal to $0$.

Diagrammatically, this gives