Should Company B enter the market in the following cases Say Company A has a monopoly producing product E, at a constant marginal cost of $3$ USD. Say the ideal number of units produced is $1$ unit which produces a profit of $2$ USD (=$P_{A}$). 

Next, Company B wants to enter the market and produce product E at a marginal cost $3$ USD, but it has initially invested $\frac{1}{2}P_{A}$ in order to enter the market. 

Question:

$1.$ Under Cournot Competition, should Company B enter the market. Why/Why not?

$2.$ Under Bertrand Competition, should Company B enter the market. Why/Why not?

Ideas: 

$1.$ No, since we are in Cournot competition, the price is determined by the total units that both Company A &amp; B produce together. If Company B were also to produce, it would drive the price down as a result of excess supply, eventually forcing the price below the marginal cost, thus not profitable

$2.$ Yes, the price is driven down until the marginal cost. Although Company B will earn $\frac{1}{2}P_{A}$ less than company A.

Do my answers suffice, or rather, are they correct, and touch on the right point?
Knowing that demand is linear, we can deduce from the two points on the demand curve, $(q,p)\in\{(1,2),(2,3)\}$, that the demand function is $p=7-2q$, which is consistent with the Company A's profit maximizing quantity being $2$ units (assuming no fixed cost). 

If B enters the market and competes in quantity, the Cournot outcome is $q_A=q_B=\frac23$ and $p=\frac{13}3$. Taking into account B's fixed cost of $1$ for entering the market, its profit is 
\begin{equation}
\left(\frac{13}3-3\right)\frac23-1=-\frac19&lt;0.
\end{equation}
Hence it's not profitable for B to enter. 

If B enters the market and competes in price, the Bertrand outcome is $p_A=p_B=3$. In this case, revenue is equal to the cost of production, so B makes zero profit (absent the entry cost). However, given the positive cost of entry, B would be making a loss if it enters the market.